i
Department of Physics
University of Puerto Rico – Rio Piedras Campus
San Juan, Puerto Rico
Electrical characterization of Al doped ZnO sol-gel films
Carlos R. Pérez
July 29, 2014
Submitted to the Department of Physics, University of Puerto Rico –
Rio Piedras Campus, in partial fulfillment of the requirements for the
degree of Master of Science
ii
Accepted by the Department of Physics, University of Puerto Rico – Río
Piedras Campus, in partial fulfillment of the requirements for the
degree of Master of Science on July 29, 2014.
Approved by:
_____________________________
Luis F. Fonseca Ph.D.
Thesis committee member
_____________________________
Wilfredo Otaño Ph.D.
Thesis committee member
_____________________________
Peter X. Feng Ph.D.
Thesis committee member
_____________________________
José F. Nieves Ph.D.
Physics Department chairman
iii
Acknowledgements
I would like to thank Dr. Wilfredo Otaño for his support and trust, without which none of
this work would have been possible.
I would also like to thank Dr. Carlos Ortiz for his guidance and continued help to
accomplish the tasks at hand.
Nelson Granda for his help in the preparation of the samples, UV-VIS measurements and
use of the laboratory equipment.
Dr. Victor Pantojas for his overall support regarding solid state physics.
Adrian Camacho for his continued help, support and company.
Ian E. Gutierrez Molina, XRD measurements, Puerto Rico Energy Center, Turabo
University.
Loraine M. Soto-Vázquez, EDS measurements, Materials Characterization Center
University of Puerto Rico.
iv
Table of contents
ACKNOWLEDGEMENTS ....................................................................................................................... IV
LIST OF TABLES ..................................................................................................................................... VI
LIST OF FIGURES ................................................................................................................................... VI
ABSTRACT ...................................................................................................................................................1
CHAPTER 1 INTRODUCTION..................................................................................................................2
CHAPTER 2 PROPERTIES OF ZNO ........................................................................................................5
2.1 CRYSTAL STRUCTURE ............................................................................................................................5
2.1.1 Lattice Parameters ........................................................................................................................8
2.2 ELECTRONIC BAND STRUCTURE .......................................................................................................... 10
2.3 CARRIER TRANSPORT........................................................................................................................... 15
2.3.1 Mobility ....................................................................................................................................... 15
2.3.2 Carrier type of undoped ZnO ...................................................................................................... 18
CHAPTER 3 THEORETICAL BACKGROUND .................................................................................... 20
3.1 CARRIER TRANSPORT........................................................................................................................... 20
3.2 MOBILITY ............................................................................................................................................ 20
3.3 SCATTERING MECHANISMS ................................................................................................................. 25
CHAPTER 4 ELECTRICAL CHARACTERIZATION TECHNIQUES .............................................. 27
4.1 FOUR-POINT PROBE VERSUS TWO-POINT PROBE CONFIGURATIONS ...................................................... 27
4.2 VAN DER PAUW RESISTIVITY CALCULATION TECHNIQUE ..................................................................... 28
4.3 HALL EFFECT MEASUREMENTS ............................................................................................................ 35
4.4 IMPORTANCE OF OHMIC CONTACTS ..................................................................................................... 38
4.4.1 Theoretical formalism of ohmic contacts .................................................................................... 38
4.4.2 Preparation of Ohmic Contacts .................................................................................................. 41
4.4.3 Ohmic contact verification.......................................................................................................... 42
CHAPTER 5 EXPERIMENTAL TECHNIQUES ................................................................................... 43
5.1 ZNO SOL-GEL SAMPLES PREPARATION................................................................................................. 43
5.2 SAMPLE HOLDER CONSTRUCTION ........................................................................................................ 44
5.3 CONTACT FORMATION FOR ELECTRICAL MEASUREMENTS ................................................................... 46
5.4 VAN DER PAUW RESISTIVITY MEASUREMENT IMPLEMENTATION ......................................................... 49
5.5 HALL EFFECT MEASUREMENTS IMPLEMENTATION .............................................................................. 53
5.6 LABVIEW PROGRAMMING ................................................................................................................... 55
5.6.1 Electrical measurement equipment and measurement overview ................................................ 55
5.6.2 Electrical measurement error calculation .................................................................................. 58
5.7 ADDITIONAL MEASUREMENTS ON ZNO SAMPLES ................................................................................ 62
5.7.1 Thickness Measurements ............................................................................................................ 62
5.7.2 X-ray diffraction measurements .................................................................................................. 63
5.7.3 Atomic force microscope measurements ..................................................................................... 64
5.7.4 Energy-dispersive X-ray spectroscopy........................................................................................ 64
5.7.5 UV-VIS transmittance measurements ......................................................................................... 64
CHAPTER 6 RESULTS ............................................................................................................................. 66
6.1 ENERGY-DISPERSIVE X-RAY SPECTROSCOPY RESULTS ........................................................................ 66
6.2 X-RAY DIFFRACTION MEASUREMENTS ................................................................................................. 67
6.3 ATOMIC FORCE MICROSCOPE MEASUREMENTS .................................................................................... 71
6.4 UV-VIS TRANSMITTANCE MEASUREMENTS ........................................................................................ 73
6.5 AL-ZNO FILMS PROFILER THICKNESS MEASUREMENTS........................................................................ 78
6.6 ELECTRICAL CHARACTERIZATION RESULTS ......................................................................................... 79
6.6.1 Resistivity measurements ............................................................................................................ 79
v
6.6.2 Modeling of resistivity results ..................................................................................................... 83
6.6.5 Hall Effect measurements ........................................................................................................... 96
6.6.4 Resistivity results of previous Al-ZnO samples ........................................................................... 97
CHAPTER 7 CONCLUSIONS AND RECOMMENDATIONS ........................................................... 102
APPENDIX ................................................................................................................................................ 107
A.1 PARTIAL DERIVATIVES PRESENT IN MEASUREMENT ERROR CALCULATIONS ..................................... 107
A.2 ADDITIONAL PLOTS FROM MEASUREMENTS PERFORMED ON THE AL-ZNO FILMS. ............................ 109
REFERENCES .......................................................................................................................................... 117
List of tables
TABLE 2.1 ZNO WURTZITE LATTICE CONTANTS [28]........................................................................................8
TABLE 2.2 ZNO ZINC BLENDE LATTICE CONTANTS [28]. ..................................................................................8
TABLE 2.3 ZNO ROCKSALT LATTICE CONTANTS [28]. ......................................................................................9
TABLE 2.4 CALCULATED AND MEASURED ENERGY GAP EG, CATION D-BAND POSITION ED AND ANION P
VALENCE BANDWIDTH WP OF ZNO [28, 68]. ........................................................................................ 11
TABLE 2.5 CARRIER CONCENTRATION AND ELECTRON MOBILITY OF ZNO [28]. ............................................ 16
TABLE 5.1 DETAILS OF AL DOPED ZNO SOLUTIONS. ...................................................................................... 43
TABLE 5.2 METALLIZATION SCHEMES FOR OHMIC CONTACTS ON ZNO [73]. ................................................. 47
TABLE 5.3 RESISTIVITY VOLTAGES NOTATION CONVENTION. ........................................................................ 51
TABLE 5.4 HALL EFFECT VOLTAGES NOTATION CONVENTION. ...................................................................... 54
TABLE 6.1 MEASURED ALUMINUM ATOMIC PERCENTAGES. ........................................................................... 66
TABLE 6.2 AL-ZNO GRAIN SIZE AND ROUGHNESS. ......................................................................................... 72
TABLE 6.3 CALCULATED BAND-GAP OF AL-ZNO FILMS. ................................................................................ 75
TABLE 6.4 THICKNESS OF AL-ZNO FILMS. ..................................................................................................... 77
TABLE 6.5 AL-ZNO PROFILER MEASURED FILM THICKNESS. .......................................................................... 78
TABLE 6.6 AL-ZNO FILMS RESISTIVITY AT ROOM TEMPERATURE. ................................................................. 82
TABLE 6.7 SUMMARY OF THE RESISTIVITY ANALYSIS OF AL-ZNO FILMS. ...................................................... 89
TABLE 6.8 AL-ZNO DONOR LEVELS. .............................................................................................................. 91
TABLE 6.9 LOCALIZATION RADIUS AND DOLS AT THE FERMI LEVEL FOR AL-ZNO FILMS. ............................ 92
TABLE 6.10 AVERAGE HOPPING DISTANCE IN MOTT VRH REGIME. ............................................................... 93
TABLE 6.11 AVERAGE HOPPING ENERGY IN MOTT VRH REGIME. .................................................................. 93
TABLE 6.12 R/A FOR AL-ZNO FILMS IN MOTT VRH REGIME.......................................................................... 93
TABLE 6.13 LOCALIZATION RADIUS OF THE CHARGE CARRIERS ON EFROS VRH REGIME. ............................. 94
List of figures
FIGURE 2.1 WURTZITE CRYSTAL STRUCTURE. ..................................................................................................5
FIGURE 2.2 WURTZITE CRYSTALLOGRAPHIC VECTORS [28]. ............................................................................6
FIGURE 2.3 ZNO ZINC BLENDE STRUCTURE. .....................................................................................................7
FIGURE 2.4 ZNO ROCKSALT STRUCTURE. .........................................................................................................7
FIGURE 2.5 LDA BULK BAND STRUCTURE OF ZNO AS CALCULATED BY USING A STANDARD
PSEUDOPOTENTIAL (PP) (LEFT PANEL) AND BY USING SIC-PP (RIGHT PANEL) [28, 68]. ....................... 11
FIGURE 2.6 COMPARISON OF CALCULATED AND MEASURED VALENCE BANDS OF ZNO. THE LEFT PANEL
SHOWS THE STANDARD LDA, WHILE THE RIGHT PANEL SHOWS SIC-PP RESULTS [28, 68]. .................. 12
FIGURE 2.7 VALENCE BAND STRUCTURE OF ZNO NEAR THE Γ POINT [28, 66]. ............................................... 13
FIGURE 2.8 BAND STRUCTURES OF VARIOUS PHASES OF ZNO [28, 69]. .......................................................... 13
FIGURE 2.9 DENSITY OF STATES FOR VARIOUS PHASES OF ZNO [28, 69]. ....................................................... 14
FIGURE 2.10 HALL MOBILITY AS A FUNCTION OF TEMPERATURE FOR BULK ZNO [28, 42]. ............................ 15
vi
FIGURE 2.11 CARRIER CONCENTRATION AS A FUNCTION OF T-1 FOR BULK ZNO [28, 42]. ............................. 16
FIGURE 3.1 VOLUME ENCLOSED BETWEEN ARBITRARILY SHAPED AREAS WITH CHARGE CARRIERS. .............. 21
FIGURE 3.2 SECTION OF SOLID MATERIAL THROUGH WHICH CURRENT FLOWS. .............................................. 23
FIGURE 4.1 TWO POINT AND FOUR POINT PROBE CIRCUIT DIAGRAMS [77]...................................................... 27
FIGURE 4.2 ARBITRARILY SHAPED FLAT SAMPLE FOR VAN DER PAUW MEASUREMENTS [77]......................... 29
FIGURE 4.3 POINTS IN INFINITE LAMELLA FOR DERIVATION AID OF VAN DER PAUW EQUATIONS. .................. 30
FIGURE 4.4 SYMMETRIC SAMPLE TYPE FOR VAN DER PAUW MEASUREMENTS [74]. ....................................... 32
FIGURE 4.5 GEOMETRICAL FACTOR F AS A FUNCTION OF RR [77]. .................................................................. 33
FIGURE 4.6 TYPICAL SAMPLE GEOMETRIES: (A) CIRCULAR, (B) CLOVERLEAF, (C) SQUARE [77]. .................. 33
FIGURE 4.7 CORRECTION FACTORS TO ACCOUNT FOR CONTACT SIZE ON SQUARE SAMPLES. .......................... 34
FIGURE 4.8 CONDUCTING SOLID SLAB. ........................................................................................................... 35
FIGURE 4.9 PARAMETERS CONSIDERED WHEN ANALYZING A METAL SEMICONDUCTOR JUNCTION: WORKFUNCTION, FERMI LEVEL, ELECTRON AFFINITY [71]. ........................................................................... 39
FIGURE 4.10 BAND BENDING IN A METAL-SEMICONDUCTOR JUNCTION [71]. ................................................. 40
FIGURE 4.11 SUMMARY OF OHMIC AND SCHOTTKY CONTACTS DEPENDING ON THE TYPE OF SEMICONDUCTOR
AND WORK-FUNCTION RELATIVE TO THE METAL [71]. .......................................................................... 41
FIGURE 5.1 SPIN-COATED AL DOPED ZNO FILM PREPARED VIA SOL-GEL METHOD. ........................................ 44
FIGURE 5.2 DIAGRAM OF SAMPLE HOLDER COMPONENTS. ............................................................................. 45
FIGURE 5.3 CRYOGENIC EQUIPMENT (LEFT), SAMPLE HOLDER CLOSE-UP (RIGHT). ........................................ 46
FIGURE 5.4 ZNO ON FUSED GLASS WITH INDIUM CONTACTS. ......................................................................... 47
FIGURE 5.5 ZNO SAMPLE WITH AL-AU CONTACTS. ........................................................................................ 48
FIGURE 5.6 SAMPLE MASKING WITH RELEVANT SIZE PARAMETERS. ............................................................... 49
FIGURE 5.7 VAN DER PAUW RESISTIVITY MEASUREMENT SCHEME. ............................................................... 52
FIGURE 5.8 HALL EFFECT MEASUREMENT SCHEME. ....................................................................................... 54
FIGURE 5.9 ELECTRICAL MEASUREMENTS EQUIPMENT CONNECTION DIAGRAM. ............................................ 56
FIGURE 5.10 RESISTIVITY MEASUREMENT ROUTINE FLOWCHART. ................................................................. 57
FIGURE 5.11 ERODED AL-ZNO THIN FILM FOR THICKNESS MEASUREMENT. .................................................. 63
FIGURE 6.1 UNDOPED ZNO EDS. ................................................................................................................... 67
FIGURE 6.2 10% AL-ZNO EDS. ..................................................................................................................... 67
FIGURE 6.3 XRD SPECTRA OF AL-ZNO FILMS. ............................................................................................... 68
FIGURE 6.4 AL-ZNO XRD PEAK LOCATION. .................................................................................................. 68
FIGURE 6.5 AL-ZNO AVERAGE CRYSTALLITE SIZE. ........................................................................................ 69
FIGURE 6.6 AL-ZNO XRD PEAK LOCATION. .................................................................................................. 70
FIGURE 6.7 AL-ZNO LATTICE PARAMETER. ................................................................................................... 71
FIGURE 6.8 UNDOPED ZNO AFM IMAGE........................................................................................................ 71
FIGURE 6.9 1%AL-ZNO AFM IMAGE............................................................................................................. 72
FIGURE 6.10 GRAIN SIZE AS A FUNCTION OF AL% AT. FOR AL-ZNO. ............................................................. 73
FIGURE 6.11 ROUGHNESS AS A FUNCTION OF AL% AT. FOR AL-ZNO. ............................................................ 73
FIGURE 6.12 OPTICAL TRANSMISSION SPECTRA OF AL-ZNO SOL-GEL FILMS.................................................. 74
FIGURE 6.13 UV ABSORPTION EDGE OF AL-ZNO FILMS. ................................................................................ 75
FIGURE 6.14 EXTRAPOLATION FOR BAND-GAP CALCULATION. ...................................................................... 76
FIGURE 6.15 OPTICAL BAND-GAP OF AL-ZNO FILMS. .................................................................................... 76
FIGURE 6.16 UNDOPED ZNO TRANSMITTANCE MAXIMUM AND MINIMUM CURVES. ....................................... 77
FIGURE 6.17 THICKNESS OF AL-ZNO FILMS. .................................................................................................. 78
FIGURE 6.18 AL-ZNO FILMS PROFILER THICKNESS MEASUREMENTS.............................................................. 79
FIGURE 6.19 RESISTIVITY OF 2% AND 3% AL-ZNO FILMS. ............................................................................ 80
FIGURE 6.20 RESISTIVITY OF 1%, 4%, 5%, AND 10% AL-ZNO FILMS. ........................................................... 81
FIGURE 6.21 AL-ZNO FILMS RESISTIVITY AT ROOM TEMPERATURE. .............................................................. 82
FIGURE 6.22 LN(Ρ) VS 1/T OF AL-ZNO FILMS. ................................................................................................ 86
FIGURE 6.23 AL-ZNO FILMS RESISTIVITY ARRHENIUS PLOT. ......................................................................... 87
FIGURE 6.24 AL-ZNO FILMS RESISTIVITY NNH PLOT. ................................................................................... 87
FIGURE 6.25 AL-ZNO FILMS RESISTIVITY MOTT VRH PLOT. ......................................................................... 88
FIGURE 6.26 AL-ZNO FILMS RESISTIVITY SHKLOVSKII-EFROS VRH PLOT. ................................................... 88
FIGURE 6.27 1% AL-ZNO RESISTIVITY DATA AND THEORETICAL FITS. .......................................................... 89
FIGURE 6.28 2% AL-ZNO RESISTIVITY DATA AND THEORETICAL FITS. .......................................................... 90
FIGURE 6.29 DIMENSIONLESS ACTIVATION ENERGY FOR N-GERMANIUM [123]. ............................................. 95
vii
FIGURE 6.30 LNW VS. LNT FOR AL-ZNO THIN FILMS...................................................................................... 96
FIGURE 6.31 RESISTIVITY OF UNDOPED ZNO WITH IN CONTACTS. ................................................................. 98
FIGURE 6.32 RESISTIVITY OF AL-ZNO WITH IN CONTACTS. ........................................................................... 98
FIGURE 6.33 AL-ZNO FILMS RESISTIVITY AT ROOM TEMPERATURE WITH IN CONTACTS. ............................... 99
FIGURE 6.34 RESISTIVITY OF UNDOPED ZNO WITH AL/AU CONTACTS. .......................................................... 99
FIGURE 6.35 RESISTIVITY OF AL-ZNO WITH AL/AU CONTACTS................................................................... 100
FIGURE 6.36 AL-ZNO FILMS RESISTIVITY AT ROOM TEMPERATURE WITH AL/AU CONTACTS. ...................... 100
viii
Abstract
Thin films of transparent conductive oxides (TCO) are of interest as components
in different applications such as transparent electrodes, solar panels as well as gas
sensors. Interest in Aluminum doped zinc oxide thin films has dramatically increased due
to the availability and relative low cost of the components, when compared to other
promising TCO materials such as indium tin oxide (ITO). Previous research on Al doped
ZnO sol-gel thin films deposited by spin coating has shown excellent hexagonal wurtzite
c-axis orientation suggesting the possibility of easy control of the chemical components
and low cost of the process.
In this work an electrical characterization platform has been developed and
implemented on Al doped ZnO sol-gel thin films deposited by spin coating on glass
substrates. For the development of the platform a sample holder was designed and built.
In addition, multiple LabVIEW™ routines that enable the automation of the electrical
characterization process were created. The platform developed enables electrical
characterization in terms of resistivity and Hall Effect measurements employing the van
der Pauw technique. Ultimately, due to the degradation of an already existing set of films
with which the platform was tested, a new set of thin films was prepared following the
same procedure used for the already existing set. The resistivity of the first and new set of
samples was measured as a function of temperature, and for the new set it was modeled
following several conduction mechanisms used by other research teams for ZnO thin
films. To complement the electrical characterization of the new set of films, EDS, AFM,
XRD, and UV-VIS measurements were also performed.
1
Chapter 1 Introduction
The use of ZnO by humans dates back centuries. It is believed that some of the
earliest uses of ZnO by humans have been in processed and unprocessed forms, mainly as
paint or medicinal ointments. Indian medical texts dating from around 500 BC, mention
the use of a substance called pushpanjan, thought to be ZnO, as a salve for eyes and open
wounds [1]. The ancient Greeks also mention the use of ZnO as a treatment for a wide
range of skin conditions [2]. Dating back to around 200BC, the Romans reacted copper
with zinc oxide through a cementation process for the production of brass [3]. Marco
Polo first described the use of ZnO in Persia in the 13th century while India recognized
Zn as a metal by 1374 and produced Zn as well as ZnO from the 12th to the 16th century.
Modern applications of ZnO, besides electronics or nanotechnology applications include:
rubber manufacture where it is used in the vulcanization process [4, 5, 6] and in medical
applications to protect the skin against fungi and ultra violet light. Other more common
day applications include: deodorizer, antibacterial [8], baby powder and cream, antidandruff shampoos, and antiseptic ointments [9, 10]. ZnO can also be found in sunscreen
lotions due to its broad UVA and UVB reflectivity [11] and because it is not absorbed by
skin which makes it nonirritating and hypoallergenic [12]. In ceramics, ZnO affects the
melting point and optical properties of ceramic glazes and enamels among other ceramic
formulations. In concrete manufacturing it improves the processing time of concrete and
its water resistance [5]. ZnO is also added to many food products because it is a source of
Zn as a nutrient [7].
Initial research and characterization efforts regarding ZnO date back to several
decades. The lattice parameter studies of ZnO date back to 1935 [13, 22]. Optical
2
properties of ZnO were investigated in detail in 1954 [15, 22]. Au Schottky barriers were
formed in 1965 [17, 22]. Its vibrational properties were studied by Raman scattering in
1966 [14, 22]. Light emitting diodes were demonstrated in 1967 [18, 22] wherein Cu2O
was used as the p-type material. Growth by chemical vapor transport of ZnO was
achieved in 1970 [16, 22]. Metal insulator semiconductor (MIS) structures were reported
in 1974 [19, 22]. ZnO/ZnTe n–p junctions were accomplished in 1975 [20, 22]. Al/Au
ohmic contacts were reported on ZnO in 1978 [21, 22].
Most of the properties of ZnO have been known for quite some time, but its
implementation in electronics as a semiconductor has been restrained due to a lack of
control of its electrical conductivity. Also the source of its native n-type conductivity,
long thought to be due to oxygen vacancies and zinc interstitials, remains a matter of
extensive debate and research. Recent density functional calculations and measurements
of optically detected electron paramagnetic resonance on high quality ZnO crystals seem
to suggest otherwise [23]. In addition to this, reports of p-type ZnO are prone to
difficulties in reproducibility and stability, and the matter still requires much research
[23].
Over the past decade there has been renewed interest and revival of research
efforts concerning ZnO for electronics applications. Some of the reasons for this are the
significant improvements in the quality and availability of ZnO single-crystal substrates
as well as epitaxial films [23]. Also fueling the renewed interest in ZnO research are its
advantages relative to GaN, such as its larger exciton binding energy of about 60 meV
compared to ~25 meV for GaN [24], resulting in bright room temperature emission.
3
Possible applications of ZnO include UV light-emitters, varistors, transparent high power
electronics, surface acoustic wave devices, piezoelectric transducers, gas-sensors, and as
window material for displays and solar cells [24].
ZnO is a candidate material for solid state blue to ultra violet optoelectronics,
including laser development, due to its wide band-gap [24]. Some applications arising
from this include: high density data storage systems, solid-state lighting (where
phosphorus excitation by blue or UV light-emitting diodes is used to produce white
light), secure communications and bio-detection [24]. Since ZnO is transparent to visible
light, transparent electronics, ultra violet optoelectronics and integrated sensors, could be
developed with it [24].In terms of its band-gap, which is about 3.37 eV at room
temperature, divalent substitution on the cation site of ZnO allows for its tuning;
reduction in the band-gap to ~3.0 eV can be achieved through cadmium substitution [24].
On the other hand, substitution of magnesium on the Zn site in epitaxial films can
increase the band-gap to ~4.0 eV while keeping a wurtzitic structure [24].
When doped with aluminum, ZnO layers can be used as transparent electrodes. Zn and Al
are much cheaper and less toxic than the generally used indium tin oxide (ITO), thus
making Al-ZnO an attractive alternative to ITO. Commercially available solar cells and
liquid crystal displays using ZnO as the front contact, are some of the recent applications
of Al doped ZnO.[25] Other applications of aluminum doped zinc oxide include flat
panel displays, smart windows, chemical sensors, UV blocking films, and transparent
window heaters among others [26].
4
Chapter 2 Properties of ZnO
2.1 Crystal structure
ZnO is a II-IV compound semiconductor which, under different conditions, can
attain the wurtzite, zinc blende and rocksalt crystal structures. The wurtzite crystal
structure of ZnO is thermodynamically stable at ambient conditions, while the zinc
blende structure can be obtained by growth of ZnO on cubic substrates. The rocksalt
crystal structure for ZnO can only be achieved under high-pressure conditions.
The wurtzite structure in ZnO is made up of two interpenetrating hexagonal closepacked substructures consisting of one type of atom each and geometrically displaced a
distance (u) with respect to each other along the c-axis ([0001] direction). For an ideal
wurtzitic structure the lattice parameters are c/a = (8/3)1/2 = 1.633 and u = 3/8 = 0.375,
where u is defined as the length of the bond parallel to the c-axis (anion–cation bond
length or the nearest-neighbor distance) divided by the c lattice parameter.
In the wurtzite structure each sub-lattice includes four atoms per unit cell with
every atom of one element surrounded by four atoms of the other or vice versa,
coordinated at the edges of a tetrahedron. The crystallographic vectors of a wurtzitic
structure
are



a = (1 / 2a, 3 / 2a,0), b = (1 / 2a,− 3 / 2a,0), c = (0,0, c) . ZnO crystals
deviate from the ideal wurtzite structure by having a smaller c/a ratio.
Figure 2.1 Wurtzite crystal structure.
5
Along the c direction, and off the c-axis, the nearest neighbor bond lengths can be
2
calculated as b = cu and b1 =
1 2 1

a +  − u  c 2 , respectively. Three types of second
3
2

nearest neighbors exist (shown in Figure 2.2) and their bond lengths can be calculated as
follows: one along the c-direction b1' = c(1 − u ) , six others b2' = a 2 + (uc ) , and three
2
2
others b =
'
3
4 2
1

a + c 2  − u  . The bond angles α and β can be calculated as follows:
3
2

−1
−1


2
2  
2
2  
4
c
1
c
1
  
  


+ 4   − u +    .
α = + cos −1  1 + 3   − u +    , β = 2 sin −1 






3
2  
2
2
a 
a 
 
 


π
Figure 2.2 Wurtzite crystallographic vectors [28].
It is important to mention that the ratio c/a correlates with the electronegativity of the
constituent atoms; showing the largest difference from the ideal c/a ratio for atoms with
the largest differences in electronegativity.
Heteroepitaxial growth on cubic substrates is the only way of stabilizing the zinc
blende structure for ZnO. It is composed of two interpenetrating face centered cubic
substructures shifted along the body diagonal by one quarter of its length. Each unit cell
6
contains four atoms and every one-type atom is tetrahedrally coordinated with four atoms
of the other type. The zinc blende structure differs from wurtzite only by the bond angle
of the second nearest neighbor atoms.
The rocksalt structure for ZnO can be obtained by transformation of the wurtzitic
structure under external hydrostatic pressure. Coulomb interactions favor ionic instead of
covalent bonds due to reduced lattice dimensions. The rocksalt structure is not stabilized
by epitaxial growth and the change from the wurtzitic phase has been reported to occur at
a pressure range of about 10GPa accompanied by a 17% reduction in volume. It is
important to mention that decompression causes the ZnO rocksalt structure to transform
back to wurtzite.
Figure 2.3 ZnO zinc blende structure.
Figure 2.4 ZnO rocksalt structure.
7
2.1.1 Lattice Parameters
Lattice constants of wurtzite ZnO obtained from calculations are in good
agreement with measurements and range from 3.2475 Å to 3.2501 Å for the a-parameter
and 5.2042 Å to 5.2075 Å for the c-parameter at room temperature. The c/a ratio and u
parameters range from 1.593 to 1.6035 and 0.383 to 0.3856, respectively. Lattice stability
and ionicity probably account for the deviation of the c/a ratio and u parameter from
those of ideal wurtzite [28].
a (Å)
c (Å)
3.2496
3.2501
3.286
3.2498
3.2475
3.2497
5.2042
5.2071
5.241
5.2066
5.2075
5.206
a (Å)
4.619
4.6
4.463
4.37
4.47
Table 2.1 ZnO wurtzite lattice contants [28].
Wurtize ZnO
c/a
u
Details
1.633
0.375 Ideal
1.6018 0.3819 Measured by XRD
1.6021 0.3817 Measured by XRD
1.595
0.383 Calculated from LCAO method
1.6021
Measured by EDXD
1.6035
Measured by XRD
1.602
Measured by XRD
1.593 0.3856 Calculated first principles LCOAO method
1.6
0.383 Calculated
[29]
[30]
[31]
[32]
[33]
[34]
[35]
[36]
Table 2.2 ZnO zinc blende lattice contants [28].
Zinc Blende (sphalerite)
Details
Calculated from LCAO method
Calculated first principles LCOAO method
Measured by RHEED, XRD, TEM
Measured by RHEED, XRD, TEM
Measured by RHEED, XRD, TEM
Ref
[31]
[35]
[37]
[37]
[37]
Ref
8
a (Å)
4.271
4.283
4.294
4.3
4.28
4.275
4.058
4.316
4.207
4.225
Table 2.3 ZnO rocksalt lattice contants [28].
Rocksalt (Rochelle salt)
Details
Measured by XRD
Measured by EDXD
Measured by XRD
Calculated first principles from LCOAO method
Measured by XRD
Measured EDXD and calculated by Coulomb Hartree-Fock perturbed
ion, GGA and LDA methods
Calculated, periodic Hartree-Fock first principles
Ref
[29]
[32]
[33]
[35]
[38]
[39]
[39]
[39]
[39]
[40]
For the zinc blende structure of ZnO, the lattice constants have been measured or
calculated by different groups. Modern ab initio calculations of the lattice constants have
yielded values of 4.60 Å and 4.619 Å. Growth of zinc blende ZnO films with a ZnS
buffer layer has been reported [37], with a lattice constant estimated to be 4.463 Å, 4.37
Å, and 4.47 Å, by analysis of the Reflection High-Energy Electron Diffraction pattern
(RHEED), by comparison with the XRD peak position, and by examination of the
transmission electron microscopy (TEM) images, respectively. X-ray diffraction results
are in close agreement with values in the range of 4.271–4.294 Å, corresponding to a
decrease in the lattice constant due to high-pressure phase transition from wurtzite to
rocksalt.
The rocksalt polytype of ZnO is obtained from phase transition of the wurtzite
structure under high-pressure conditions. For ZnO in the rocksalt form, the lattice
constant is reduced to a range of 4.271–4.294 Å. The predicted lattice parameters
obtained through calculation techniques such as HFPI, GGA, and HF; are in the range of
4.058–4.316 Å, and are around +-5% the experimental values obtained by X-ray
diffraction.
9
2.2 Electronic Band Structure
Several experimental and theoretical techniques have been employed through the
years for the determination of the band structure of ZnO. Some of them are X-ray or UV
reflection/absorption or emission, photoelectron spectroscopy (extension of the
photoelectric effect to the X-ray region known as PES), and angle-resolved photoelectron
spectroscopy (ARPES).
Theoretical calculations of the band structure of ZnO are mostly performed using
Density Functional Theory (DFT), specifically the Local Density Approximation (LDA).
LDA considers the exchange correlation energy, in regions of a material undergoing slow
charge density variations, to be the same as for a locally uniform electron gas of the same
charge density. This approximation, though simple, is surprisingly accurate, and is at the
center of most modern DFT codes, even working reasonably well for systems with
rapidly varying charge density. Some of the drawbacks include: under prediction of
atomic ground state energies and ionization energies, and over prediction of binding
energies [70].
For ZnO, LDA calculations are challenging because of the cationic d electrons. If
treated as core electrons, the calculations yield lattice constants that underestimate
experimental values by as much as 18% for wurtzitic ZnO, while their inclusion in the
valence band yields very accurate calculated lattice constants. Still, LDA calculations are
somewhat problematic, properly considering the d electrons still results in underestimated
band-gap and overestimated occupation of cationic d bands [28].
LDA calculations underestimate the band-gap of ZnO to be as low as Eg = 0.23
eV compared to the experimental value of Eg = 3.37 eV. Introduction of self-interaction
10
corrections into the LDA calculations can improve the quantitative description of a
system with strongly localized states such as 3d electrons. Figure 2.5 shows the standard
LDA result on the left panel, the right panel shows the band structure as calculated with
self-interaction corrected pseudo potential (SIC-PP) included in the LDA. LDA/SIC-PP
results yield d bands that are shifted down in energy considerably and the band-gap is
opened drastically. Comparing the experimental results to the standard LDA calculation,
it’s noted that the bands are artificially high and lead to strong p-d interactions with the
oxygen 2p bands. On the other hand, the LDA-SIC-PP results match better the
experimental results. Figure 2.6 shows a comparison of calculated and measured valence
bands of ZnO
Figure 2.5 LDA bulk band structure of ZnO as calculated by using a standard pseudopotential (PP)
(left panel) and by using SIC-PP (right panel) [28, 68].
Table 2.4 Calculated and measured energy gap Eg, cation d-band position Ed and anion p valence
bandwidth Wp of ZnO [28, 68].
LDA-PP
LDA-SIC-PP Experiment
Eg (eV)
Ed (eV)
0.23
-5.1
3.77
-8.9
3.4
-7.8
Wp (eV)
-3.99
-5.2
-5.3
11
Figure 2.7 shows the valence band structure of ZnO near the Γ point. The open
circles represent the results calculated using the atomic sphere approximation (ASA) and
linearized muffin-tin orbital (LMTO) methods within the LDA [66]. The solid lines are
fits to the Rashba-Sheka-Pikus effective Hamiltonian. As shown in Figure 2.7, the band
structure of ZnO reveals three very closely spaced valence bands; the heavy-hole, lighthole, and crystal field split-off bands labeled A, B, and C respectively. These three
valence bands result from the spin–orbit coupling and crystal field splitting near the zone
center (Γ point) [28]. The calculated splitting energies between the A, B, and C bands are
EAB = 10 meV and EBC = 34 meV [66], while the experimental values are EAB = 9.5meV
and EBC = 39.7meV [67].
Figure 2.6 Comparison of calculated and measured valence bands of ZnO. The left panel shows the
standard LDA, while the right panel shows SIC-PP results [28, 68].
12
Figure 2.7 Valence band structure of ZnO near the Γ point [28, 66].
Figure 2.8 Band structures of various phases of ZnO [28, 69].
Some of the calculation methods already discussed have also been used for the
electronic structure of other phases of ZnO, the results of which can be seen in Figure
2.8. The results were obtained using LDA and the PBE96 forms of the Generalized
Gradient Approximations methods (GGA) with optimized Gaussian basis sets and
expansion of the crystal orbitals and periodic electron density. The left pane shows the
13
results for the B4 (wurtzite) phase, the middle pane shows B1 (rocksalt), and the right
pane shows B2 (CsCl). The results were calculated over a range of unit cell volumes [69].
Compression of wurtzitic ZnO produces a reduction in height of the peak at the
top of the upper valence band accompanied by a downshift in energy. There, also occurs
a broadening and an up-shift in energy of the oxygen 2s and zinc 3d peaks as can be seen
in Figure 2.9. Additionally a splitting occurs in the zinc 3d states.
Figure 2.9 Density of states for various phases of ZnO [28, 69].
Transformation of ZnO to the rocksalt structure (B1) results in a reduction in
height of the peak near the valence band maximum. There is also a narrowing of the zinc
3d peak and a reduction in energy of the oxygen 2s derived states. Compression of the
rocksalt phase of ZnO broadens the upper valence band and increases the splitting of the
zinc 3d peak in the densitiy of states (DOS). Additionally, upon compression of the
rocksalt phase of ZnO, there is a broadening and a downshift in energy of the oxygen 2s
14
derived band as well as a broadening of the upper valence band. As the pressure is
increased in the range under consideration the band-gap also increases.
2.3 Carrier transport
2.3.1 Mobility
Room temperature electron mobility of ZnO has been predicted to be around 300
cm2/V-s using Monte Carlo simulations [41]. For single crystal ZnO in bulk form, the
highest electron mobility at room temperature was measured to be around 205 cm2/V-s
and the carrier concentration was found to be 6.0x1016 cm-3 [42]. The growth method
employed was vapor-phase transport. Figure 2.10 shows Hall mobility as a function of
temperature for bulk ZnO. Table 2.5 summarizes the carrier concentration and electron
mobility results obtained by various groups.
Figure 2.10 Hall mobility as a function of temperature for bulk ZnO [28, 42].
15
Figure 2.11 Carrier concentration as a function of T-1 for bulk ZnO [28, 42].
Table 2.5 Carrier concentration and electron mobility of ZnO [28].
16
The mobility data was fitted by solving the Boltzmann transport equation using
Rode’s method [42], considering scattering mechanisms such as: polar optical phonon
scattering, acoustic phonon scattering through deformation and piezoelectric potentials,
and Coulomb scattering from ionized impurities or defects. The measured resistivity was
2.8x104 Ω-cm at 8K and it was argued to be due to the carriers “freeze-out” effect with
conduction being dominated by hopping. The electronic transport was determined to be
due to mixed band and hopping conduction for temperatures ranging from 15K to 40K.
For temperatures above 40K, the data was fitted using the charge balance equation with
transport in the conduction band only [42]; the maximum value of mobility was found to
be around 2000 cm2/V-s at 50 K. The theoretical carrier concentration as a function of
inverse temperature shown in Figure 2.11 [42] was obtained employing a two-donor
charge-balance equation given by [42]
n + NA = ∑
i
N Di
1+ n
φi
where
φi = ( g 0i / g1i ) N c' exp(α i / k )T 3 / 2 exp(− E D 0i / kT )
g 0i is the unoccupied state degeneracy of donor i
g1i is the occupied state degeneracy of donor i
N c' is the effective conduction band of states at T = 1K
α i is the temperature coefficient given by E Di = E D 0i − α iT where E Di is the activation
energy of donor i.
17
2.3.2 Carrier type of undoped ZnO
For some time it has been known that undoped wurtzitic ZnO becomes n-type due
to the presence of intrinsic or extrinsic defects. It was generally accepted that native
defects such as Zn-on-O antisite, Zn interstitial and O vacancies were the cause of the
naturally n-type nature of undoped ZnO [28, 52]. However, recent first principles
calculations based on Density Functional Theory seem to suggest otherwise. Though
oxygen vacancies have been thought as responsible for the n-type conductivity of ZnO,
the argument is mainly based on indirect evidence. Examples of this are the observations
made by researchers showing that the electrical conductivity increases as the oxygen
partial pressure decreases during the sample formation [23]. A decrease in oxygen
pressure could likely promote oxygen vacancies in ZnO, but when hydrogen is present it
also becomes more likely that hydrogen can incorporate onto oxygen sites [23]. Even
though oxygen vacancies have the lowest formation energy among defects that behave as
donors, density-functional calculations indicate that oxygen vacancies are very deep
donors and cannot contribute to n-type conductivity [23, 56, 57, 59]. Other point defects
such as Zn interstitials and Zn antisites have also been found to be unlikely causes of the
n-type conductivity in as-grown ZnO crystals [23, 57, 59]. The density functional
calculations have been corroborated by optically detected electron paramagnetic
resonance measurements on high quality ZnO crystals [23, 54, 55, 56, 57, 59, 60, 61].
The cause of the n-type conductivity in undoped ZnO seems to be the unintentional
incorporation of impurities acting as shallow donors [23]. Hydrogen, being present in
almost all growth and processing environments, seems to be the most likely culprit [23,
18
53, 58]. DFT calculations have shown that interstitial hydrogen forms a strong bond with
oxygen in ZnO acting as a shallow donor [23, 53].
However, even though interstitial hydrogen acts as a shallow donor when bonded to
oxygen, it is highly mobile [62, 63] and can easily diffuse out of samples; contradicting
the stability of the n-type conductivity at higher temperatures [23, 64, 65]. Another
possibility is hydrogen substituting for oxygen in ZnO and acting as a shallow donor [23,
58]. Hydrogen as a substitutional impurity for oxygen in ZnO is much more stable than
interstitial hydrogen, which can explain both the stability and variation of the n-type
conductivity with oxygen partial pressure in ZnO [23, 58].
19
Chapter 3 Theoretical Background
3.1 Carrier transport
Semiconductor materials such as ZnO often operate under non-equilibrium
conditions. Therefore it is important to discuss the dynamics of charge carriers in
semiconductor materials to better understand the underlying mechanisms present in their
electrical behavior. As a starting point, the electrical conductivity will be modeled
following the Drude-Sommerfeld model, which takes into account the following
considerations concerning the electrons from a gas of electrons:
(i) electrons are considered free moving particles in space with a momentum and an
energy, (ii) electrons are subject to instantaneous collision events (e.g. with other
particles such as electrons, or atom cores or with defects in the crystal), (iii) the
probability of a collision during an interval of time dt is proportional to dt, (iv) the
particles reach their thermal equilibrium only through these collisions. [71].
3.2 Mobility

An electron with charge –q under a uniform electric field E will experience a force given
by:


(Eqn. 3.1)……………………………… F = −qE
Incorporating equation 3.1 into Newton’s second law yields m


dv

= −qE where v and
dt
m are the electron’s velocity and mass respectively. Now let’s consider two parallel and
identically shaped areas A and A’ through which electrical charges flow perpendicularly

as shown in Figure 3.1. The distance between the areas is given by v dt and the volume
20

contained within the two areas is given by A v dt . The total electrical charge passing
through this volume can be calculated as:

(Eqn 3.2)…………………………….. Q = nqA v dt
where n and q are the concentration of charge carriers (charge carriers per unit volume),
and their unit charge, flowing through the volume, respectively. The current density is
determined by the number of electrons passing through the surface A during a time

Q
interval dt and is given by J =
or in vector form and introducing equation 3.2:
Adt


(Eqn. 3.3)……………………………… J = −nqv
where the minus sign reflects the fact that electrons move in opposite direction to the
applied electric field and current density vector. Because of collisions, the average
velocity of the electron gas due to the applied electric field must be considered instead of
the instantaneous velocity of each electron. This average velocity is called drift velocity
and can be calculated by considering the average time between electron collisions
Figure 3.1 Volume enclosed between arbitrarily shaped areas with charge carriers.
21
or scattering events named electron relaxation time τ. The electron relaxation time
enables the calculation of the drift velocity by integration of Newton’s second law in the
form previously presented:




dv
qE
qE
=−
→ dv = −
dt
dt
m
m

(Eqn. 3.4)…………………. vdrift =


qE
qτ 
∫t =0 − m dt → vdrift = − m E
t =τ
As can be seen in equation 3.4 the drift velocity of the electrons is proportional to the
applied electric field. The proportionality factor is called mobility and is given by:
(Eqn 3.5)………………………………... µ =
qτ
m


Using equation 3.5 the drift velocity equation can be written as vdrift = − µE .
Mobility is expressed in units of cm2⋅V-1⋅s-1, and represents the velocity gained by an
electron per unit electric field strength, with units given by velocity (cm⋅s-1) divided by
electric field strength (V⋅cm-1). The drift current density can now be expressed as
previously stated but introducing the quantities just defined:


(Eqn. 3.6)………………………….... J drift = −nqvdrift

nq 2τ 
(Eqn. 3.7)…………………………… J drift =
E
m
Equation 3.7 shows that the current density is proportional to the applied electric field
and as before, the proportionality factor has a name of its own, known as conductivity
and given by:
22
(Eqn. 3.8)………………………….. σ =
nq 2τ
= nqµ
m
and with this expression for conductivity the current density can be expressed as


J drift = σE . The inverse of conductivity is called resistivity and is given by:
(Eqn. 3.9)………………………….. ρ =
m
1
=
2
nq τ nqµ
When performing electrical measurements, the resistance of the material is often
calculated from voltage and current, and is used to obtain material dependent
characteristics such as resistivity. Next we discuss the relationship between resistance and
resistivity.

Consider a cylindrical section of a solid material, with an applied electric field E
in the axial direction, length L and cross-sectional area A, through which a current density

J flows as shown in Figure 3.2.
Figure 3.2 Section of solid material through which current flows.

The voltage difference along the section is given by V = L E since the applied electric
field is constant. The electric current along the section can be obtained from the current
 V



we get
density magnitude and is given by I = A J . With J = σ E and E =
L
23
 AσV
IL ρL
=
I . By comparing with Ohm’s Law
I = Aσ E =
, and rearranging V =
Aσ
A
L
V = IR we can see that:
(Eqn. 3.10)……………………………... R =
ρL
A
where R is the resistance of the solid section of material through which current flows.
To extend the ongoing discussion to the case of materials carrying both positive
(holes) and negative (electrons) charge carriers as in semiconductors, we take into
consideration the drift current density vector as being composed of drift current densities



from both electrons and holes simultaneously, J drift = J drift −h + J drift −e , where the
subscripts h and e denote the drift current densities corresponding to holes and electrons
respectively. By considering the direction in which the positive and negative charge
carriers move with respect to the applied electric field the following equations are
obtained:


(Eqn 3.11)…………………………… vdrift −h = µ h E


(Eqn 3.12)…………………………... vdrift −e = − µ e E
for holes and electrons respectively. We also get:


(Eqn 3.13)…………………………. J drift −h = pqvdrift −h


(Eqn 3.14)…………………………. J drift −e = −nqvdrift −e
where p and n denote the density of positive and negative charge carriers respectively.
Finally introducing equations 3.11 and 3.12 into equations 3.13 and 3.14 yields:


J drift −h = pqµ h E
24


J drift −e = nqµ e E





J drift = pqµ h E + nqµ e E = q ( pµ h + nµ e ) E = σE
(Eqn. 3.15)………………………... σ = q( pµ h + nµ e )
where σ is the conductivity of the material.
Since resistivity is the inverse of conductivity, with equation 3.15 resistivity can be
expressed as:
(Eqn 3.16)………………………… ρ =
1
q ( pµ h + nµ e )
3.3 Scattering Mechanisms
The scattering processes within the solid under consideration, shown in figure 3.2,
affect the relaxation time of the electrons which in turn modify the current drift velocity.
Scattering processes can be categorized as elastic and inelastic. In elastic processes
electrons undergo a change in momentum but not in energy, while in inelastic processes
both momentum and energy change due to energy interactions with the lattice. When
scattering processes are present in a solid, the total relaxation time will be given by
Matthiessen’s rule:


(Eqn. 3.17)…………………………... τ T =  ∑ τ i 
 i 
−1
where τi represents the relaxation time for each scattering process. Matthiessen’s rule
assumes that while the electrons undergo one scattering process they are not affected by
another. The scattering processes affecting ZnO are those governing electron transport in
III-IV semiconductors and are briefly discussed next.
25
Ionized impurity scattering is caused by long range Coulomb potentials attributed
to charged centers of defects or intentionally doped impurities. Donors and/or acceptors,
typically ionized, are introduced as dopants in semiconductors and because of the charge
of the ionized dopants, Coulombic forces deflect electrons or holes that approach them
[72, 73].
Polar longitudinal optical phonon scattering occurs when lattice vibrations induce
electric fields through polarization and in turn the fields interact with the moving charges.
The field inducing lattice vibrations are caused by the ionic nature of the bonds in polar
semiconductors [73].
Acoustic phonon scattering is present when strain caused by acoustic phonons
induces changes in energy of the electron band edges. In turn the wave vectors of the
phonons are increased with increasing scattering rate [73].
Piezoelectric scattering occurs in semiconductors where electric fields arise from
unit cell distortions caused when the lattice is subjected to strain in certain directions.
This type of scattering happens only in compound semiconductors because of their polar
nature and is small in most cases but non negligible at low temperatures when other
scattering mechanisms are weak [72, 73].
Scattering through defects can occur in semiconductors with high dislocation density and
native defects. When a dislocation line is introduced in a semiconductor crystal, acceptor
centers are also introduced. In n-type semiconductors conduction band electrons are
captured by the dislocation lines, which eventually become negatively charged. These
negatively charged dislocation lines eventually scatter electrons traveling across them
thus reducing mobility [73].
26
Chapter 4 Electrical Characterization Techniques
4.1 Four-point probe versus two-point probe configurations
Four-point probe configurations are preferred over two-point probe configurations
for electrical characterization of semiconductor devices because they provide a method to
eliminate unaccounted probe contact effects. Figure 4.1 shows the circuit diagram
representation of two-point and four-point probe configurations.
Figure 4.1 Two point and four point probe circuit diagrams [77].
For the two-point probe configuration the total resistance in the circuit is given
by:
(Eqn. 4.1)……………………… RT = 2 RW + 2 RC + RDUT
where RW, RC and RDUT are the wire resistance, contact resistance and the resistance of the
device under study respectively. With the two-point probe configuration it’s impossible
to determine RC and RDUT separately since the voltage-drop measured includes the drops
in voltage across the wires, probe contacts and device under study combined. To
27
overcome this problem a four-point probe method is used; current is passed through two
of the probes and voltage is measured across the other two connected in parallel.
Although the voltage path in the four-point probe configuration contains the resistance of
the wires and probe contacts, the current flowing through it is small due to the high input
impedance of the voltmeter. Since the current is small, the voltage-drops across the wires
and contacts are negligibly and the measured voltage is essentially the drop in voltage
across the device under study.
4.2 Van der Pauw resistivity calculation technique
A collinear arrangement of the measuring probes is the most commonly used
four-point probe configuration. However, this technique places limitations in the size of
the samples, since the probes have to be placed in the center of the sample sufficiently far
from its edges for the resistivity calculations to be valid. Another four-point probe
technique available for electrical characterization is the one developed by van der Pauw
[74]. Through the use of conformal mapping he showed that the resistivity of flat
arbitrarily shaped samples could be measured without knowledge of the current pattern
provided that specific conditions were met. The conditions for the implementation of the
van der Pauw technique are: the contacts must be placed at the circumference of the
sample, they must be sufficiently small compared to the area of the sample, in turn, the
sample must be of uniform thickness throughout, and its surface must not contain isolated
holes (i.e. must be singly connected). The derivation of the van der Pauw equations is
shown next.
28
As a starting point let’s consider a flat arbitrarily shaped piece of conducting
material with contacts numbered 1, 2, 3, and 4 as shown in Figure 4.2. For this sample
current enters through terminal 1 and exits through terminal 2. Simultaneously voltage is
measured across terminals 3 and 4. The resistance can be then defined as:
(Eqn. 4.2)……………………………... R12,34 =
V34
I12
where I12 and V34 represent the current and voltage measurements previously discussed.
Figure 4.2 Arbitrarily shaped flat sample for van der Pauw measurements [77].
For a sample meeting the four criteria previously established van der Pauw proved that
the following equation holds:
 − πR12,34t 
 − πR23, 41t 
 + exp
 = 1
(Eqn. 4.3)………………... exp
ρ
ρ




where t is the sample thickness and:
R23, 41 =
V41
I 23
is the resistance obtained after interchanging the voltage and current terminals with
respect to the original configuration and ρ is the resistivity of the sample. Next we derive
equation 4.3.
29
Let us consider a lamella of infinite extension and thickness t to which a current
of magnitude 2I is applied at point 1 as shown in figure 4.3(a). At a distance r from point
1, the magnitude of the current density J is given by:
(Eqn. 4.4)……………………………… J =
2I
2πrt
The electric field generated extends in the radial direction and its magnitude is given by:
(Eqn. 4.5)…………………………… E = ρJ =
ρI
πrt
The potential difference between two other points 3 and 4 can be calculated as:
4
V4 − V3 = − ∫ Edr =
3
=
ρI 3 dr ρI
ρI
(ln(a + b ) − ln(a + b + c ))
= ln r |34 =
∫
tπ 4 r tπ
tπ
ρI
ρI a + b + c
a+b
ln
= − ln
tπ a + b + c
tπ
a+b
Figure 4.3 Points in infinite lamella for derivation aid of van der Pauw equations.
If we consider only one half of the lamella, since no current flows perpendicular to line
1234, we can halve the current considered, as shown in figure 4.3(b), and the result just
30
derived is still valid [74]. Next we consider a current flowing from point 2 as seen in
figure 4.3(c). Following the same reasoning as before we arrive at the following equation
for the potential:
V4 − V3 =
ρI b + c
ln
tπ
b
By superimposing both cases we get the potential between points 3 and 4 due to a current
I entering through point 1 and exiting through point 2:
V4 − V3 =
a + b + c  ρI  (a + b)(b + c) 
ρI  b + c

− ln
 ln
 = ln
tπ 
b
a + b  tπ  (a + b + c)b 
Dividing by the current we obtain an expression for the resistance as:
R12,34 =
 tπR12,34  (a + b + c)b
V34 ρ  (a + b)(b + c) 
 or exp −
 =
= ln
.
ρ
I12 tπ  (a + b + c)b 

 (a + b)(b + c)
Similarly we get:
 tπR23, 41 
ac
 =
exp −
ρ  (a + b)(b + c)

Finally adding these two equations we get:
 tπR12,34 
 tπR23, 41 
 + exp −
 =1
exp −
ρ 
ρ 


In general an explicit expression for the resistivity cannot be derived from the equation
above, but if the sample has an axis of symmetry and contacts 1 and 3 are placed on such
axis, while contacts 2 and 4 are symmetric with respect to this axis as shown in figure
4.4, then the equation simplifies to:
(Eqn. 4.6)………………………….... ρ =
π
tR12,34
ln 2
31
Figure 4.4 Symmetric sample type for van der Pauw measurements [74].
For samples without the symmetry just presented it is more convenient to write the
resistivity equation in the form:
(Eqn. 4.7)…………………….. ρ =
π  R12,34 + R23, 41 
 f
t
ln 2 
2

where f is a geometrical correction factor calculated from iteration of:

 ln 2  
 exp

 Rr − 1 
f  
f


 =
(Eqn. 4.8)………………... 
ar cosh

2
 Rr + 1  ln 2






where: Rr =
R12,34
R23, 41
Figure 4.5 shows a plot of f as a function of Rr. Next we derive the expression for the
geometrical factor f.
32
Figure 4.5 Geometrical factor f as a function of Rr [77].
Figure 4.6 Typical sample geometries: (a) Circular, (b) Cloverleaf, (c) Square [77].
Let’s start by introducing the following substitutions for simplicity:
tπR12,34 = x1 and tπR23, 41 = x2 .
Equation 4.3 then becomes:
 x 
 x 
(Eqn. 4.9)…………………….. exp − 1  + exp − 2  = 1
 ρ
 ρ
Also let’s express x1 and x2 as:
x1 =
1
((x1 + x2 ) + (x1 − x2 )) and x2 = 1 ((x1 + x2 ) − (x1 − x2 ))
2
2
Now equation 4.7 can be expressed as:
33
 x + x2 
 x − x2 
 x − x2  
 exp − 1
exp − 1
 + exp 1
  = 1
2 ρ 
2ρ 


 2ρ  
which is the same as:
 x +x 
x −x  1
exp − 1 2  cosh 1 2  =
2ρ 

 2ρ  2
By expressing the exponent as
x1 + x2 ln 2
we get:
=
2ρ
f


 ln 2 
(Eqn. 4.10)……………….. exp −
 cosh

f




x1

−1

ln 2  1
x2
= ,
 2
x1
f
+1

x2

which is the equation plotted in figure 4.5
As previously stated, the van der Pauw technique assumes negligibly small
contacts placed on the periphery of the sample, however in practice contacts are of finite
size and their influence can be accounted for by the introduction of correction factors.
Figure 4.7 shows the correction factors considering the finite size for contacts placed on
the corners and mid points of the sample sides [77], [78].
Figure 4.7 Correction factors to account for contact size on square samples.
34
The errors introduced by non-ideal contacts can be eliminated by the use of the
cloverleaf sample type shown in figure 4.6(b), but preparation of such a sample shape can
be quite complicated and square or circular samples are often employed. One of the
advantages of the van der Pauw technique is that it enables the use of smaller samples
than those required by collinear four-point probes due to their edge-to-probe distance
requirements, as well as minimum probe spacing limitations of such configurations.
4.3 Hall Effect measurements
For the development of the Hall Effect equations let’s start by considering a solid
slab containing only positive charge carriers as shown in figure 4.8.
Figure 4.8 Conducting solid slab.


An electric E field is applied in the +x direction with the current density vector J in the

same direction and an applied magnetic field B in the +z direction. Under these
circumstances a positive charge carrier will initially move along the +x direction and
experience a force called the Lorentz force given by:
35



(Eqn. 4.11)………………………… FLorentz = qv p × B


where q , v p , and B are the carrier charge, carrier drift velocity and applied magnetic
field respectively. The Lorentz force is perpendicular to both drift velocity and magnetic
field as suggested by the cross product and in the present development points in the -y
direction. For our case of a conducting solid slab, the positive carriers will start
accumulating towards the -y direction, due to the Lorentz force, and simultaneously
negative charges will start to appear on the opposite +y direction, due to the absence of
positive charge carriers.
The separation of charges due to the Lorentz force will give rise to an electric

field E Hall denominated Hall electric field, pointing in the opposite direction with respect
to the above-mentioned Lorentz force. At equilibrium the Lorentz force and the force on
the charge carriers due to the Hall electric field balance each other and we can write:





FLorentz + FHall = 0 = qv p × B + qE Hall
From the equation above, the Hall electric field can be obtained and is given by:



(Eqn. 4.12)…………………………. E Hall = −v p × B
For the case under consideration the drift velocity points in the +x direction, while the
magnetic field points in the +z direction. Hence by the cross product the Hall electric
field points towards the +y direction and its magnitude given by:
E Hall , y = v p , x Bz


Recalling the current density vector J p = pqv p , which in the case under consideration
reduces to J p , x = pqv p , x , we can write the Hall electric field as follows:
36
(Eqn. 4.13)…………………………. E Hall , y =
J p , x Bz
pq
Equation 4.13 is typically rearranged and to give the carrier concentration in the solid and
is called the Hall constant. It is given by:
(Eqn. 4.14)………………………. RHall =
E Hall , y
J p , x Bz
=
1
pq
and it is positive for conducting materials with positive charge carriers. With the above
expression we can introduce the Hall mobility given by:
(Eqn. 4.15)…………………………. µ Hall , p = σRHall
For materials with negative carriers an identical derivation scheme as the one developed
for positive charge carriers leads to the following equations:



 
FLorentz + FHall = 0 = −qvn × B − qE Hall ,

 
E Hall = −vn × B , E Hall , y = vn , x Bz ,
and recalling J n , x = −nqvn , x we get:
(Eqn. 4.16)………………………… E Hall , y = −
(Eqn. 4.17)……………………... RHall = −
J n , x Bz
nq
E Hall , y
J n , x Bz
=−
1
nq
(Eqn. 4.18)…………………………. µ Hall ,n = σ RHall
As equation 4.17 suggests the Hall coefficient for materials with negative charge carriers
is negative, as one would expect given that the direction of the applied magnetic and
electric fields remains the same as for the case of positive charge carriers. As can be seen,
the sign of the Hall coefficient is an indicator of the type of material under analysis, p-
37
type or n-type, corresponding to positive and negative majority charge carriers/
respectively.
4.4 Importance of ohmic contacts
Ohmic contacts are defined as having a linear and symmetric relationship between
current and voltage for both positive and negative voltage values [79]. Achieving ohmic
low resistance contacts between the measuring equipment and the sample under
consideration is necessary to ensure that the results obtained are caused by the bulk
resistivity of the material and not the contact resistance itself. Selection of metals that are
appropriate for the semiconductor under study can enable the formation of low resistance
ohmic contacts. The criteria necessary for the contact material selection are provided in
the following section.
4.4.1 Theoretical formalism of ohmic contacts
Under thermal equilibrium, two dissimilar materials brought into contact will
have the same Fermi energy. In the case of a metal in contact with a semiconductor, band
bending will compensate the difference in Fermi energies. Electrons will flow from the
material in which they have higher energy into the material where lower energy electrons
are present. The fact that the Fermi level in a metal is located inside an energy band
enables electrons to be emitted or received by the metal and charge redistribution takes
place, eventually generating an electric field. The difference in the work functions of the
38
metal and semiconductor determines both the direction of electron flow and the amount
of band bending that will take place.
Figure 4.9 Parameters considered when analyzing a metal semiconductor junction: Work-function,
Fermi Level, Electron affinity [71].
When the work-function of the metal is greater than that of the semiconductor
(Φm > Φs), the energy bands of the semiconductor must shift downwards and electrons
are transferred into the metal. For the case where the work function of the metal is less
than that of the semiconductor (Φm < Φs), the energy bands of the semiconductor must
shift upwards and electrons are transferred from the metal into the semiconductor. In both
cases the shift in the energy bands of the semiconductor occurs in order for the Fermi
energies to align. By conducting a similar analysis as for a p-n junction, the signs of the
carriers as well as the direction of the built-in electric field can be determined for a metalsemiconductor junction. Since electrons are transferred through the junction, what
happens next depends on the whether the semiconductor is n-type or p-type.
39
Figure 4.10 Band bending in a metal-semiconductor junction [71].
If the semiconductor is n-type and its work function is greater than that of the
metal (Φm < Φs), then electrons are transferred into the semiconductor, thus providing
more majority carriers and a depletion region is not formed. The resulting contact will not
exhibit rectifying behavior, but ohmic properties instead. On the other hand, if the work
function of the semiconductor is less than that of the metal (Φm > Φs), then electrons will
be extracted from the semiconductor into the metal thus creating a depletion region near
the junction and exhibiting rectifying or Schottky behavior when an external voltage is
applied.
For the case of a p-type semiconductor the reverse happens. If its work function is
greater than that of the metal (Φm < Φs), then electrons will be transferred into the
semiconductor thus creating a region of negative charge accumulation near the junction
and thus behaving as a Schottky barrier. When the work function of the semiconductor is
less than that of the metal (Φm > Φs), electrons are then transferred from the
semiconductor into the metal. The electrons will be taken from the p-type dopants, which
in turn become ionized and create more holes. Since more holes are created, more
40
majority carriers accumulate near the junction and a depletion region is not created. Thus
the majority carriers are free to flow in either direction under the influence of an external
voltage.
Figure 4.11 Summary of ohmic and Schottky contacts depending on the type of semiconductor and
work-function relative to the metal [71].
4.4.2 Preparation of Ohmic Contacts
As discussed in the previous section, the appropriate selection of a contact metal
will determine if the contact formed with the semiconductor is ohmic or rectifying. If the
type of the semiconductor under study is known (p-type or n-type), as well as its workfunction, then selecting an appropriate metal should produce ohmic contacts. Otherwise,
if the type and work-function are not known, a literature review should be performed in
order to select and test candidate metals that will produce ohmic contacts. Once the metal
or metals have been selected, the contact formation can be done through any available
process, (Sputtering, PLD, melting etc.), after appropriately masking the samples under
study to conform to the measuring technique to be implemented.
41
4.4.3 Ohmic contact verification
The contacts made on the sample under study should be checked for ohmic
behavior before any measurement is made. Depending on the measuring technique being
employed (voltage source or current source being used), an I-V or V-I plot should be
produced through an appropriate range of the source. The power dissipated through the
sample should be kept below 5mW (preferably under 1mW), to prevent Joule heating of
the sample. Power dissipation through the sample can be easily calculated from:
(Eqn. 4.19)…………………………….... P = VI
The maximum voltage or current to be applied to the sample is obtained from:
1
1
P 2
(Eqn. 4.20)………………….. Vmax = (PR ) 2 , I max =   ,
R
where P is the maximum power to be dissipated, 1mW or 5mW, and R is the sample’s
resistance [80].
42
Chapter 5 Experimental Techniques
5.1 ZnO sol-gel samples preparation
Al doped ZnO sol-gel samples made as part of a previous research project, were
electrically characterized employing the van der Pauw technique discussed in chapter 4.
However the thickness of those samples near the edges, where electrical contacts should
be placed, was not uniform. Thickness variations were evidenced by light scattering
observed when holding the samples at certain angles with respect to a light source. Such
variations are not appropriate for the van der Pauw technique and in order to correctly
perform the electrical characterization measurements, new samples were prepared using
the same preparation procedure.
For the solution preparation, depending on the Al doping level desired (see Table
5.1), appropriate amounts of aluminum nitrate nonahydrate Al(NO3)3·9H2O and zinc
acetate dihydrate Zn(CH3OO)2·2H2O were combined in solution with 0.54ml of
Monoethanolamine (MEA) and 12ml of Methoxyethanol. MEA was used as a solution
stabilizer and to promote homogeneity while Methoxyethanol served as the main solvent.
Table 5.1 Details of Al doped ZnO solutions.
43
The solution was placed in a water bath at a temperature range of 60°C-70°C while
stirring for 3 hours. The solutions were deposited using a needle-less syringe with a 0.2
μm filter. For each sample, seven layers made up of three drops each, were deposited
onto fused glass substrates. After deposition of each three-drop layer, the samples were
spun at 1400-1600 rpm for 6 seconds and 3000 rpm for 30 seconds. After each layer and
subsequent spinning, the samples were placed on a hot plate at 300°C for 2 minutes to
evaporate the solvent and organic residue from the precursors used. After all the layers
were deposited, the samples were heated in a furnace at a temperature of 700°C for 2
hours. This annealing temperature was reached through a 350°C/hour ramp and
subsequently slowly cooled.
Figure 5.1 Spin-coated Al doped ZnO film prepared via sol-gel method.
5.2 Sample holder construction
A four-point probe sample holder was designed and built to enable the electrical
characterization of thin film samples. The sample holder was incorporated into existing
cryogenic equipment enabling measurements in the 30K-320K temperature range, with
the possibility of going as low as 15K under certain conditions. The design consists of an
aluminum base (25mm x 25mm) with four posts electrically insulated from it. Attached
44
to the posts are small arms made out of brass with tungsten needles attached at the ends.
The arms have radial and angular degrees of freedom and have silver coated copperberyllium springs to maintain contact to the sample even when thermal contraction occurs
at low temperatures. The size of the sample holder allows measurement of samples
ranging in dimensions from 5mm x 5mm (25mm2) to 15mm x 15 mm (225mm2). All
materials used in the sample holder are non-magnetic to enable Hall Effect measurements
under strong magnetic fields. Additionally, an aluminum nitride (AlN) sheet adhered to
the sample holder surface, serves as the platform where the samples are placed to perform
the measurements. Aluminum nitride was selected due to its high electrical insulation,
good thermal conductivity and non-toxicity.
Figure 5.2 Diagram of sample holder components.
45
Figure 5.3 Cryogenic equipment (left), Sample holder close-up (right).
5.3 Contact formation for electrical measurements
As mentioned in section 4.4, selecting appropriate metals to form ohmic contacts
to the semiconductor under study is essential to successfully measure its resistivity. Table
5.2 presents various metals and combinations of metals and their characteristics when
forming contacts with ZnO. For the Al doped ZnO thin films under study, indium and
aluminum-gold contacts were deposited on the samples through various methods. As a
first trial, indium contacts measuring around 1.5mm x 1.5mm in size were cut from
sheets and subsequently pressed at the edges of the samples. The size of the contacts was
selected as a compromise between one that could be easily applied to the samples while
following the size requirements presented in figure 4.7. To promote adherence, the
samples with pressed indium contacts were annealed at 300°C for 5 minutes in an air
atmosphere and slowly cooled to prevent cracking. Another method employed to promote
contact adherence was placing the samples on a hot plate at 200°C for 30 seconds. This
process produced ohmic contacts as confirmed by V-I plots. Figure 5.4 shows a ZnO
sample with indium contacts deposited.
46
Figure 5.4 ZnO on fused glass with indium contacts.
Table 5.2 Metallization schemes for ohmic contacts on ZnO [73].
47
Another combination tried on the samples was the deposition of layered
aluminum and gold contacts on the edges of the samples via sputtering. To mask the
samples, fused glass substrates rotated 45 degrees were carefully placed on top of the
samples during the sputtering process, yielding triangular contacts at the edges. The base
pressure for sputtering was in the 7 x10-6 to 9 x10-6 Torr range and the deposition
pressure between 10 mTorr and 13 mTorr. The distance from the Al and Au targets was
around 3.5 cm and 6.5 cm, respectively. The aluminum and gold targets used measured 2
inches in diameter. The power applied to the targets during deposition was 50W and the
deposition time ranged between 2 to 3 minutes. The contacts produced were about 100nm
thick for both the Al and Au layers. Figure 5.5 shows a ZnO sample with Al-Au contacts
deposited at the edges.
Figure 5.5 ZnO sample with Al-Au contacts.
For square samples it is recommended to form contacts at the edges. The size of the
contacts should fall in the size range shown in figure 4.7 in order to minimize errors in
the resistivity measurements.
Following a simple geometrical analysis, the appropriate dimensions of the mask
can be calculated for a desired contact size, given the sample dimensions. Let the samples
48
be squared of sides l, let the desired contact size be s and let the sides of the square mask
be l’. It is assumed that the sample and mask are centered and the mask is rotated 45
degrees with respect to the sample as shown in figure 5.6. By applying simple geometry
for right triangles, it is determined that the height h of the triangle formed by the contact
is given by h =
l 2 − l'
. Also, application of the Pythagorean relation to the triangle
2
( )
2
2
formed by the contact yields s = h + (s cos 45°) = h + s
2
2
2
2
= h2 +
s2
, thus
2
s = h 2 . Finally, substitution of the expression for h previously derived yields
l ' = 2 (l − s ) .
Figure 5.6 Sample masking with relevant size parameters.
5.4 Van der Pauw resistivity measurement implementation
Implementation of the van der Pauw resistivity equation for real world
measurements requires some technical considerations that are presented in this section.
49
Recall a sample with contacts at the periphery, numbered 1 through 4, as shown in figure
4.2 from chapter 4. For such a sample the van der Pauw resistivity and geometrical factor
are given by:
ρ=
π  R12,34 + R23, 41 
 f
t
ln 2 
2


 ln 2  
 exp

 Rr − 1 
f  
f



 =
ar cosh

2
 Rr + 1  ln 2






Rr =
R12,34
R23, 41
where R12,34 is obtained by sending current through contacts 1-2 and measuring voltage
through contacts 3-4 and R23,41 is obtained by sending current through contacts 2-3 and
measuring voltage through contacts 4-1. Since a current source is employed, current is an
input of the measurements and can be taken out as an average from the terms in
parenthesis as well as from the resistance ratio used to calculate the geometrical factor f:
(Eqn. 5.1)…………….. ρ =
V
V 
π
t  34 + 41  f =
t (V34 + V41 ) f
2 ln 2  I12 I 23 
2 I ln 2
π
V34
(Eqn. 5.2)……………..………….. Rr =
V41
I12
I 23
=
V34
=Q
V41
As can be seen from the resistivity equation, the voltage measurements are
centered in a corner of the sample, specifically around corner 4 in the case under
50
discussion. The same equation can be repeated for the other diagonally located corner of
the sample and an average of the two values for resistivity can be taken:
(Eqn. 5.3)………….. ρ a =
π
π
t (V34 + V41 ) f and ρ b =
t (V12 + V23 ) f
2 I ln 2
2 I ln 2
(Eqn. 5.4)……………………..……….. ρ avg =
ρ a + ρb
2
Table 5.3 gives the notation convention used to simplify the equations for the resistivity
measurements and figure 5.7 shows the implemented measurement scheme. The eight
voltages shown in Table 5.3 ensure that the sample is measured in all possible
configurations.
Table 5.3 Resistivity voltages notation convention.
Designation
Current
Applied
Between
Voltage
Measured
Between
V1
1-2
3-4
V2
2-1
3-4
V3
2-3
4-1
V4
3-2
4-1
V5
3-4
1-2
V6
4-3
1-2
V7
4-1
2-3
V8
1-4
2-3
51
Figure 5.7 Van der Pauw resistivity measurement scheme.
To account for thermoelectric potential offsets a voltage delta method is implicitly
used when performing measurements for resistivity calculations. This is evidenced by the
current reversal for each measurement configuration. Recalling the van der Pauw
resistivity equation and implementing the scheme presented in Table 5.3 the practical
form of the resistivity equations become:
(Eqn. 5.5).. ρ a =
π  V2 − V1 V4 − V3 
π  V6 − V5 V8 − V7 
t
t
+
+
 f a and ρ b =
 fb
2 I ln 2  2
2 
2 I ln 2  2
2 
thus:
(Eqn. 5.6)…. ρ a =
π
4 I ln 2
t (V2 − V1 + V4 − V3 ) f a and ρ b =
π
4 I ln 2
t (V6 − V5 + V8 − V7 ) f b
Similarly the voltage ratios used to iterate for the geometrical factor fa and fb now
become:
(Eqn. 5.7)...….. Qa =
V
V −V
V −V
V34 V34 − V43 V1 − V2
=
=
and Qb = 34 = 12 21 = 5 6 .
V41 V23 − V32 V7 − V8
V41 V41 − V14 V3 − V4
52
The redundancy of measurements for the van der Pauw resistivity calculation
enables quality check of the sample as well as of the contacts made in the sample. To
ensure the quality of the contacts the resistances calculated by reversing the current
should be within 5% or less [80]. The checks for contact quality are summarized by the
following equations:
(Eqn. 5.8)…………………….. R21,34 = R12, 43 R43,12 = R34, 21
(Eqn. 5.9)…………………….. R32, 41 = R23,14 R14, 23 = R41,32 .
Additionally to ensure uniformity of the electrical properties of the sample the following
sums of resistances should be within 5% or less [80]:
(Eqn. 5.10)…………….…….. R21,34 + R12, 43 = R43,12 + R34, 21
(Eqn. 5.11)…..………………. R32, 41 + R23,14 = R14, 23 + R41,32
5.5 Hall Effect measurements implementation
Practical implementation of the Hall Effect for actual measurements follows a
similar reasoning as the van der Pauw resistivity measurements. Since current and
voltage are sent and measured perpendicular to each other, there are four possible
configurations for a four contact sample. In practice there are eight measurements made,
four with magnetic flux in the forward direction and four in the reverse direction. In the
same way that a new notation convention was introduced for resistivity measurements, a
new notation convention is introduced for Hall Effect measurements, as shown in Table
5.4. Figure 5.8 shows the Hall Effect measurement scheme.
53
Table 5.4 Hall Effect voltages notation convention.
Designation
V1
V2
V3
V4
V5
V6
V7
V8
Flux
+B
+B
+B
+B
-B
-B
-B
-B
Current
Voltage
Applied Measured
Between Between
1-2
3-4
2-1
3-4
2-3
4-1
3-2
4-1
3-4
1-2
4-3
1-2
4-1
2-3
1-4
2-3
Figure 5.8 Hall Effect measurement scheme.
The practical implementation of the Hall coefficients derived in chapter 4 is as follows:
(Eqn. 5.12)…………...….. RHall
VHall
VHall
EHall
W
W = VHall t
=
=
=
I
I
IB
JB
B
B
Wt
A
where VHall is the voltage measurement perpendicular to the flow of source current, I is
the source current, t is the thickness of the sample, B is the magnetic flux perpendicular to
the plane of the sample and W is the width of the sample. The modifications to the
equation above involve subtracting voltages to eliminate any thermoelectric potentials
54
present at the time of the measurement. The measurements are also averaged taking into
account the magnetic flux in the forward and reverse directions. Adapting the notation
conventions presented in Table 5.4 the Hall coefficient equations become:
(Eqn. 5.13)...… R Hall − a =
t  V2 − V1 V5 − V6 
t
(V2 − V1 + V5 − V6 )
+
=

IB  2
2  2 IB
(Eqn. 5.14)…... R Hall −b =
t  V4 − V3 V7 − V8 
t
(V4 − V3 + V7 − V8 )
+
=

IB  2
2  2 IB
+ RHall −b 
R
(Eqn. 5.15)…………………. Ravg =  Hall −a

2


It is important to remember that the sign of the Hall coefficient will indicate the type of
sample under study, n-type (negative Hall coefficient) or p-type (positive Hall
coefficient).
5.6 LabView programming
5.6.1 Electrical measurement equipment and measurement overview
As already discussed, resistivity measurements of thin films using the van der
Pauw technique, require a voltage and source current switching scheme difficult to
perform if done manually. For this reason successful implementation of the van der Pauw
measuring technique can be greatly improved by the automation of the process.
Automation of the process is achieved by connecting the current source as well as the
voltage and current measuring equipment to a computer-controlled switching equipment.
The following is a list of the equipment used for the electrical measurements:
55
•
Keithley 2182A Nanovoltmeter
•
Keithley 6485 PicoAmmeter
•
Keithley 220 Current Source
•
Keithley 7001 Switch System
•
Keithley 7065 Hall Effect Card
•
Varian V4005 4” Electromagnet
•
Varian V2901 Regulated Magnet Power Supply
Figure 5.9 shows the connection diagram of the electrical measurements equipment. As
can be seen in the figure, four contacts placed in the edges of the sample (for square
samples), are connected to the switch system. The rest of the equipment, (current source,
voltmeter and ammeter), are connected to the switch system. To supply programming
commands, a computer is connected to all of the instruments via GPIB cables.
Figure 5.9 Electrical measurements equipment connection diagram.
56
The automation of the measurement equipment already mentioned was achieved
through the use of LabView routines. Multiple routines for resistivity measurements, Hall
Effect measurements, and ohmic contact checks, among others, were developed as part of
the research to aid the characterization process. Figure 5.10 shows a flow diagram of how
the resistivity routine works.
Figure 5.10 Resistivity measurement routine flowchart.
57
5.6.2 Electrical measurement error calculation
As previously discussed resistivity and Hall Effect calculations require
measurements of current and voltage and subsequent calculations. For each of the current
and voltage measurements the error was calculated as the quotient between the standard
deviation and the average value. These errors were then combined taking into
consideration that each of the voltage and current measurements were independent of
each other. Considering a quantity f that depends on quantities a and b namely f(a,b), the
standard deviation of f can be calculated as [93]:
 ∂f 
 ∂f 
(Eqn. 5.16)………………… σ f =   σ a2 +   σ b2
 ∂b 
 ∂a 
2
2
where σa and σb are the standard deviations of quantities a and b respectively. To
calculate the standard deviation in the van der Pauw resistivity calculations, the notation
introduced in section 5.4 will be used. Also considering that ρa = ρa(V1,V2,V3,V4,I1,I2,I3,I4)
and ρb = ρb(V5,V6,V7,V8,I5,I6,I7,I8) we have:
σρ
(Eqn. 5.17)………
a
  ∂ρ
=   a
  ∂V1

 ∂ρ
+  a
 ∂I1
2
2
2
2
2
 2  ∂ρ a  2  ∂ρ a  2  ∂ρ a  2
 σ V3 + 
 σ V4
 σ V2 + 
 σ V1 + 
V
V
V
∂
∂
∂
 4
 2

 3
 2  ∂ρ a
 σ I1 + 
 ∂I 2

1/ 2
2
2
2
 2  ∂ρ a  2  ∂ρ a  2 
 σ I3 + 
 σ I 4
 σ I 2 + 

I
I
∂
∂
 4

 3

Similarly for ρb we obtain:
58
σρ
(Eqn. 5.18)…………
b
2
2
2
  ∂ρ  2
 ∂ρ b  2  ∂ρ b  2  ∂ρ b  2
2
b

 σ
 σ +
 σ +
 σ +
= 
  ∂V5  V5  ∂V6  V6  ∂V7  V7  ∂V8  V8

1/ 2
2
2
2
2
 ∂ρ b  2  ∂ρ b  2  ∂ρ b  2  ∂ρ b  2 
 σ I8
 σ I 7 + 
 σ I 6 + 
 σ I5 + 
+ 

 ∂I 8 
 ∂I 7 
 ∂I 6 
 ∂I 5 

Using equation 5.6 without averaging the currents we get:
(Eqn. 5.19)………………... ρ a =
V V V V 
t  2 − 1 + 4 − 3  f a
4 ln 2  I 2 I1 I 4 I 3 
(Eqn. 5.20)…………….…. ρ b =
V V V V 
t  6 − 5 + 8 − 7  f b
4 ln 2  I 6 I 5 I 8 I 7 
π
π
The partial derivatives in equation 5.17 and 5.18 are given by:
(Eqn. 5.21)……..
 V V V V  ∂f
∂ρ a
∂  V2 V1 V4 V3 
 − + −  f a + k  2 − 1 + 4 − 3  a
=k
∂Vn
∂Vn  I 2 I1 I 4 I 3 
 I 2 I1 I 4 I 3  ∂Vn
(Eqn. 5.22)……..
 V V V V  ∂f
∂ρ a
∂  V2 V1 V4 V3 
 − + −  f a + k  2 − 1 + 4 − 3  a
=k
∂I n
∂I n  I 2 I1 I 4 I 3 
 I 2 I1 I 4 I 3  ∂I n
(Eqn. 5.23)……..
 V V V V  ∂f
∂ρ b
∂  V6 V5 V8 V7 
 − + −  f b + k  6 − 5 + 8 − 7  b
=k
∂Vn
∂Vn  I 6 I 5 I 8 I 7 
 I 6 I 5 I 8 I 7  ∂Vn
(Eqn. 5.24)……..
 V V V V  ∂f
∂ρ b
∂  V6 V5 V8 V7 
 − + −  f b + k  6 − 5 + 8 − 7  a
=k
∂I n
∂I n  I 6 I 5 I 8 I 7 
 I 6 I 5 I 8 I 7  ∂I n
where the n subscript goes from 1 to 4 for ρa and from 5 to 8 for ρb and k =
π
t . Since
4 ln 2
the geometrical factors fa and fb are iterated after calculation of Qa and Qb respectively,
the uncertainty of fa and fb will be taken as the uncertainty of the left side of the equation:

 ln 2  
 exp
 
f
 Q −1 
f



(Eqn. 5.25)……………… 
 =
ar cosh

2
 Q + 1  ln 2






59
V1
where Q = Qa =
V3
I1
I3
−
V2
−
V4
I2
V5
and Q = Qb =
I4
V7
I5
I7
−
V6
−
V8
I6
for fa and fb respectively.
I8
With these considerations we get:
(Eqn. 5.26)………
∂f a
∂
=
∂Vn ∂Qa
 Qa − 1  ∂Qa ∂f a
∂
 ⋅

,
=
 Qa + 1  ∂Vn ∂I n ∂Qa
 Qa − 1  ∂Qa
 ⋅

 Qa + 1  ∂I n
(Eqn. 5.27)………
∂f b
∂
=
∂Vn ∂Qb
 Qb − 1  ∂Qb ∂f b
∂
 ⋅

,
=
 Qb + 1  ∂Vn ∂I n ∂Qb
 Qb − 1  ∂Qb
 ⋅

 Qb + 1  ∂I n
Once the derivatives shown above are calculated, the results can be substituted back into
equations 5.21-5.24 and 5.17-5.18 to calculate the standard deviation of the resistivity.
Recalling that the average resistivity is given by ρ avg =
ρ a + ρb
, the standard deviation of
2
the average resistivity is calculated as:
(Eqn. 5.28)……………………... σ ρavg =
(
1
σ ρ 2 + σ ρb 2
2 a
)
1/ 2
and the normalized error is calculated as:
(Eqn. 5.29)………………………... Error = σ ρavg / ρ avg
The calculation of the standard deviation of the Hall Effect coefficients follows
the same approach as for the van der Pauw resistivity. Recalling equations 5.13 and 5.14
without averaging currents we get:
(Eqn. 5.30)……………….... RHall −a =
t  V2 V1 V5 V6 
 − + − 
2 B  I 2 I1 I 5 I 6 
(Eqn. 5.31)………………… RHall −b =
t  V4 V3 V7 V8 
 − + − 
2 B  I 4 I 3 I 7 I 8 
60
Comparison of equations 5.19 and 5.20 with 5.30 and 5.31 respectively suggests that the
partial derivatives for the Hall coefficients can obtained by replacing the indices 4 and 5
from ρa by 5 and 6 respectively for RHall-a, and indices 6, 5, 8, 7 from ρb by 4, 3, 7, 8
respectively for RHall-b. The partial derivatives to be evaluated are:
(Eqn. 5.32)…………...…..
∂RHall −a
∂  V2 V1 V5 V6 
 − + − 
=k
∂Vn
∂Vn  I 2 I1 I 5 I 6 
(Eqn. 5.33)…………….....
∂  V2 V1 V5 V6 
∂RHall −a
 − + − 
=k
∂I n
∂I n  I 2 I1 I 5 I 6 
(Eqn. 5.34)…………….....
∂RHall −b
∂  V4 V3 V7 V8 
 − + − 
=k
∂Vn
∂Vn  I 4 I 3 I 7 I 8 
(Eqn. 5.35)…………….....
∂RHall −b
∂  V4 V3 V7 V8 
 − + − 
=k
∂I n
∂I n  I 4 I 3 I 7 I 8 
where k =
t
.
2B
Once the derivatives are calculated the standard deviation of the Hall coefficients will be
given by:
σR
Hall − a
(Eqn. 5.36)…
1/ 2
2
2
2
2
 ∂RHall −a  2  ∂RHall −a  2  ∂RHall −a  2  ∂RHall −a  2 
 σ I6
 σ I5 + 
 σ I1 + 
 σ I 2 + 
+ 

 ∂I1 
 ∂I 2 
 ∂I 6 
 ∂I 5 

σR
Hall −b
(Eqn. 5.37)…
2
2
2
2
  ∂R
 2  ∂RHall −a  2  ∂RHall −a  2  ∂RHall −a  2
Hall −a

σ
 σ +
 σ +
 σ +
= 
  ∂V1  V1  ∂V2  V2  ∂V5  V5  ∂V6  V6

2
2
2
2
  ∂R







R
R
R
∂
∂
∂
2
2
2
Hall
b
Hall
b
Hall
b
Hall
b
−
−
−
−
 σ2
 σ +
 σ +
 σ +
=  
  ∂V3  V3  ∂V4  V4  ∂V7  V7  ∂V8  V8

2

 ∂R
+  Hall −b  σ I23
 ∂I 3 
1/ 2
2
2
2
 ∂RHall −b  2  ∂RHall −b  2  ∂RHall −b  2 
 σ I8
 σ I7 + 
 σ I 4 + 
+ 

 ∂I 4 
 ∂I 8 
 ∂I 7 

The standard deviation of the average Hall coefficient is then given by:
61
(Eqn. 5.38)……………….... σ RHall −avg =
(
1
σ RHall −a 2 + σ RHall −b 2
2
)
1/ 2
and the normalized error is calculated as:
(Eqn. 5.39)…………………….. Error = σ RHall −avg / RHall −avg
Once the standard deviation of the resistivity and the Hall coefficients has been
calculated the standard deviation of the Hall mobility can be calculated invoking equation
5.16. The Hall mobility is given by:
(Eqn. 5.40)…………………………. µ Hall =
RHall −avg
ρ avg
Equation 5.16 then yields:
(Eqn. 5.41)………. σ µHall
 ∂µ Hall
= 
 ∂R
 Hall −avg
2
 ∂µ
 2
 σR
+  Hall
Hall
avg
−
 ∂ρ

 avg

2
 2
 σρ
 avg

And the partial derivatives in equation 5.41 yield:
RHall −avg
1 ∂µ Hall
∂µ Hall
,
=−
=
2
ρ avg
∂RHall −avg ρ avg ∂ρ avg
5.7 Additional measurements on ZnO samples
5.7.1 Thickness Measurements
The thickness of the ZnO samples was measured using a KLA Tencor Alpha Step
IQ profiler. Since the ZnO samples were originally produced without a height step, film
material had to be removed from the surface. The removal of material was achieved by
partially dipping the samples into HCl acid. An Al-ZnO sample prepared for thickness
measurements is shown in figure 5.11.
62
Figure 5.11 Eroded Al-ZnO thin film for thickness measurement.
5.7.2 X-ray diffraction measurements
To investigate the crystallite orientation of the ZnO films, x-ray diffraction (XRD)
measurements were performed. The XRD measurements were performed using a Bruker
D8 Advance diffractometer. The samples were measured in the 15°
≤ 2θ ≤ 50° range
using a Cu Kα , λ =0.154 nm probe beam. The crystallite size and lattice parameter of the
ZnO films were determined by calculating the full width half maximum (FWHM) of the
XRD peaks. To obtain the FWHM of the XRD spectra they were fitted to Gaussian
functions of the form:
 ( x − b) 2
(Eqn. 5.42)……………….…. f ( x) = a exp −
2c 2


 + d

where a, b, c, and d are constants and FWHM = 2 2 ln 2 ⋅ c . The crystallite size was
calculated using the Scherrer equation:
(Eqn. 5.43)…………………………… d =
Kλ
β cos θ
where d is the crystallite size, K is a dimensionless factor close to unity related to the
shape of the crystallites, λ is the X-ray wavelength, β is the (FWHM) in radians and θ is
the Bragg angle. The lattice parameter was calculated using the following equation for
hexagonal closed packed structures:
(Eqn. 5.44)…………………...
1
4  h 2 + hk + k 2  l 2

 + 2
=
d 2 3 
c2
 c
63
where d is the spacing between planes in the lattice, c is the lattice parameter and h, k,
and l are the Miller indices.
5.7.3 Atomic force microscope measurements
The topography of the ZnO samples was investigated using a VEECO CP-II
atomic force microscope. The measurements were performed using the non contact mode
of the equipment.
5.7.4 Energy-dispersive X-ray spectroscopy
To study the chemical composition of the ZnO samples Energy-dispersive X-ray
spectroscopy (EDS) measurements were performed. The samples were analyzed using a
JEOL JSM-6480LV scanning electron microscope. The acceleration voltage used was
20kV.
5.7.5 UV-VIS transmittance measurements
The transmittance of the ZnO samples was investigated using a Perkin-Elmer Lambda 35
UV/VIS Spectrometer, for a wavelength range of 200-1100 nm. The spectrometer is a
double beam model and a blank substrate was used as 100% transmittance reference. The
transmittance data was analyzed using the envelope method [94] to calculate the
thickness of the films using the following equations:
(Eqn. 5.45)……………………... t =
(Eqn. 5.46)…………………. N (λ ) =
λ1λ2
2[n(λ1 )λ2 − n(λ2 )λ1 ]
(ns2 + 1)
(T − Tmin )
+ 2ns max
2
TmaxTmin
64
1
(Eqn. 5.47)………………….. n(λ ) =  N (λ ) + (N (λ ) 2 − ns2 ) 2 


1
2
where n(λ1), n(λ2), are the refractive indices of the film at two adjacent transmittance
maxima or minima with wavelengths λ1, and λ2 respectively, t is the thickness of the film,
ns is the refractive index of the substrate, and Tmax and Tmin are the maximum and
minimum transmittance values at the same wavelength from the envelope curves. The
absorption coefficient of the films was calculated using the relation:
1
(Eqn. 5.48)…………………………. α (λ ) = − ln(T )
t
where t is the thickness of the films and T is the normalized transmittance. The optical
band-gap of the films was estimated assuming a direct transition between valence and
conduction bands with the expression:
(Eqn. 5.49)……………………… αhν = K (hν − E g )
1
2
where K is a constant, h is Planck’s constant, ν is the light’s frequency and Eg is
determined by extrapolating the straight line portion of the spectrum to αhν = 0 [95].
65
Chapter 6 Results
6.1 Energy-dispersive X-ray spectroscopy results
The results of the EDS measurements of the ZnO sol-gel films are shown in
figures 6.1 and 6.2 for undoped and 10% at. intended Al content. The EDS results for the
remaining samples can be found in section A.2 from the appendix. The EDS spectra show
peaks corresponding to silicon Kα at 1.740 keV, oxygen Kα at 0.523 keV, zinc Lα at
1.012 keV, Kα at 8.637 keV and Kβ at 9.57 keV, and aluminum Kα at 1.486 keV. The
silicon detected is due to the glass substrates used, the oxygen is due to the substrates as
well as from the zinc oxide films, the zinc is due to the zinc oxide films and the
aluminum is due to the intentional doping. Table 6.1 shows the aluminum content in at.%
measured in the samples using the K peaks. The measured aluminum content of the
samples seems to suggest the loss of zinc during the annealing process performed at
700°C.
Table 6.1 Measured aluminum atomic percentages.
Intended Al%
Measured Al%
0.00%
-
1.00%
5.13%
2.00%
8.95%
3.00%
7.91%
4.00%
9.15%
5.00% 10.00%
10.83% 13.22%
66
Figure 6.1 Undoped ZnO EDS.
Figure 6.2 10% Al-ZnO EDS.
6.2 X-ray diffraction measurements
Figure 6.3 shows the X-ray diffraction measurements results performed on the AlZnO sol-gel films. The results show a single peak at around 2θ = 35°, shown in more
detail in figure 6.4. The closest match to the peak shown corresponds to (0 0 2) planes of
wurtzitic ZnO typically located between 2θ = 34° and 2θ = 35°. The results suggest that
the films are polycrystalline and preferentially oriented along the c axis. The XRD results
also show a decrease in peak intensity with increasing Al%. This is most likely due to a
decrease in crystallite size induced by the substitution of Zn by Al into the ZnO lattice
[96]. This in turn introduces stress into the ZnO lattice, due to the smaller size of Al3+
(53pm) compared to Zn2+ (74pm), which limits the growth of crystallites. By employing
the Scherrer equation and fitting the XRD spectra to Gaussian functions, the FHWM was
obtained and the average crystallite size of the films calculated with a maximum standard
deviation of 3.59nm (11.65%) corresponding to undoped ZnO, the results of which are
shown in figure 6.5. It can be seen that the crystallite size of the films decreases with
increasing Al content.
67
10%
Intensity (au.)
5%
4%
3%
2%
1%
0%
10
15
20
25
30
35
40
45
50
55
2θ (deg)
Figure 6.3 XRD spectra of Al-ZnO films.
10%
Intensity (a.u.)
5%
4%
3%
2%
1%
0%
32
33
34
35
36
37
38
2θ (deg)
Figure 6.4 Al-ZnO XRD peak location.
68
Size (nm)
Al-ZnO Crystallite Size
29
27
25
23
21
19
17
15
0%
2%
4%
6%
8%
10%
12%
Al%at.
Figure 6.5 Al-ZnO average crystallite size.
Figure 6.4 shows a close up view of the XRD results in the vicinity of the peak for
the ZnO films of varying Al content. A plot of the location of the peak is also shown in
figure 6.6 where the maximum standard deviation is 0.079deg (0.23%) corresponding to
undoped ZnO. The results might suggest an overall trend in the shift of the peak toward
higher angles with increasing Al content. However the standard deviation of the
measurements prohibits drawing any definitive conclusion. The shift of the XRD peak
location, if real, could be due to uniform stress introduced into the lattice by the
incorporation of Al3+, which is smaller than Zn2+, into the lattice [97, 98]. The results
show that the XRD peak location of the 2% Al film coincides with the location of the
peak of the undoped film. A maximum XRD peak shift occurs at 4% Al, while the peaks
of the films containing 1%, 5%, and 10% Al lie close to each other.
69
XRD Peak Location
35.2
2θ (deg)
35.15
35.1
35.05
35
34.95
0%
2%
4%
6%
8%
10%
12%
Al% at.
Figure 6.6 Al-ZnO XRD peak location.
The calculated lattice parameter of the films is shown in figure 6.7 with a
maximum standard deviation of 0.011Angs (0.22%) corresponding to undoped ZnO. The
results show a maximum at 2% Al content and a minimum at aluminum content of 4%.
The results seem to mirror the displacement peak of the XRD results, suggesting that
crystal cells with smaller lattice parameter shift the XRD peak to higher values. The c
lattice parameter values calculated for all films are lower than the lattice parameter value
for pure ZnO crystals of 5.21Å [73] due to the incorporation of Al into the lattice.
However, both the shape of the plot and the standard deviation of the calculations prevent
drawing any trend between lattice parameter size and aluminum content present in the
films.
70
Al-ZnO lattice parameter
5.125
c (nm)
5.12
5.115
5.11
5.105
5.1
5.095
0%
1%
2%
3%
4%
5%
6%
7%
8%
9%
10% 11%
Al% at.
Figure 6.7 Al-ZnO lattice parameter.
6.3 Atomic force microscope measurements
The results of the AFM measurements of the Al-ZnO films are shown in figures
6.8 and 6.9 for undoped and 1% at. Al. The AFM results for the remaining samples can
be found in section A.2 from the appendix. The images show an overall decrease in grain
size as the aluminum content increases. This is possibly due to lattice strain caused by the
incorporation of aluminum preventing further growth of the grains. Another possible
explanation is the segregation of aluminum to the grain boundaries also preventing
further grain growth [99, 100, 101].
Figure 6.8 Undoped ZnO AFM image.
71
Figure 6.9 1%Al-ZnO AFM image.
Table 6.2 summarizes the results for grain size and roughness of the Al-ZnO
films. The results are shown graphically in figures 6.10 and 6.11 for grain size and
roughness respectively. The maximum standard deviation in figures 6.10 and 6.11 is
19.7nm (21%) and 0.92nm (27%) respectively and both corresponding to 4% Al. The
grain size of the films shows a considerable decrease in size between undoped and Al
doped samples. A somewhat linear decrease in grain size is observed between 1% and
3% Al doping level. An increase in grain size is shown between 3% and 4% followed by
another almost linear decrease between 4% and 10% Al doping. The roughness of the
films, shown in figure 6.11, shows a sharp decrease between the undoped and doped
samples and remains almost constant with further increase of Al doping.
Table 6.2 Al-ZnO grain size and roughness.
Al%
0%
1%
2%
3%
4%
5%
10%
Grain Size (nm)
132.76
75.90
71.06
63.60
93.75
94.50
72.05
Rrms (nm)
8.80
3.88
2.70
3.06
3.44
3.21
3.61
72
Al-ZnO Grain Size
200
150
size (nm) 100
50
0
0%
2%
4%
6%
8%
10%
12%
Al% at.
Figure 6.10 Grain size as a function of Al% at. For Al-ZnO.
Al-ZnO Roughness
10
8
6
Rrms (nm)
4
2
0
0%
2%
4%
6%
8%
10%
12%
Al% at.
Figure 6.11 Roughness as a function of Al% at. for Al-ZnO.
6.4 UV-VIS transmittance measurements
The results of transmittance measurements performed on the ZnO films are shown
in figure 6.12. The ranges of transmittance of the films in the visible region are: 85%-
73
99%, 79%-90%, 88%-99%, 84%-97%, 88%-98%, 86%-97%, 86%-97% for samples of
Al content of 0, 1, 2, 3, 4, 5, and 10% at. respectively.
The transmittance spectra show a drop in UV absorption around 380nm for all
samples. Figure 6.13 shows the UV absorption edge of the Al-ZnO films in more detail.
The UV absorption edge moves to lower wavelengths when comparing the spectra of 0%
and 10% Al doped films. For the samples with Al doping of 1% through 5%, the UV
absorption edge lies between those of the 0% and 10% Al doped samples. Organizing the
films in descending order of UV absorption edge wavelength yields: 3%, 2%, 4%, 1%,
5% Al at.
120
100
80
T%
0 pct
1 pct
2 pct
3 pct
4 pct
5 pct
10 pct
60
40
20
0
0
200
400
600
800
1000
1200
Wavelength (nm)
Figure 6.12 Optical transmission spectra of Al-ZnO sol-gel films.
74
100
90
80
70
0 pct
1 pct
2 pct
3 pct
4 pct
5 pct
10 pct
60
T%
50
40
30
20
10
0
370
372
374
376
378
380
382
384
386
388
390
Wavelength (nm)
Figure 6.13 UV absorption edge of Al-ZnO films.
Figure 6.14 shows (αE)2 as a function of energy for the Al-ZnO samples, where α
is the absorption coefficient of the films. The optical band-gap was obtained by
extrapolating the linear part of the plot to (αE)2 = 0 [95]. The calculated optical band-gap
of the Al-ZnO films is shown in table 6.3 and figure 6.15 where the maximum standard
deviation is 0.0047 eV (0.142%) corresponding to 10% at. Al.
Table 6.3 Calculated band-gap of Al-ZnO films.
Al%
0%
1%
2%
3%
4%
5%
10%
Eg (eV)
3.274
3.266
3.281
3.276
3.277
3.278
3.281
75
Figure 6.14 Extrapolation for band-gap calculation.
3.290
3.285
3.280
Eg (eV) 3.275
3.270
3.265
3.260
0%
2%
4%
6%
8%
10%
12%
Al% at.
Figure 6.15 Optical band-gap of Al-ZnO films.
Figure 6.16 shows the transmittance maximum and minimum curves for undoped
ZnO used in the envelope method [94]. The curves for the Al doped samples can be
found in section A.2 of the appendix. The thickness of the films was calculated
76
considering the maxima and minima of the fringes in the spectra far from the absorption
edge and the results are shown in table 6.4 and figure 6.17. The maximum standard
deviation for figure 6.17 is 10.22nm (2.4%) corresponding to 10% Al. The calculated
thickness of the films shows an increase with increasing Al content. This is most likely
due to the increased viscosity of the solutions with increasing Al content, which in turn
resulted in thicker layers during the spin-coating process.
120
100
80
T%
60
40
20
0
350
450
550
650
750
850
950
1050
1150
wavelength (nm)
Figure 6.16 Undoped ZnO transmittance maximum and minimum curves.
Table 6.4 Thickness of Al-ZnO films.
Al%
0%
1%
2%
3%
4%
5%
10%
t (nm)
260.3285
289.2232
326.1762
322.0641
322.5445
357.4382
421.4105
77
Al-ZnO film thickness
450
t (nm)
400
350
300
250
200
0%
2%
4%
6%
8%
10%
12%
Al% at.
Figure 6.17 Thickness of Al-ZnO films.
6.5 Al-ZnO films profiler thickness measurements
In addition to the thickness calculations derived from the transmittance data of the
Al-ZnO films, the thickness of the films was measured using a surface profiler. The
results are shown in table 6.5. A plot of the results is also shown in figure 6.18.
Table 6.5 Al-ZnO profiler measured film thickness.
Al%
0%
1%
2%
3%
4%
5%
10%
t (nm)
212
239
255
317
307
330
411
78
Al-ZnO film thickness (Profiler)
500
t (nm)
400
300
200
100
0
0%
2%
4%
6%
8%
10%
12%
Al% at.
Figure 6.18 Al-ZnO films profiler thickness measurements.
The measured thickness results are in close agreement with the values calculated
from the transmittance data shown in figure 6.17 and table 6.4. It’s important to mention
that the scan length of the measurements was 1000μm, performed over three parallel
scans on the Al-ZnO films. The profiler thickness results were performed to confirm the
calculations obtained from the transmittance data.
6.6 Electrical characterization results
6.6.1 Resistivity measurements
As previously discussed, the electrical resistivity of the Al-ZnO films was
calculated through the van der Pauw technique using a four point probe system. However
Hall Effect measurements could not be performed because the Hall Voltages were of the
same magnitude as the noise present and could not be isolated. It’s important to mention
that the resistivity measurements were performed on already existing Al-ZnO samples as
79
well as on a newly made set. The measurements were performed in a cryogenic chamber
at temperatures in the range of 30K to 320K. The chamber was kept in the dark at a
pressure below 7.6 mTorr.
The resistivity of the undoped ZnO film from the newly prepared set of samples
resulted too high for the equipment to measure and as a consequence the resistivity was
measured at room temperature only. The results of the doped ZnO films with Al content
2%, and 3% at. are shown in figure 6.19. For the sample containing 2% Al, the error in
the measurements is between 5% and 10% for temperatures between 80K and 140K, and
lower than 5% for temperatures above 140K. The error associated with the 3% Al
samples is between 5% and 15% for temperatures between 30K and 130K, and lower
than 2% for temperatures above 130K. As can be seen in figure 6.19, the samples
containing 2% and 3% Al are highly resistive over the whole temperature range and
exhibit a typical semiconductor behavior becoming more resistive as temperature is
decreased.
Al-ZnO Resistivity
5000000
4500000
4000000
ρ (Ohm-cm)
3500000
3000000
ZnO Al 2%
ZnO Al 3%
2500000
2000000
1500000
1000000
500000
0
0
50
100
150
200
250
300
350
Temperature (K)
Figure 6.19 Resistivity of 2% and 3% Al-ZnO films.
80
Al-ZnO Resistivity
80000
70000
ρ (Ohm-cm)
60000
50000
ZnO Al 1%
ZnO Al 4%
ZnO Al 5%
ZnO Al 10%
40000
30000
20000
10000
0
0
50
100
150
200
250
300
350
Temperature (K)
Figure 6.20 Resistivity of 1%, 4%, 5%, and 10% Al-ZnO films.
The resistivity results of the films containing 1%, 4%, 5%, and 10% Al are shown
in figure 6.20. The error in the measurements of the samples containing 1% and 10% Al
is below 1% for the entire temperature range. For the samples containing 4% and 5% Al,
the error lies between 2% and 7% for temperatures between 40K and 70K and below 1%
for all other temperatures. The resistivity of the films shown in figure 6.20, while lower
than the samples containing 2% and 3% Al, is still high through the entire temperature
range when compared to the results obtained by other research teams for Al-ZnO sol-gel
films [96, 99, 101, 102, 103]. However a few instances of high resistivity ZnO sol-gel
films have been reported for films annealed at 700°C containing 1% at. Al [102], and
undoped ZnO annealed at 500°C [104].
The resistivity of the Al-ZnO films as a function of Al content at room
temperature is shown in figure 6.21. As can be seen in the figure, the resistivity of
81
aluminum doped film is significantly lower than the resistivity of the undoped ZnO film.
The minimum resistivity is achieved at 1% Al content, while a maximum is observed at
2% Al content. The conductivity of undoped ZnO is typically attributed to oxygen
vacancies and interstitial zinc providing additional electrons to transport charge [99, 101].
The decrease in resistivity of the ZnO films when doped with Al is attributed to Al3+
substituting for Zn2+ and thus providing additional electrons [96]. Beyond 1% Al doping
the increase in resistivity is attributed the segregation of aluminum in the form of Al2O3
to grain boundaries acting which acts as a trap for free electrons and increases the
electrical barrier a grain boundaries [99, 102].
Al-ZnO resistivity at RT
ρ (Ohm-cm)
8000
6000
4000
2000
0
0%
2%
4%
6%
8%
10%
12%
Al %at.
Figure 6.21 Al-ZnO films resistivity at room temperature.
Table 6.6 Al-ZnO films resistivity at room temperature.
Al %
0%
1%
2%
3%
4%
5%
10%
ρ (Ohm-cm)
6500.00
317.59
2296.74
1842.35
341.07
515.20
459.88
82
6.6.2 Modeling of resistivity results
The resistivity of the Al-ZnO films was analyzed following existing models used
by other research teams [105-110]. The models considered are: Arrhenius [111], Mott
Nearest Neighbor Hopping [112], Mott Variable Range Hopping [20], and ShklovskiiEfros Variable Range Hopping [113]. The Arrhenius equation is an empirical expression
used to analyze the effect of temperature on rates of reactions. It can be used to model the
temperature variation of diffusion coefficients, population of crystal vacancies, creep
rates, and many other thermally-induced processes. Arrhenius resistivity is given by:
 E 
(Eqn. 6.1)………………………… ρ Arr = ρ 0 exp a 
 k BT 
where ρ0 is a constant independent of temperature, Ea is the activation energy, and kB is
the Boltzmann constant.
In disordered semiconductors at low temperatures electrical conduction can be
dominated by hopping mechanisms. In hopping conduction carrier transport occurs due to
electrons hopping between localized states within the band-gap. Transport of electrons
between localized states within the band-gap includes:
(i). Electron hops from a state to another with higher energy. This is thermally assisted
tunneling and depends on temperature.
(ii). Electron hops between states with equal energy. This is a tunneling process non
dependent on temperature.
(iii). Electron hops to a state of lower energy. This is a tunneling process with emission of
a phonon and is independent of temperature.
Nearest Neighbor Hopping (NNH), is characterized by electron hops from
localized states to the nearest localized states with an energy Ehop above the former state.
83
This process is rate limited by the thermal energy of the electrons. At low temperatures
electrons in the localized states cannot be thermally activated to the conduction band. The
energy difference of two neighbor states is usually smaller than the activation energy
required to reach the conduction band.
Nearest Neighbor Hopping (NNH) is given by [121, 122]:
 E hop
(Eqn. 6.2)……………………….. ρ NNH = CT exp
 k BT



where C is a constant independent of temperature, Ehop is the hopping energy, and kB is
the Boltzmann constant.
Mott Variable Range Hopping (VRH), is characterized by electron hops from
localized states to the nearest energy state, but not to the nearest state in terms of
distance. At very low temperatures, the probability of electron thermal activation between
states close in space but far in energy decreases and eventually electrons can hop between
states which are farther in space but closer in energy. As temperature is decreased the
characteristic hopping length increases. Mott Variable Range Hopping is given by:
(Eqn. 6.3)…………………….. ρVRH = CT
1/ 2
  T0 1 / 4 
exp   
 T  


where C is a constant independent of temperature and T0 is a characteristic temperature
given by:
(Eqn. 6.4)………………………….. T0 =
β
k B N ( E F )a 3
where β = 21 [113], N(EF) is the density of localized states (DOLS) around the Fermi
level, and a is the localization radius of charge carriers.
84
At even lower temperatures than Mott VRH, Coulomb interactions between
electrons become important. This results in a constant nonzero value of the density of
localized states around the Fermi level. A modification to Mott VRH was introduced by
Shklovskii and Efros [113], and is given by:
(Eqn. 6.5)……………………. ρVRH = CT
1/ 2
  T0 1 / 2 
exp   
 T  


where C is a constant independent of temperature and T0 is a characteristic temperature
given by:
(Eqn. 6.6)……………………………… T0 =
βe 2
κk B a
where β = 2.8 [113], κ = εε0 = 8.5ε0 [110, 114] is the static dielectric permittivity, e is the
electron charge, and a is the localization radius of charge carriers.
To determine which conduction mechanism applies to the Al-ZnO films, attempts
were made to linearize the data according to the different models: ln( ρ )vs
1
for
T
1
1
 ρ 
 ρ 
ρ 1
Arrhenius, ln vs for NNH, ln 1 / 2 vs 1 / 4 for VRH (Mott), and ln 1 / 2 vs 1 / 2
T  T
T  T
T  T
for VRH (Shklovskii-Efros). It was found that none of the models fitted the data through
the entire temperature range. However the resistivity data could be divided into regions
were different models could be applied. Figure 6.22 shows a plot of ln( ρ )vs
1
for the
T
Al-ZnO films, where the slopes are indicative of regions with different activation
energies.
85
Figure 6.22 ln(ρ) vs 1/T of Al-ZnO films.
The variation of activation energy with temperature suggests regions following
different conduction mechanisms. The data in the high temperature ≥190
range K,
showed Arrhenius resistivity. For the temperature range 190 K < T≤ 130 K, the data was
well fitted to NNH resistivity. Between 130K < T≤ 80 K, Mott VRH best described the
resistivity behavior. Below 80 K, the data was fitted to Shklovskii-Efros VRH. Figures
6.23 through 6.26 show the linearization of the resistivity data according to the models
and temperature ranges just mentioned.
86
Figure 6.23 Al-ZnO films resistivity Arrhenius plot.
Figure 6.24 Al-ZnO films resistivity NNH plot.
87
Figure 6.25 Al-ZnO films resistivity Mott VRH plot.
Figure 6.26 Al-ZnO films resistivity Shklovskii-Efros VRH plot.
88
Table 6.7 Summary of the resistivity analysis of Al-ZnO films.
Al%
1%
2%
3%
4%
5%
10%
Arrhenius
Ea (eV)
0.068
0.093
0.093
0.077
0.077
0.059
NNH
Ehop (eV)
0.044
0.066
0.074
0.056
0.054
0.045
VRH Mott
T0 (K)
4.80x105
3.15x106
1.64x107
2.07x106
1.56x106
6.79x105
VRH Efros
T0 (K)
196.84
4292.92
504.92
667.90
530.28
344.46
Table 6.7 summarizes the results of the resistivity analysis for the Al-ZnO films. The
Arrhenius activation energy (Ea) corresponds to the temperature range 190K ≤ T ≤ 320K.
The Nearest Neighbor Hopping energy (Ehop) applies to the temperature range 130K≤ T
≤ 180K. While the characteristic temperatures (T0) correspond to the temperature ranges
80K ≤ T ≤ 120K and 30K ≤ T ≤ 70K, for VRH Mott and VRH Efros respectively.
Figures 6.27 and 6.28 show the fitted data with the parameters of table 6.7 for the
samples containing 1% and 2% at Al respectively. The fits for the remaining samples can
ρ (Ω-cm)
be found in section A.2 of the appendix.
20000
18000
16000
14000
12000
10000
8000
6000
4000
2000
0
Data
Arr
NNH
VRH Mott
VRH Efros
0
50
100
150
200
250
300
350
T (K)
Figure 6.27 1% Al-ZnO resistivity data and theoretical fits.
89
700000
600000
Data
Arr
NNH
VRH Mott
VRH Efros
ρ (Ω-cm)
500000
400000
300000
200000
100000
0
0
50
100
150
200
250
300
350
T (K)
Figure 6.28 2% Al-ZnO resistivity data and theoretical fits.
The Arrhenius activation energies presented in table 6.7 are comparable to
energies reported by other research teams. Majumder et al. [107] studied Al-ZnO sol-gel
films deposited on sapphire substrates and calculated activation energies in the range of
0.04eV-0.14 eV for films containing 0%, 2%, 3%, 4%, 5%, and 10% at. aluminum at
temperatures between 25°C-175° C. Natsume et al. [110] reported activation energies of
0.02eV-0.04 eV in the temperature range of 250K-300K for undoped ZnO sol-gel films
deposited onto glass substrates annealed between 500°C-575°C. The closest known donor
ionization energy levels of ZnO pertaining to films doped with aluminum are shown in
table 6.8. The donors are shown using the Kröger-Vink notation [115]: i = interstitial site,
Zn = zinc, O = oxygen and V = vacancy. The superscripted terms indicate charges, where
a dot indicates positive charge and a cross indicates zero charge.
90
Table 6.8 Al-ZnO donor levels.
Donor
Al
●●
Zni /Zni●
Zni●/Znix
VO●/VOx
Ionization
Energy (eV)
0.12
0.15
0.05
0.05
Ref.
[116, 23 ]
[117]
[117]
[117]
The Arrhenius activation energies from table 6.7 in the range of 0.059eV0.093eV, might suggest zinc interstitials or oxygen vacancies as the main contributors of
the conductivity for that temperature range. However, given the suspected zinc loss noted
in the EDS results, a large amount of zinc interstitials is unlikely. The hopping energies
from table 6.7 corresponding to NNH, are closer to the lowest energies from table 6.8.
However for that temperature region (NNH) all films, except the ones with 1% and 10%
Al doping are above the oxygen vacancy ionization energy, which might suggest several
conduction mechanisms acting simultaneously giving the appearance of a higher than
expected single hopping energy.
At lower temperatures than where NNH conduction fits the data, Mott VRH was
employed. Natsume et al. [110] reported Mott VRH conduction below 250K but no
details of the fit parameters were provided. P. Sagar et al. [109] studied Al-ZnO sol-gel
films deposited on glass substrates and reported Mott VRH for temperatures in the range
of 150K-400K for samples of 0%-0.8% wt. Al content. They argued that the preexponential factor C in Mott VRH from equation 6.3 is given by:
 (8πk B )1 / 2
(Eqn. 6.7)………………….. C = 
 3eυ
ph



1



 aN ( E ) 
F



1/ 2
where e is the charge of the electron and υph is the phonon energy at the Debye
temperature and has a value of ~1013 s-1 [109, 112, 118, 119]. However, upon close
91
inspection the units of the constant C in equation 6.7 do not match Ω-cm/K1/2, and e2
instead of e is required. Solving equation 6.4 T0 =
β
for N(EF) and inserting
k B N ( E F )a 3
into equation 6.7, yields the localization radius of charge carriers. Finally substituting a
into equation 6.4 yields N(EF). Table 6.9 shows the results for a and N(EF) (DOLS), in
the Mott VRH regime.
Table 6.9 Localization radius and DOLS at the Fermi level for Al-ZnO films.
Al%
1%
2%
3%
4%
5%
10%
a (nm)
112.39
14.71
0.01
3.3
11.15
52.18
N(EF) (eV-1-cm-3)
3.57x1014
2.43x1016
4.87x1025
3.27x1018
1.13x1017
2.53x1015
All the values for the localization radius of the Al-ZnO films presented in table
6.9, except for 1% Al, are smaller than the grain size for each film obtained from AFM
scans. The crystallite size calculated from XRD for the films with 1% and 10% aluminum
content are smaller than their respective calculated localization radius shown in the table
above. S. Bandyopadhyay et al. [109] calculated (DOLS) in the range of 2.75×1016 eV1
cm-3- 1.24×1021 eV-1cm-3 for Al-ZnO sol-gel samples dip coated onto glass substrates
with aluminum content of 0.7%, 1%, 1.7%, 2% at. through a temperature range of 100K200K. Their films were annealed at temperatures between 500°C-600°C.
The average hopping distance and the average hopping energy in Mott VRH are
given by equations 6.8 and 6.9 respectively, the results of which are shown in tables 6.10
and 6.11.
92


9a

(Eqn 6.8)……………………..…. R = 
 8πk B T ⋅ N ( E F ) 
(Eqn 6.9)…………………………... W =
1/ 4
4
3πR N ( E F )
3
Table 6.10 Average hopping distance in Mott VRH regime.
Al%
1%
2%
3%
4%
5%
10%
80 K
357.52
74.86
0.05
15.14
47.61
180.99
R (nm) VRH Mott
90 K
100 K
347.14
338.12
72.68
70.80
0.05
0.05
14.70
14.32
46.23
45.03
175.74
171.17
110 K
330.16
69.13
0.05
13.98
43.97
167.14
120 K
323.05
67.64
0.05
13.68
43.02
163.54
Table 6.11 Average hopping energy in Mott VRH regime.
Al%
1%
2%
3%
4%
5%
10%
80 K
0.026
0.042
0.063
0.037
0.035
0.028
80 K
0.007
kT (eV)
W (eV) VRH Mott
90 K
100 K
0.028
0.031
0.045
0.049
0.069
0.074
0.041
0.044
0.038
0.041
0.031
0.034
90 K
100 K
0.008
0.009
110 K
0.033
0.053
0.080
0.048
0.044
0.036
110 K
0.009
120 K
0.035
0.056
0.085
0.051
0.047
0.038
120 K
0.010
Table 6.12 R/a for Al-ZnO films in Mott VRH regime.
Al%
1%
2%
3%
4%
5%
10%
80 K
3.18
5.09
7.69
4.58
4.27
3.47
90 K
3.09
4.94
7.47
4.45
4.15
3.37
R/a
100 K
3.01
4.81
7.28
4.33
4.04
3.28
110 K
2.94
4.70
7.10
4.23
3.94
3.20
120 K
2.87
4.60
6.95
4.14
3.86
3.13
The average hopping energies presented in table 6.11 are in the range of those
obtained by P. Sagar et al. [109]. According to [109, 120], the conditions for Mott VRH
are: W>kT and R/a>1. For the Al-ZnO films under consideration table 6.11 and 6.12
93
show that those conditions are met. However, for all films except the ones containing 3%,
4%, and 5% Al, the average hopping distances shown in table 6.10 are greater than the
grain sizes observed from AFM measurements. Also, the crystallite size from XRD for
all films except for 4% and 3% Al content are smaller than the hopping distances from
table 6.10.
Table 6.13 Localization radius of the charge carriers on Efros VRH regime.
Al%
1%
2%
3%
4%
5%
10%
a (nm)
351.41
16.11
137.00
103.57
130.44
200.81
The resistivity data corresponding to the Efros VRH regime can be analyzed with
equation 6.6 T0 =
βe 2
, to yield the localization radius of the charge carriers. As can be
κk B a
seen on Table 6.13, the results show that with the exception of the film with 2% Al, the
localization radii for Efros VRH are larger than both the crystallite and grain size of the
films.
A dimensionless activation energy proposed by Zabrodskii and Zinov’eva [123],
is another powerful tool to verify if a VRH conduction mechanism is present in the thin
films under study. The dimensionless activation energy is given by:
(Eqn 6.10)………………………….. W ≡ T −1
d ln ρ
dT −1
Additionally the following equation should be satisfied if VRH Mott or VRH Efros is at
work:
94
(Eqn 6.11)…………………….... ln W = − s ln T + const
A plot of lnW vs. lnT should give a straight line over the temperature range where VRH is
present, with its slope s = ½ for VRH Efros or s = ¼ for VRH Mott. Figure 6.29 shows
the plot just discussed for n-germanium [123], and it can be seen that at low temperatures
the plots have a slope of around -0.5 yielding s = 0.5 indicative of Efros VRH. As it can
be seen in figure 30, the same plots for the Al-ZnO thin films in the VRH temperature
range do not yield straight lines with slopes indicative of VRH in the temperature range
where it was considered in the modeling.
Figure 6.29 Dimensionless activation energy for n-germanium [123].
95
Al-ZnO
2
1.5
1
0.5
1%
2%
3%
4%
5%
10%
0
ln W -0.5
-1
-1.5
-2
-2.5
-3
3
3.5
4
4.5
5
ln T
Figure 6.30 lnW vs. lnT for Al-ZnO thin films.
The results of the dimensionless activation energy of the Al-ZnO thin films,
coupled with the hopping distance results for the Mott VRH regime and the localization
radii for both VRH models considered, might suggest that even though the resistivity data
can be fitted according to Efros and Mott VRH for certain temperature ranges, other
conduction mechanism or a combination of more than one, perhaps influenced by grain
boundaries, could be responsible for the observed resistivity of the Al-ZnO films at low
temperatures.
6.6.5 Hall Effect measurements
To further investigate the conduction mechanisms present in the Al-ZnO films,
considerable efforts to conduct Hall Effect measurements were undertaken. However
given the highly resistive nature of the films, no definitive Hall Voltage measurements
were detected.
96
6.6.4 Resistivity results of previous Al-ZnO samples
As has already been mentioned, the resistivity of previously made Al-ZnO films
was measured prior to the measurements performed on the new set of samples. However,
when the electrical measurements of the previously made Al-ZnO films were performed,
the measurement system was still under development. Some of the concerns with the
initial measurements are that temperature readings were not stable enough, the
temperature sensor was not located close enough to the samples under study and there
was no error calculation of the electrical measurements. Because of these reasons, the
data was not analyzed in terms of conduction mechanisms. Nonetheless, is important to
present those results given how different they are to the latest ones obtained. Two sets of
measurements were carried out. One set of measurements was done using melted indium
contacts while the other set of measurements was performed using aluminum/gold
sputtered contacts. The results of the resistivity measurements with indium contacts are
shown in figures 6.31 through 6.33 and with aluminum/gold sputtered contacts in figures
6.34 through 6.36.
97
Al-ZnO Resistivity
3000
2500
ρ (Ohm-cm)
2000
1500
ZnO Al 0%
1000
500
0
0
50
100
150
200
250
300
350
Temperature (K)
Figure 6.31 Resistivity of undoped ZnO with In contacts.
Al-ZnO Resistivity
70
60
ρ (Ohm-cm)
50
ZnO Al 1%
ZnO Al 2%
ZnO Al 3%
ZnO Al 5%
ZnO Al 10%
40
30
20
10
0
0
50
100
150
200
250
300
350
Temperature (K)
Figure 6.32 Resistivity of Al-ZnO with In contacts.
98
Al-ZnO Resistivity at RT
ρ (Ohm-cm)
30
20
10
0
-10
0%
2%
4%
6%
8%
10%
12%
Temperature (K)
Figure 6.33 Al-ZnO films resistivity at room temperature with In contacts.
Al-ZnO Resistivity
35000
30000
ρ (Ohm-cm)
25000
20000
ZnO Al 0%
15000
10000
5000
0
0
50
100
150
200
250
300
350
Temperature (K)
Figure 6.34 Resistivity of undoped ZnO with Al/Au contacts.
99
Al-ZnO Resistivity
60
50
ρ (Ohm-cm)
40
ZnO Al 1%
ZnO Al 2%
ZnO Al 3%
ZnO Al 5%
ZnO Al 5%
30
20
10
0
0
50
100
150
200
250
300
350
Temperature (K)
Figure 6.35 Resistivity of Al-ZnO with Al/Au contacts.
Al-ZnO Resistivity at RT
ρ (Ohm-cm)
190
140
90
40
-10
0%
2%
4%
6%
8%
10%
12%
Temperature (K)
Figure 6.36 Al-ZnO films resistivity at room temperature with Al/Au contacts.
100
The results obtained using In and Al/Au contacts are much less resistive than the
results of the newly prepared set of ZnO films presented in section 6.6.1. The role of Al
doping decreasing the resistivity of ZnO in comparison to undoped films is still present in
the three sets of measurements. The results with the In contacts show a maximum in
resistivity among the doped samples at 5% at. While for the films with Al/Au contacts the
maximum resistivity among the doped samples occurs at 10%at. The maximum
resistivity among the doped, newly prepared films from section 6.6.1 occurred at 2% at.
The results for the measurements with In and Al/Au contacts are closer to each other than
to the results of the newly prepared set of ZnO films.
101
Chapter 7 Conclusions and recommendations
The purpose of this work was to develop an electrical testing platform for the
electrical characterization of thin films, including resistivity as a function of temperature
(Van der Pauw) and applied magnetic field (Hall Effect), and to perform the electrical
characterization of Al-ZnO sol-gel thin films. As part of the platform a sample holder
was designed and built, computer programming was developed and implemented to
control the measurement equipment, and several other technical issues had to be solved.
The system was tested for previously prepared Al-ZnO films and a new set of samples
was prepared and characterized for comparison. .
EDS measurements performed on the new set of samples revealed higher than
intended percentage of Al compared to Zn, possibly due to the loss of Zn during the
annealing process. XRD results suggest the films are polycrystalline and preferentially
oriented along the c axis. The results also showed a decrease in peak intensity with Al
content most likely caused by a decrease in crystallite size due to the substitution of Zn
by Al into the ZnO lattice or loss of crystal orientation due to an increase in amorphous
phase. The average crystallite size of the films, derived from the XRD results, decreased
with increasing Al content.
AFM measurements showed an overall decrease in grain size with aluminum
content. The decrease in grain size is possibly due to lattice strain caused by the
incorporation of aluminum into the ZnO lattice or due to its segregation to grain
boundaries both preventing further grain growth [99, 100, 101, 102].
The optical band-gap of the films, calculated from UV-VIS transmittance results,
was between 3.27eV and 3.28eV. The band-gap of the Al doped samples was higher than
102
the undoped film with the exception of 1% at. Al. The band-gap showed an increasing
tendency with aluminum content from 3% to 10% Al. The highest band-gap was obtained
for 2% Al (3.281 eV) and the minimum for 1% Al (3.266 eV).
The resistivity of the Al-ZnO films was measured as a function of temperature.
The measurements were performed on previously made films as well as on the newly
prepared set. For both sets of samples, aluminum doping reduced the electrical resistivity,
with a minimum resistivity at 1% Al. The newly made Al-ZnO films were significantly
more resistive (around 141 times more resistive for undoped ZnO at room temperature)
than the first set of samples suggesting variations of resistivity over time or variations
during their preparation.
Resistivity as a function of temperature was modeled employing Arrhenius,
nearest neighbor hopping and variable range hopping conduction, applicable to different
temperature ranges. Even though the models were able to reproduce the data within
certain error margins, the modeling constants obtained yielded parameters with
unreasonable values in some cases. In the case of Arrhenius conduction, the activation
energies obtained did not correspond with any known donor energy levels for Al-ZnO.
Additionally, for the portion of the data modeled using variable range hopping, the
hopping distances calculated exceeded in most cases both the crystallite and grain sizes
obtained from XRD calculations and AFM scans respectively. For these reasons it is
believed that a combination or an altogether different conduction mechanism might be
dominant on the films studied.
There are several possible explanations for the increased resistivity of the newly
prepared set in comparison with the already existing set of films first tested. Through
103
experience in the laboratory it has been noted that the samples need a stabilization period
of around 24 hours before performing electrical measurements. During this period the
samples need to be in vacuum conditions and total darkness. The resistivity of the films
has been observed to increase at vacuum conditions possibly due to gas desorption from
the surface or hydrocarbon adsorption from oil backstreaming into the chamber from the
vacuum pump. It is thus recommended to test incorporating a dry vacuum pump into the
cryogenic system to determine if oil adsorption plays a role in the increased resistivity of
the films with vacuum time. It has also been noted that exposure of ZnO films to light
substantially decreases their resistivity, requiring hours of stabilization time before
performing electrical measurements. Such precautions were not taken for the
measurements on the first set of Al-ZnO films tested with In and Al/Au contacts
separately.
S. Bandyopadhyay et al. [108] reported a continuous decrease in the resistivity at
room temperature of Al-ZnO sol-gel films of 1% and 2% at. over a period of more than
200 days (a decrease of 4 and 12 times the initial resistance for 1% and 2% at. Al doped
films). They attributed these changes to possible desorption of surface H2O or negatively
charged O2 on the surface. In the present case, the resistivity of the newly prepared films
presented in section 6.6.1 was measured about a week after they were prepared while the
resistivity of the first set of samples was measured several months after they were made.
Thus, the evolution of the samples through time might explain the difference in resistivity
between the sets of samples measured as part of this work. However the reason for the
changes in resistivity with time is still a potential subject for continued research.
104
Due to the apparent zinc loss observed in the EDS results, the Al-ZnO films
present higher percentage of Al compared to Zn than intended. This could also explain
the high resistivity of the newly prepared set of films because of the possibility of
segregation of aluminum to grain boundaries, working as an electrical barrier, instead of
being incorporated into the zinc oxide lattice thus increasing electrical conductivity.
In conclusion, an electrical testing platform was developed and used to perform
the electrical characterization of two sets of Al-ZnO sol-gel thin films prepared under
similar conditions. The platform allows to measure resistivity as a function of
temperature using the van der Pauw technique, and resistivity as a function of applied
magnetic field using Hall Effect measurements. The electrical properties of Al-ZnO solgel thin films were studied using this platform which, combined with EDS, XRD, UVVIS and AFM measurements, showed that the incorporation of the aluminum atoms to
the synthesis process increases the electrical conductivity of the samples while
maintaining good optical transmission in the desired optical range.
The XRD and AFM results of the films after the addition of the aluminum atoms
are consistent with their incorporation into the ZnO lattice resulting in smaller crystallite
size and increased electrical conductivity. However, modeling of the electrical response
as a function of temperature shows a complex behavior suggesting that several
conduction
mechanisms might be present. EDS showed that Zn is lost during the
synthesis process.
Due to the small crystallite size, it is possible that Al is also
incorporated to the grain boundaries and film surface affecting the conduction
mechanisms of the films.
105
To better understand the conduction mechanism present in the Al-ZnO sol-gel
films it is recommended to perform X-ray photoelectron spectroscopy (XPS)
measurements as a complement to the electrical characterization. These measurements
can give information about the nature of the bonds of the elements present in the samples.
It is necessary to know whether the zinc present has been well incorporated into the ZnO
lattice or if there is a large amount of it in metallic form. Also XPS measurements would
indicate whether Al has been segregated to the grain boundaries or incorporated into the
ZnO structure.
It is also recommended to perform photoluminescence (PL) measurements on the
Al-ZnO films to detect the energy of the impurity levels. This information is necessary to
compare with the activation energies calculated when modeling the resistivity results and
either validate or discard the conduction mechanisms considered in the analysis.
Since producing Al-ZnO films with good electrical conductivity is desirable for
applications in electronics, variations in the synthesis of the sol-gel films and their effect
on electrical properties should be further studied. Several research teams have studied the
effects of temperature, atmosphere composition, and time variations, on evaporation of
organic residues and annealing, for sol-gel Al-ZnO films [96, 99, 101, 102, 103, 104].
The teams cited have reported changes on electrical and structural properties attributed to
the variation of the synthesis parameters mentioned above.
106
Appendix
A.1 Partial derivatives present in measurement error calculations
Recalling equation 5.17 the standard deviation of ρa is given by:
σρ
(Eqn. A.1)………
a
  ∂ρ
=   a
  ∂V1

 ∂ρ
+  a
 ∂I1
2
2
2
2
 2  ∂ρ a  2  ∂ρ a  2  ∂ρ a  2
 σ V3 + 
 σ V4
 σ V2 + 
 σ V1 + 
 ∂V4 
 ∂V2 

 ∂V3 
2
 2  ∂ρ a
 σ I1 + 
 ∂I 2

2
 2  ∂ρ a
 σ I 2 + 

 ∂I 3
1/ 2
2
2
 2  ∂ρ a  2 
 σ I3 + 
 σ I 4

I
∂
 4


also, recalling equations 5.19, 5.21, and 5.22 given respectively by:
ρa =
V V V V 
t  2 − 1 + 4 − 3  f a
4 ln 2  I 2 I1 I 4 I 3 
π
 V V V V  ∂f
∂ρ a
∂  V2 V1 V4 V3 
 − + −  f a + k  2 − 1 + 4 − 3  a
=k
∂Vn
∂Vn  I 2 I1 I 4 I 3 
 I 2 I1 I 4 I 3  ∂Vn
 V V V V  ∂f
∂ρ a
∂  V2 V1 V4 V3 
 − + −  f a + k  2 − 1 + 4 − 3  a
=k
∂I n
∂I n  I 2 I1 I 4 I 3 
 I 2 I1 I 4 I 3  ∂I n
then the partial derivatives in the equations above are:
(Eqn. A.2)…..
(Eqn. A.3)...
1 ∂  V2 V1 V4 V3  1
∂  V2 V1 V4 V3 
 − + − =
 − + −  = − ,
I1 ∂V2  I 2 I1 I 4 I 3  I 2
∂V1  I 2 I1 I 4 I 3 
1 ∂
∂  V2 V1 V4 V3 
 − + −  = − ,
I 3 ∂V4
∂V3  I 2 I1 I 4 I 3 
(Eqn. A.4)…...
(Eqn. A.5).….
 V2 V1 V4 V3  1
 − +
−  =
 I 2 I1 I 4 I 3  I 4
V
∂  V2 V1 V4 V3  V1 ∂  V2 V1 V4 V3 
 − + −  = − 22
 − + −  = 2 ,
I2
∂I1  I 2 I1 I 4 I 3  I1 ∂I 2  I 2 I1 I 4 I 3 
V
∂  V2 V1 V4 V3  V3 ∂  V2 V1 V4 V3 
 − + −  = 2 ,
 − + −  = − 42
I4
∂I 3  I 2 I1 I 4 I 3  I 3 ∂I 4  I 2 I1 I 4 I 3 
107
∂f
2
1 V V 
∂  Qa − 1  ∂Qa
∂  Qa − 1 
 ⋅
 =


(Eqn. A.6).. a =
=
⋅  3 − 4 
2
∂V1 ∂V1  Qa + 1  ∂Qa  Qa + 1  ∂V1 (Qa + 1) I1  I 3 I 4 
(Eqn. A.7)..
−1
2
1 V V 
∂  Qa − 1  ∂Qa
∂f a
∂  Qa − 1 
 ⋅

 =

=−
⋅  3 − 4 
=
2
∂V2 ∂V2  Qa + 1  ∂Qa  Qa + 1  ∂V2
(Qa + 1) I 2  I 3 I 4 
∂f
∂
(Eqn. A.8)…. a =
∂V3 ∂Qa
 Qa − 1  ∂Qa
2
1 V V  V V 
 ⋅

=−
⋅  1 − 2  ⋅  3 − 4 
2
(Qa + 1) I 3  I1 I 2   I 3 I 4 
 Qa + 1  ∂V3
∂f
∂
(Eqn. A.9).… a =
∂V4 ∂Qa
 Qa − 1  ∂Qa
2
1 V V  V V 
 ⋅

=
⋅  1 − 2  ⋅  3 − 4 
2
 Qa + 1  ∂V4 (Qa + 1) I 4  I1 I 2   I 3 I 4 
∂f
∂
(Eqn. A.10)……... a =
∂I1 ∂Qa
 Qa − 1  ∂Qa
2
V V V 
 ⋅

=−
⋅ 12  3 − 4 
2
(Qa + 1) I1  I 3 I 4 
 Qa + 1  ∂I1
∂f
∂
(Eqn. A.11)……… a =
∂I 2 ∂Qa
(Eqn. A.12)…
∂f a
∂
=
∂I 3 ∂Qa
∂f
∂
(Eqn. A.13)... a =
∂I 4 ∂Qa
 Qa − 1  ∂Qa
2
V V V 
 ⋅

=
⋅ 22  3 − 4 
2
(Qa + 1) I 2  I 3 I 4 
 Qa + 1  ∂I 2
−1
−2
−2
−1
−1
 Qa − 1  ∂Qa
2
V V V  V V 
 ⋅

=
⋅ 32  1 − 2  ⋅  3 − 4 
2
(Qa + 1) I 3  I1 I 2   I 3 I 4 
 Qa + 1  ∂I 3
−2
 Qa − 1  ∂Qa
2
V V V  V V 
 ⋅

=−
⋅ 42  1 − 2  ⋅  3 − 4 
2
(Qa + 1) I 4  I1 I 2   I 3 I 4 
 Qa + 1  ∂I 4
−2
The same calculations in equations are performed for ρb by substituting indices 1, 2, 3, 4,
and a by 5, 6, 7, 8, and b respectively.
Recalling the Hall coefficient given by equation 5.31:
RHall −a =
t  V2 V1 V5 V6 
 − + − 
2 B  I 2 I1 I 5 I 6 
the partial derivatives to be evaluated are:
108
(Eqn. A.14)…………...….
∂RHall −a
∂
=k
∂Vn
∂Vn
(Eqn. A.15)…………….....
 V2 V1 V5 V6 
 − + − 
 I 2 I1 I 5 I 6 
∂RHall −a
∂  V2 V1 V5 V6 
 − + − 
=k
∂I n
∂I n  I 2 I1 I 5 I 6 
Shown explicitly the derivatives become:
(Eqn. A.16)…
1 ∂  V2 V1 V5 V6  1
∂  V2 V1 V5 V6 
 − + −  = − ,
 − + − =
I1 ∂V2  I 2 I1 I 5 I 6  I 2
∂V1  I 2 I1 I 5 I 6 
(Eqn. A.17)…
1 ∂
∂  V2 V1 V5 V6 
 − + −  = − ,
I 6 ∂V5
∂V6  I 2 I1 I 5 I 6 
(Eqn. A.18)…
V
∂  V2 V1 V5 V6  V1 ∂  V2 V1 V5 V6 
 − + −  = 2 ,
 − + −  = − 22
I2
∂I1  I 2 I1 I 5 I 6  I1 ∂I 2  I 2 I1 I 5 I 6 
(Eqn. A.19)…
V
∂  V2 V1 V5 V6  V6 ∂  V2 V1 V5 V6 
 − + −  = 2 ,
 − + −  = − 52
I5
∂I 6  I 2 I1 I 5 I 6  I 6 ∂I 5  I 2 I1 I 5 I 6 
 V2 V1 V5 V6
 − +
−
 I 2 I1 I 5 I 6
 1
 =
 I5
For RHall-b the derivatives shown above can be modified by replacing subscripts 1, 2, 5,
and 6 by 3, 4, 7, and 8 respectively.
A.2 Additional plots from measurements performed on the Al-ZnO films.
120
100
80
T%
60
40
20
0
350
450
550
650
750
850
950
1050
1150
wavelength (nm)
Figure A.1 1% Al-ZnO transmittance maximum and minimum curves.
109
120
100
80
T%
60
40
20
0
350
450
550
650
750
850
950
1050
1150
wavelength (nm)
Figure A.2 2% Al-ZnO transmittance maximum and minimum curves.
120
100
80
T%
60
40
20
0
350
450
550
650
750
850
950
1050
1150
wavelength (nm)
Figure A.3 3% Al-ZnO transmittance maximum and minimum curves.
120
100
80
T%
60
40
20
0
350
450
550
650
750
850
950
1050
1150
wavelength (nm)
Figure A.4 4% Al-ZnO transmittance maximum and minimum curves.
110
120
100
80
T%
60
40
20
0
350
450
550
650
750
850
950
1050
1150
wavelength (nm)
Figure A.5 5% Al-ZnO transmittance maximum and minimum curves.
120
100
80
T%
60
40
20
0
350
450
550
650
750
850
950
1050
1150
wavelength (nm)
Figure A.6 10% Al-ZnO transmittance maximum and minimum curves.
Figure A.7 1% Al-ZnO EDS.
Figure A.8 2% Al-ZnO EDS.
111
Figure A.9 3% Al-ZnO EDS.
Figure A.10 4% Al-ZnO EDS.
Figure A.11 5% Al-ZnO EDS.
112
Figure A.12 2%Al-ZnO AFM image.
Figure A.13 3%Al-ZnO AFM image.
Figure A.14 4%Al-ZnO AFM image.
Figure A.15 5%Al-ZnO AFM image.
113
ρ (Ω-cm)
Figure A.16 10%Al-ZnO AFM image.
5000000
4500000
4000000
3500000
3000000
2500000
2000000
1500000
1000000
500000
0
Data
Arr
NNH
VRH Mott
VRH Efros
0
50
100
150
200
250
300
350
T (K)
Figure A.17 3% Al-ZnO resistivity data and theoretical fits.
114
ρ (Ω-cm)
90000
80000
70000
60000
50000
Data
Arr
NNH
40000
30000
20000
VRH Mott
VRH Efros
10000
0
0
50
100
150
200
250
300
350
T (K)
Figure A.18 4% Al-ZnO resistivity data and theoretical fits.
80000
70000
ρ (Ω-cm)
60000
Data
Arr
50000
40000
NNH
VRH Mott
VRH Efros
30000
20000
10000
0
0
50
100
150
200
250
300
350
T (K)
Figure A.19 5% Al-ZnO resistivity data and theoretical fits.
115
25000
ρ (Ω-cm)
20000
Data
Arr
NNH
15000
10000
VRH Mott
VRH Efros
5000
0
0
50
100
150
200
250
300
350
T (K)
Figure A.20 10% Al-ZnO resistivity data and theoretical fits.
116
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