Magneto transport in 2D electron gases in semiconductor

Motivation
Introduction
Model
Results
Conclusions
Magneto transport in 2D electron gases in
semiconductor heterostructures
A. Kunold1
1 Area
de Física Teórica y Materia Condensada
Departamento de Ciencias Básicas
UAM-A
ICCMSE, Athens-Greece, April, 2014
Kunold, Torres
Magneto transport in 2D electron gases...
Motivation
Introduction
Model
Results
Conclusions
Outline
1
Motivation
2
Introduction
3
Model
4
Results
5
Conclusions
Kunold, Torres
Magneto transport in 2D electron gases...
Motivation
Introduction
Model
Results
Conclusions
Outline
1
Motivation
2
Introduction
3
Model
4
Results
5
Conclusions
Kunold, Torres
Magneto transport in 2D electron gases...
Motivation
Introduction
Model
Results
Conclusions
Outline
1
Motivation
2
Introduction
3
Model
4
Results
5
Conclusions
Kunold, Torres
Magneto transport in 2D electron gases...
Motivation
Introduction
Model
Results
Conclusions
Outline
1
Motivation
2
Introduction
3
Model
4
Results
5
Conclusions
Kunold, Torres
Magneto transport in 2D electron gases...
Motivation
Introduction
Model
Results
Conclusions
Outline
1
Motivation
2
Introduction
3
Model
4
Results
5
Conclusions
Kunold, Torres
Magneto transport in 2D electron gases...
Motivation
Introduction
Model
Results
Conclusions
Motivation
Magneto transport in Hall-like systems give rise to
interesting phenomena: MIRO, HIRO, MHIRO, PIRO,
Oscillations under a periodic modulation.
The method I am presenting today is successful in
explaining their most important features.
Most of this systems can be modelled by the Hamiltonian
H + Vi + Vp where H0 is the Hamiltonian of a charged
particle in uniform magnetic and and arbitrarily time
dependent electric fields, Vi is an impurity potential and Vp
is a periodical modulation.
The electric field might be very large in some cases and it
can not be treated as a perturbation.
MIRO, microwave’s electric field ER ≈ 200V /m
HIRO, Hall field EH ≈ 135V /m
Kunold, Torres
Magneto transport in 2D electron gases...
Motivation
Introduction
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Motivation
We would like to use this method to study a periodical
modulation to study Bloch Oscillations and THz radiation.
Many applications.
Bloch Oscillations are Ocillations of electrons in a
periodical potential subjected to a constant force (electric
field).
Bloch Oscillations can not be observed in regular
semiconductors because the period of Bloch Oscillations is
two small compared to the period of scattering events.
Kunold, Torres
Magneto transport in 2D electron gases...
Motivation
Introduction
Model
Results
Conclusions
Hall effect
Experimental setup of
the Hall effect
Quantized Hall
resistence
Rxy =
1 ~
,
j e2
j = 1, 2, . . .
Shubinikov-de Haas
oscillations
Kunold, Torres
Magneto transport in 2D electron gases...
Motivation
Introduction
Model
Results
Conclusions
Hall effect
Experimental setup of
the Hall effect
Quantized Hall
resistence
Rxy =
1 ~
,
j e2
j = 1, 2, . . .
Shubinikov-de Haas
oscillations
Kunold, Torres
Magneto transport in 2D electron gases...
Motivation
Introduction
Model
Results
Conclusions
MIRO
Microwave
Induced
Resistence Oscillations (MIRO)
arise in GaAs − AlGaAs
heterostructures under microwave radiation.
Kunold, Torres
Magneto transport in 2D electron gases...
Motivation
Introduction
Model
Results
Conclusions
MIRO
MIRO are oscillations of
Rxx as a function of the
magnetic field.
MIRO can occur in a B
weaker than the onset of
the Shubinikov-de Haas
oscillations (SdH) where
Landau levels are not
yet resolved.
Hall resistence shows
classical linear
dependence on B.
Kunold, Torres
M.
A. Zudov, et al. Phys. Rev. B 64,
201311(R) (2001) R. G. Mani, et
al., Nature 420, 646 (2002)
Magneto transport in 2D electron gases...
Motivation
Introduction
Model
Results
Conclusions
MIRO
Their period is controlled
by the ratio of the
microwave frequency to
the electron cyclotron
frequency
=
~ω
,
~ωc
where ωc = eB/m∗ and
m∗ is the effective mass
of the conduction band
electrons.
MIRO are periodical in
1/B
Kunold, Torres
Magneto transport in 2D electron gases...
Motivation
Introduction
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Results
Conclusions
MIRO
Their period is controlled
by the ratio of the
microwave frequency to
the electron cyclotron
frequency
=
~ω
,
~ωc
where ωc = eB/m∗ and
m∗ is the effective mass
of the conduction band
electrons.
MIRO are periodical in
1/B
Kunold, Torres
Magneto transport in 2D electron gases...
Motivation
Introduction
Model
Results
Conclusions
Theoretical models for MIRO
Some of the models (Ryzhii,Durst,Vavilov) suggest that the
main cause of MIRO is photon-assisted scattering by
impurities or disorder. Alternative explanations (Dmitriev)
propose that MIRO arise from MW induced distribution
function fluctuations. 1
1
V. I. Ryzhii, R. Suris, J Phys. Cond. Matt. 15, 6855 (2003), A. C. Durst, S.
Sachdev, N. Read, S. M. Girvin, Phys. Rev. Lett. 91 086803 (2003), J. Shi, X.
C. Xie, Phys. Rev. Lett. 91, 086801 (2003), X. L. lei, S. Y. Liu, Phys. Rev.
Lett. 91, 226805 (2003), M. G. Vavilov, I. L. Aleiner, Phys. Rev. B 69, 035303
(2004), I. A. Dmitriev, A. D. Mirlin, D. G. Polyakov, Phys. Rev. Lett. 91,
226802 (2003); M. Torres, A. Kunold, Phys. Rev. B 71 115313 (2005) I. A.
Dmitriev, M. G. Vavilov, I. L. Aleiner, A. D. Mirlin, D. G. Polyakov,
cond-mat/0310668 (2003); cond-mat/0409590 (2004). A. V. Andreev, I. L.
Aleiner, A. J. Millis, Phys. Rev. Lett. 91, 056803 (2003)
Kunold, Torres
Magneto transport in 2D electron gases...
Motivation
Introduction
Model
Results
Conclusions
MIRO
A.
Kunold, M. Torres,
Phys. Rev. B 71,
115313
(2005),
phys. stat. sol. (a)
204, 467-471 (2007)
Kunold, Torres
Magneto transport in 2D electron gases...
Motivation
Introduction
Model
Results
Conclusions
MIRO
6
ω1 /2π = 31GHz
ω1 /2π = 47GHz
5
4
ρxx [Ohm]
3
2
1
0
-1
α ≈ 1/5
-2
-3
0
0.02
0.04
0.06
0.08
0.1
A.
Kunold, M. Torres,
Phys. Rev. B 71,
115313
(2005),
phys. stat. sol. (a)
204, 467-471 (2007)
B [T]
Kunold, Torres
Magneto transport in 2D electron gases...
Motivation
Introduction
Model
Results
Conclusions
MIRO
6
ω1 /2π = 31GHz
ω1 /2π = 47GHz
5
4
ρxx [Ohm]
3
2
1
0
-1
α ≈ 1/5
-2
-3
0
2
4
6
8
10
12
A.
Kunold, M. Torres,
Phys. Rev. B 71,
115313
(2005),
phys. stat. sol. (a)
204, 467-471 (2007)
ω/ωc
Kunold, Torres
Magneto transport in 2D electron gases...
Motivation
Introduction
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Results
Conclusions
HIRO
Hall field-induced
resistence oscillations
(HIRO) are observed
under constant
longitudinal current.
Under constant
longitudinal current
EH =
1 Ix B
ne wb
Kunold, Torres
Magneto transport in 2D electron gases...
Motivation
Introduction
Model
Results
Conclusions
HIRO
Hall field-induced
resistence oscillations
(HIRO) are observed
under constant
longitudinal current.
Under constant
longitudinal current
EH =
1 Ix B
ne wb
Kunold, Torres
Magneto transport in 2D electron gases...
Motivation
Introduction
Model
Results
Conclusions
HIRO
The period is controled
by
~ωH
=
~ωc
where
ωH = γ (2π/n)1/2
Ix
ew
Kunold, Torres
W.
Zhang and H.-S. Chiang and M.
A. Zudov and L. N. Pfeiffer and
K. W. West, Phys. Rev. B 75
041304(R) (2007)
Magneto transport in 2D electron gases...
Motivation
Introduction
Model
Results
Conclusions
HIRO
The period is controled
by
~ωH
=
~ωc
where
ωH = γ (2π/n)1/2
Ix
ew
Kunold, Torres
W.
Zhang and H.-S. Chiang and M.
A. Zudov and L. N. Pfeiffer and
K. W. West, Phys. Rev. B 75
041304(R) (2007)
Magneto transport in 2D electron gases...
Motivation
Introduction
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Results
Conclusions
HIRO
A. Kunold, M.Torres,
Physica
B
403
(2008) 3803-3808,
Physica
B
425
(2013) 78?82
Kunold, Torres
Magneto transport in 2D electron gases...
Motivation
Introduction
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Results
Conclusions
HIRO
76 5 4
ǫ = ωH /ωc
3
2
1
rxx [Ω]
2
Jx = 0.8A/m
(a)
1.5
0.0
0.5
B [kG]
1.0
0.3
1.5
A. Kunold, M.Torres,
Physica
B
403
(2008) 3803-3808,
Physica
B
425
(2013) 78?82
∆rxx [Ω]
Jx = 0.8A/m
(b)
-0.3
1
2
3
4
5
6
7
8
ǫ = ωH /ωc
Kunold, Torres
Magneto transport in 2D electron gases...
Motivation
Introduction
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HIRO
A. Kunold, M.Torres,
Physica
B
403
(2008) 3803-3808,
Physica
B
425
(2013) 78?82
Kunold, Torres
Magneto transport in 2D electron gases...
Motivation
Introduction
Model
Results
Conclusions
HIRO
B=416G
∆rxx [Ω]
0.5
0
-0.5
-1
(a)
1
2
3
4
5
6
ǫ = ωH /ωc
A. Kunold, M.Torres,
Physica
B
403
(2008) 3803-3808,
Physica
B
425
(2013) 78?82
B=732G
∆rxx [Ω]
0.5
0
-0.5
-1
(b)
1
2
3
ǫ = ωH /ωc
Kunold, Torres
Magneto transport in 2D electron gases...
Motivation
Introduction
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Conclusions
MIRO+HIRO=MHIRO
MHIRO are observed
under the combined
effect of microwave and
hall field excitations.
W. Zhang, M. A. Zudov, L. N.
Pfeiffer, and K. W. West Phys.
Rev. Lett 98, 106804 (2007)
Kunold, Torres
Magneto transport in 2D electron gases...
Motivation
Introduction
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Conclusions
MIRO+HIRO=MHIRO
MHIRO are observed
under the combined
effect of microwave and
hall field excitations.
W. Zhang, M. A. Zudov, L. N.
Pfeiffer, and K. W. West Phys.
Rev. Lett 98, 106804 (2007)
Kunold, Torres
Magneto transport in 2D electron gases...
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MIRO+Periodical Modulation
Another version of MIRO
is observed in
modulated 2D electron
gas
Yuan ZQ, Yang CL, Du RR,
et al., Phys.
Rev.
B
74 075313 (2006), Torres M,
Kunold A, Journal of PhysicsCondensed Matter 18 40294045 (2006)
Kunold, Torres
Magneto transport in 2D electron gases...
Motivation
Introduction
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MIRO+Periodical Modulation
Another version of MIRO
is observed in
modulated 2D electron
gas
Yuan ZQ, Yang CL, Du RR,
et al., Phys.
Rev.
B
74 075313 (2006), Torres M,
Kunold A, Journal of PhysicsCondensed Matter 18 40294045 (2006)
Kunold, Torres
Magneto transport in 2D electron gases...
Motivation
Introduction
Model
Results
Conclusions
Periodical modulation in strong electric fields
a = 300nm and we have over a million antiquantum dots.
Kunold, Torres
Magneto transport in 2D electron gases...
Motivation
Introduction
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Hamiltonian
The Hamiltonian for electrons in a 2D modulation potential,
perpendicular magnetic field and inplane electric field in the
effective mass approximation
H = H0 + V
where
H0 =
1
(p + eA)2 + eE (t) · x
∗
2m
Kunold, Torres
Magneto transport in 2D electron gases...
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Impurity potential and modulation potential
The perturbative potential is given by
V = Vi + Vm
where
Vi (r) =
ni Z
X
i
d 2q
(2π)2
V (q) exp [iq · (r − r i )]
is the potential for impurities and
2π
2π
Vm (r) = V0 cos
x + λ cos
y
a
a
is the potential for the modulation.
Kunold, Torres
Magneto transport in 2D electron gases...
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Current density
The current density is given by
Z
1 ˆ
ˆ + eA
hJi = e dtTr ρˆ (t)
p
2m∗
where ρ is the density matrix. It follows the von Neumann
equation
h
i
∂
ˆ0 + V
ˆ , ρˆ
ˆt ρˆ = H
ρˆ = p
∂t
h
i
h
i
ˆ0 + V
ˆ −p
ˆ −p
ˆt , ρˆ = 0
ˆt , ρˆ = H
H
i
Kunold, Torres
Magneto transport in 2D electron gases...
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Solution Method
Von Neumann’s equation
ˆ −p
ˆt by a series
The idea is to "reduce" the Floquet operator H
of unitary operators U1 , U2 , ... Un such that
ˆ H
ˆ −p
ˆt
U
ˆ = U
ˆ n ...U
ˆ 2U
ˆ1
U
ˆ † = −p
ˆt
U
Von Neumann’s equation transforms into
h
i
h i
ˆ H
ˆ −p
ˆ† = U
ˆ H
ˆ −p
ˆ †, U
ˆ ρˆ (t) U
ˆ†
ˆt , ρˆ (t) U
ˆt U
U
h
i
ˆ ρˆ (t) U
ˆ† = 0
ˆt , U
= p
Kunold, Torres
Magneto transport in 2D electron gases...
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Solution Method
Density matrix
h
i
ˆ ρˆ (t) U
ˆ † is basically the time derivative of U
ˆ ρˆ (t) U
ˆ † thus
ˆt , U
p
ˆ = ρˆ (0)
ˆ ρˆ (t) U
ˆ † = Cnt
U
and
ˆ † ρˆ (0) U
ˆ
ρˆ (t) = U
Kunold, Torres
Magneto transport in 2D electron gases...
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Solution Method
Thermodynamical averages
Now we are ready to calculate the current densities as
1 ˆ
ˆ + eA
hJi = e dtTr ρˆ (t)
p
2m∗
Z
1 †
ˆ
ˆ
ˆ
ˆ + eA
= e dtTr U ρˆ (0) U
p
2m∗
Z
1 ˆ
ˆ†
ˆ U
ˆ
= e dtTr ρˆ (0)
U
p
+
e
A
2m∗
ˆ p
ˆ † than U
ˆ † ρˆ (0) U.
ˆ
ˆ U
ˆ + eA
It is a lot easier to calculate U
Z
Kunold, Torres
Magneto transport in 2D electron gases...
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Solution Method
Unitary transformations
The next two transformations separate the Hamiltonian into the
x and y parts
ˆ 1 = exp (iαxˆ yˆ )
U
ˆ 1p
ˆ† = p
ˆt U
ˆt
U
1
ˆ 1 xˆ U
ˆ † = xˆ
U
1
ˆ 1 yˆ U
ˆ † = yˆ
U
1
ˆ
ˆ
ˆx U1† = p
ˆx − ~αyˆ
U1 p
†
ˆ 1p
ˆ = p
ˆy U
ˆy − ~αxˆ
U
1
eB
α =
2~
Kunold, Torres
Magneto transport in 2D electron gases...
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Solution Method
Unitary transformations
ˆ 2 = exp (iβ p
ˆx p
ˆy )
U
†
ˆ 2p
ˆ = p
ˆt U
ˆt
U
2
ˆ 2 xˆ U
ˆ † = xˆ + ~β p
ˆy
U
2
†
ˆ 2 yˆ U
ˆ = yˆ + ~β p
ˆx
U
2
ˆ 2p
ˆ† = p
ˆx U
ˆx
U
2
ˆ 2p
ˆ† = p
ˆy U
ˆy
U
2
β = −
Kunold, Torres
1
eB~
Magneto transport in 2D electron gases...
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Solution Method
Unitary transformations
The next unitary transformation corresponds to displacements
in space, momentum and energy U2 = U2t U2x U2y where
i
U3t = exp S (t) ,
~
i
i
ˆx ,
U3x = exp πx (t) xˆ exp λx (t) p
~
~
i
i
ˆ
ˆ
U3y = exp πy (t) y exp λy (t) py ,
~
~
with time dependent transformation parameters S (t), λx (t),
πx (t), λy (t) and πy (t) → classical equations an electron.
Kunold, Torres
Magneto transport in 2D electron gases...
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Solution Method
Unitary transformations
This unitary operator yields the following transformation rules
U3 xˆ U3† = xˆ + λx (t) ,
ˆx U3† = p
ˆx − πx (t) ,
U3 p
U3 yˆ U3† = yˆ + λy (t) ,
ˆy U3† = p
ˆy − πy (t) ,
U3 p
ˆt U3† = p
ˆt + S˙ − λ˙ x πx − λ˙ y πy
U3 p
ˆx + π˙ y yˆ + λ˙ y p
ˆy .
+π˙ x xˆ + λ˙ x p
It removes linear terms leaving just the Hamiltonian of an
oscillator.
Kunold, Torres
Magneto transport in 2D electron gases...
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Solution method
Unitary Transformations
The next transformation removes the quantum oscillator, it is
the quantum version of Arnold’s transformation
δ
1 2
2
ˆ
U4x = exp i
∆xˆ + p
2~
∆ x
and yields the following transformation rules
†
U4x xˆ U4x
†
ˆx U4x
U4x p
†
ˆt U4x
U4x p
1
ˆx sin δ,
p
∆
ˆx cos δ − ∆xˆ sin δ,
= p
= xˆ cos δ +
ˆt
= p
Kunold, Torres
Magneto transport in 2D electron gases...
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Solution method
The last transformation removes the perturbation potential To
the second order in time dependent perturbation theory, the
time evolution operator is given by
i
U5 (t) = 1 +
~
Z
t
ds1 V (s1 )
−∞
2 Z t Z s1
i
+
ds2 V (s1 ) V (s2 )
~
−∞ ∞
Kunold, Torres
Magneto transport in 2D electron gases...
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Current density
We plug in the time evolution operator
Z
1 †
ˆ
ˆ + eA
hJi = e dtTr U (t) ρ (0) U (t)
p
2m∗
hJi = hJic + hJii + hJim
hJic : Drude part
hJii : Impurity part
hJim : Modulation part
Kunold, Torres
Magneto transport in 2D electron gases...
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Classical part of the density current
The x and y components of the classical part of the density
current are
"
#
δ4
nea
Jx (Ex , Ey ) = αKx 1 +
, α=
,
2
2π
Kx2 + η 2
s√
"
#
2V
δ4
2π
Jy (Ex , Ey ) = αKy 1 −
δ=
2
a
m
K 2 + η2
y
2πτ e Ey + ωc τ Ex
2πτ e Ex − ωc τ Ey
, Ky =
,
Kx =
2
am 1 + (ωc τ )
am 1 + (ωc τ )2
Kunold, Torres
Magneto transport in 2D electron gases...
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Results
We suppose the lateral current
"
Jy = αKy 1 −
#
δ4
Ky2 + η 2
2
=0
This gives us a relation between Ex and Ey . This leads to two
possible results
Ky = 0, &
Ky2 + η 2
Kunold, Torres
2
− δ4 = 0
Magneto transport in 2D electron gases...
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Results
How do we choose from these possibilities?
We define a Lyapunov function to test the stability of the
equilibriums
Z
U=
3
(Jx dKx + Jy dKy )
50 mT
100 mT
300 mT
2.5
Σxx HWL
60
2
Uy
50
1.5
40
30
1
20
0.5
10
0.05
0
-2
-1.5
-1
-0.5
0
0.5
1
1.5
0.10
0.15
0.20
0.25
2
Ky (T Hz)
Kunold, Torres
Magneto transport in 2D electron gases...
0.30
BHTL
Motivation
Introduction
Model
Results
Conclusions
Results
How do we choose from these possibilities?
We define a Lyapunov function to test the stability of the
equilibriums
Z
U=
(Jx dKx + Jy dKy )
T =1K
1.2
U
1
0.8
0.6
T =6K
0.4
-1.5
-1
-0.5
0
0.5
1
1.5
Ky (THz)
Kunold, Torres
Magneto transport in 2D electron gases...
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Results
6
5
4
σxx (Ω)
8V
9V
10V
11V
12V
13V
14V
15V
17V
0.01V
0.1V
1V
2V
3V
4V
5V
6V
7V
3
2
1
0
0.2
0.25
0.3
0.35
0.4
B(T )
Kunold, Torres
Magneto transport in 2D electron gases...
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Results
18
16
14
Ex(kV /m)
12
10
8
6
4
2
0
-2
0.22 0.24 0.26 0.28
0.3
0.32 0.34 0.36 0.38
0.4
B(T )
Kunold, Torres
Magneto transport in 2D electron gases...
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Conclusions
We have studied the magneto transport in a 2D periodical
potential under strong bias electric field.
Our model takes into account exactly the magnetic and
electric field and considers the modulation potential to
second order in time dependent perturbation theory.
The observed peak structure in the differential conductivity
seems to originate in a spontaneous symmetry breaking.
Bloch oscillations?
Kunold, Torres
Magneto transport in 2D electron gases...
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Acknowledgements
CONACyT
Departamento de Ciencias Básicas de la
UAM-Azcapotzalco.
Kunold, Torres
Magneto transport in 2D electron gases...
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Collaborators
Wei Pan (SNL)
Manuel Torres (IF-UNAM)
Francisco López (UAM)
Kunold, Torres
Magneto transport in 2D electron gases...
Motivation
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Thank you
THANK YOU
Kunold, Torres
Magneto transport in 2D electron gases...

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