Linearly dispersing plasmons in monolayer transition metal
dichalcogenides
David Abergel
with Konstantyn Kechedzhi.
Nordita
March 4th 2014
D.S.L. Abergel (Nordita)
Plasmons in TMDs
3/4/14
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Linearly dispersing plasmon
0.05
MoS2
µ = -0.8eV
α = 20meV
40
20
0.03
0
0.02
Re ε(q,ω)
ω (eV)
0.04
Transition metal dichalcogenides with
Zeeman field support additional
plasmon modes.
-20
The extra mode has linear dispersion.
0.01
-40
A(q,ω) (arb. units)
0
8
0
0.1
0.2
0.3
q (nm-1)
0.4
Spectral function has clear peak
corresponding to linear mode.
0.5
Existence of additional plasmon
requires some tuning of parameters.
q = 0.05 nm-1
-1
q = 0.10 nm-1
q = 0.15 nm-1
q = 0.20 nm
6
Possible to induce additional plasmon
via proximity effect from strong
ferromagnetic layer.
4
2
00
See arXiv:1402.5274.
0.01
D.S.L. Abergel (Nordita)
0.02
ω (eV)
0.03
0.04
Plasmons in TMDs
3/4/14
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Transition metal dichalcogenides
SOC splits valence band.
Time reversal symmetry demands
opposite shift in each valley.
Result is valley-spin locked optical
excitations.
K.F. Mak et al., Nat. Nano. 7, 494 (2012).
D.S.L. Abergel (Nordita)
Plasmons in TMDs
3/4/14
3/8
Transition metal dichalcogenides
SOC splits valence band.
Time reversal symmetry demands
opposite shift in each valley.
Result is valley-spin locked optical
excitations.
K.F. Mak et al., Nat. Nano. 7, 494 (2012).
We add Zeeman field:
H = ξatσ · k +
∆
σz − 1
σz − ξsλ
+ sα
2
2
And calculate the plasmon spectrum.
K′
D.S.L. Abergel (Nordita)
Plasmons in TMDs
K
3/4/14
3/8
Theoretical details
In random phase approximation:
χ0 (q, ω) =
(q, ω) = 1 − V (q)χ0 (q, ω)
and
χ0 (q, ω) =
Z
fξsβk − fξsβk+q
d2 k X
Fξsβ (k, k + q)
.
4π 2
ω + iη + Eξsβk − Eξsβk+q
ξ,s,β
In limit q/kF 1 we can write an analytical expression for χ0 :
i
X h ζs I qvω
+ iJ qvω
χ0 (q, ω) ≈ −
s
Fs
Fs
where vF s is band-dependent Fermi velocity, ζs = kF s /(2πvF s ), and
a
K↑
I(a) = Θ(1 − a) − Θ(a − 1) √
−1
2
µ
a −1
a
J (a) = −Θ(1 − a) √
K′ ↓
1 − a2
D.S.L. Abergel (Nordita)
Plasmons in TMDs
3/4/14
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Variation with chemical potential
0.05
D=
e2
(kF ↑ vF ↑ + kF ↓ vF ↓ )
2κ
D.S.L. Abergel (Nordita)
-0.7
0.03
0.02
E (eV)
with
0.04
ω (eV)
First plasmon has dispersion
√
ω1 (q) = D q
0.01
-0.8
00
Plasmons in TMDs
0.1
0.2
q (nm-1)
0.3
K’↓
k
K↑
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Variation with chemical potential
(a) 0.05
D=
e2
(kF ↑ vF ↑ + kF ↓ vF ↓ )
2κ
-0.7
0.03
0.02
E (eV)
with
0.04
ω (eV)
First plasmon has dispersion
√
ω1 (q) = D q
(b)
0.01
-0.8
00
0.1
0.2
q (nm-1)
0.3
K’↓
k
K↑
Second plasmon has dispersion
ω2 (q) ≈ vF ↓ q
Re εRPA
100
0
-100
-200
-300
0
D.S.L. Abergel (Nordita)
Plasmons in TMDs
0.02
0.04
0.06
q (nm-1)
0.08
3/4/14
0.1
5/8
Variation with chemical potential
(a) 0.05
D=
e2
(kF ↑ vF ↑ + kF ↓ vF ↓ )
2κ
-0.7
0.03
0.02
E (eV)
with
0.04
ω (eV)
First plasmon has dispersion
√
ω1 (q) = D q
(b)
0.01
-0.8
00
0.1
0.2
q (nm-1)
0.3
K’↓
k
K↑
Second plasmon has dispersion
ω2 (q) ≈ vF ↓ q
Re εRPA
100
0
-100
-200
-300
0
D.S.L. Abergel (Nordita)
Plasmons in TMDs
0.02
0.04
0.06
q (nm-1)
0.08
3/4/14
0.1
5/8
Variation with chemical potential
0.05
D=
e2
(kF ↑ vF ↑ + kF ↓ vF ↓ )
2κ
-0.7
0.03
0.02
E (eV)
with
0.04
ω (eV)
First plasmon has dispersion
√
ω1 (q) = D q
0.01
-0.8
00
0.1
0.2
q (nm-1)
0.3
K’↓
k
K↑
Second plasmon has dispersion
ω2 (q) ≈ vF ↓ q
Re εRPA
100
0
-100
-200
-300
0
D.S.L. Abergel (Nordita)
Plasmons in TMDs
0.02
0.04
0.06
q (nm-1)
0.08
3/4/14
0.1
5/8
Variation with chemical potential
0.05
D=
e2
(kF ↑ vF ↑ + kF ↓ vF ↓ )
2κ
-0.7
0.03
0.02
E (eV)
with
0.04
ω (eV)
First plasmon has dispersion
√
ω1 (q) = D q
0.01
-0.8
00
0.1
0.2
q (nm-1)
0.3
K’↓
k
K↑
Second plasmon has dispersion
ω2 (q) ≈ vF ↓ q
Re εRPA
100
0
-100
-200
-300
0
D.S.L. Abergel (Nordita)
Plasmons in TMDs
0.02
0.04
0.06
q (nm-1)
0.08
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0.1
5/8
Spectral function
A(q, ω) = −ImχRPA (q, ω)
0.05
MoS2
µ = -0.8eV
α = 20meV
40
20
0.03
0
0.02
Re ε(q,ω)
ω (eV)
0.04
-20
A(q,ω) (arb. units)
The spectral function is
8
q = 0.05 nm-1
q = 0.10 nm-1
q = 0.15 nm-1
q = 0.20 nm-1
6
4
2
0.01
-40
0
0
0.1
0.2
0.3
q (nm-1)
0.4
0.5
00
0.01
0.02
ω (eV)
0.03
0.04
Vertical lines mark edge of two continua.
Quasiparticle peak corresponding to linear plasmon clear between the two continua.
Lineshape not Lorentzian
Peak narrows and increases in height above quasiparticle background as q increases.
D.S.L. Abergel (Nordita)
Plasmons in TMDs
3/4/14
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Prospects for observation
Zeeman field required is of the order of meV.
Magnetic field to produce this Zeeman is ∼ 20T if applied
perpendicular to the TMD plane.
I
But this will necessarily cause Landau levels to form, which have to
be taken into account in the calculation.
Rotating the field in plane means requiring stronger field ∼ 40T
due to anisotropy of TMD g-factor.
Another approach is to use proximity effect of ferromagnetic
material.
I
I
Proximity-induced Zeeman field has been shown in Bi2 Se3 /EuS
heterostructures.
Strained, ultra thin LaSrMnO films have Curie temperature
∼ 500K.
D.S.L. Abergel (Nordita)
Plasmons in TMDs
3/4/14
7/8
Summary
TMDs in a Zeeman field show additional plasmon mode.
I
Linear dispersion
Appropriate tuning of parameters needed.
Experimental technology probably within reach to see such
plasmons in heterostructures with ferromagnetic layers.
See arXiv:1402.5274.
0.05
MoS2
µ = -0.8eV
α = 20meV
40
20
0.03
0
0.02
Re ε(q,ω)
ω (eV)
0.04
-20
0.01
-40
0
D.S.L. Abergel (Nordita)
0
0.1
0.2
0.3
q (nm-1)
Plasmons in TMDs
0.4
0.5
3/4/14
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