Zero resistance states from surface acoustic
waves
Malcolm Kennett (SFU)
Collaborators:
Nigel Cooper (University of Cambridge)
John Robinson (Lancaster University)
Vladimir Fal’ko (Lancaster University)
References:
M.K., J.R., N.C., & V.F., Phys. Rev. B 71, 195240 (2005).
J.R., M.K., N.C., & V.F., Phys. Rev. Lett. 93, 036804 (2004).
Overview
• Background and Motivations
Two dimensional electron gases
Microwave induced zero resistance states
• Classical Effects
Weiss oscillations (spatial periodicity)
Weiss-like oscillations (spatial and temporal periodicity)
• Quantum Effects
Theories for zero resistance states
Quantum effects due to SAWs
• Role of disorder potential
• Summary and Future Work
Two dimensional electron gases
• Confining electrons in a plane allows control
of charges with gates – basis of Si
electronics industry
• Electrons confined at interface
– quasi 2 dimensional
Heterostructure
Dopant layer
2DEG
• Separation between donor atoms and 2DEG
reduces impurity scattering, increases
mobility (modulation doping)
• Past 25 years have seen incredible
increases in electron mobility in these
structures – about 104 higher than bulk
GaAs
• Has led to new physics:
Quantum Hall Effect,
Fractional Quantum Hall Effect,
and Zero Resistance States
Donor
atoms
Conduction Band
+
+++
+
Electron
Energy
2DEG
Fermi energy
Valence Band
GaAs
AlGaAs
GaAs
Transport in 2DEGs
• Integer (von Klitzing, 1980) and
Fractional Quantum Hall Effects
(Tsui, Stormer, and Gossard, 1982)
• Transport has proved a very useful
tool for investigating new physics
in 2DEGs
• QHE – DC transport tells us about
ground states of 2D electron
systems
• Ground state is not the only way
to characterize a state of matter
– collective excitations
• Can DC transport be used as a
probe for excited states?
Probing with finite frequency
electric fields
Surface Acoustic Waves
• Surface acoustic waves (SAWs) in GaAs generate Electric field through
piezoelectric coupling
• SAW Electric field screened by 2DEG
x
• Frequency ω = sq, s ' 3000 ms−1
• Potential φ(x, t) = φ0 sin(qx − ωt) =⇒ E = E0 cos(qx − ωt)x̂
• σxx (ω, q) present in acoustic attenuation and velocity shift
Geometric commensurability
• Study of metals using acoustic attenuation in a magnetic field (Pippard, 1957)
(ω ' 0 =⇒ Geometrical commensurability )
2Rc
λ
Geometric Commensurability
between SAW wavelength (λ) and
electron cyclotron radius (Rc):
nλ = 2Rc
Rc = vF /ωc
• In 2DEGs :
1) Weiss oscillations
2) Measurement of cyclotron radius of composite fermions close to ν = 1/2
(Willett et al, 1993) (ω ∼ 2π × 6 GHz, λ ∼ 0.6 µm)
Microwave-induced magnetoresistance
oscillations
Rxx (Ohm)
H
B
ω
E
0.3
ω
Vxx
4
SdH
2
dark
0
0.0
0.5
10
Rxx (Ohm)
ε=1
(a)
Iac
6
3/2
8
6
T=1K
f = 57 GHz
1.0
1.5
B (kG)
Rxy (kOhm)
3 5/2 2
0.2
0.1
0.0
2.0
(b) T = 1 K
80 100 120
B (G)
4
35 GHz
2
57 GHz
90 GHz
0
0.0 0.2 0.4 0.6 0.8 1.0 1.2
ωc/ω
Samples:
• Very high quality 2DEGs
• Density ne ∼ 3 × 1011 cm−2
• Mobility µ ∼ 2.5 × 107 cm2 V−1 s−1
=⇒ ωcτ 1
• High filling fraction ν ∼ 50 − 100
Observations
Low magnetic fields compared to QHE • Sequence of peaks depending on
Dimensionless parameters ωτ and ωωc • Zero resistance states!
Non-linear effect from ac Electric field manifested in dc resistance
ω
ωc
Motivation
• Interesting effects when high
quality 2DEGs are illuminated with
microwaves – magnetoresistance
oscillations and zero resistance
states
(Mani et al.; Zudov et al.)
• Surface acoustic waves (SAWs) give a way to probe finite
frequency and wavenumber excitations in 2DEGs by creating
modulated electric fields
• SAWs allow the study of high quality 2DEGs on small
lengthscales (λ lmf p) and SAW-induced non-equilibrium states
Typically, λSAW aB , λF , and vF s = ω/q
• Study the non-linear effects of SAWs on DC resistivity of a
2DEG – leads to prediction of SAW induced zero resistance
states
Introduction
The problem: (Classically)
Electrons in 2D moving in a magnetic
field in the presence of a spatially and
temporally varying electric field, with
scattering off disorder
Electric field: E(ω, q)
• Microwaves (ω 6= 0, |q| = 0)
• Patterned gates (ω = 0, q 6= 0)
• Surface acoustic waves (SAWs)
(ω 6= 0, q 6= 0), s = ωq
Disorder
B
Cyclotron
orbits
2DEG
E ( ω, q)
• In absence of modulated electric field: Drude resistivity tensor
ρij = ρ0
1 ωc τ
−ωcτ 1
,
m∗
ρ0 = 2
,
e ne τ
ωc =
eB
m∗
Classical Effects –Weiss oscillations
• Non-linear correction to DC resistivity due to a static periodic potential:
Can be understood semi-classically (ν 1, disorder broadened Landau
levels leads to roughly flat density of states): Beenakker (1991)
Geometric commensurability effect on
resistance
Dimensionless parameter qRc
Channeling of electron orbits
Purely Classical Effect – E × B drift
Guiding centre drift due to SAWs
• Lorentz force: F = −e(E + v × B), with E = Eωq cos(qx − ωt)x̂, B = Bẑ
• Guiding centre co-ordinates (X, Y )
( x, y )
vy
X = x− ,
ωc
vx
Y = y+ ,
ωc
Rc
Guiding
centre ( X, Y )
eB
ωc = ∗
m
• Unperturbed cyclotron motion:
x0(t) = X0 + Rc cos (ωc t − ψ) ,
Ẋ = 0,
Ẏ ' −
y0(t) = Y0 + Rc sin (ωc t − ψ)
Eωq
cos(qx0(t) − ωt)
B
Resistance Correction due to SAWs
• Resistivity change: SAW induced drift of electron cyclotron orbit – enhanced
diffusion
Z ∞
Z
short−range scattering ∞ −t/τ Ẏ (t)Ẏ (0) dt
Ẏ (t)Ẏ (0) X ,ψ dt
δDyy =
−→
e
0
0
0
• Need to average over phase (ψ) and location (X0) of orbit
Final Result:
∞
δρxx δDyy vF2 ω 2 τ 2 2 X
Jp(qRc )2
=
=
E
,
2τ 2
ρ0
D0
4s2
1
+
(ω
−
pω
)
c
p=−∞
δρxy = 0,
δρyy = 0
and when qRc 1 ,
∞
pπ
δρxx
vF2 ω 2 τ 2 2 X cos2(qRc + 2 − π4 )
'
E
,
ρ0
4πqRcs2 p=−∞ 1 + (ω − pωc )2τ 2
with dimensionless SAW amplitude E =
SAW
eascr Eωq
(Thomas
F
Fermi screening)
As with Weiss oscillations, resistivity correction is always positive
Numerical Results
• Comparison of SAW induced magnetoresistance with Weiss oscillations
0.6
Static modulation
Dynamic modulation
0.5
δρxx/ρ0
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
ωc/ω
1
1.2
1.4
• Parameters: ql = 600, (l = vF τ ), E = 0.005 – weak modulation
• Static: ωτ = 0
• Dynamic: ωτ = 20
Kinetic equation
• Distribution function
f (t, x, φ, ) = fT () +
X
ωq
e
−iωt+iqx
X
m
fωq
()eimφ
m
• Kinetic equation in relaxation-time approximation (short-range potential)
f − f 0 f 0 − fT ( − F (t, x))
L̂f (t, x, φ, ) = −
,
−
τ
τ
in
∂
∂
eE
∂
∂
L̂ =
+ v cos φ + ωc −
sin φ
+ ev cos φ
∂t
∂x
p
∂φ
∂
• Elastic scattering time τ (momentum relaxation ),
Inelastic scattering time τin (energy relaxation ); τin τ
Solution:
i) Take Fourier harmonics
iii) Use solution for fωq to find
correction to f00
ii) Solve for fωq
iv) f00 gives dc current, and hence
DC resistance
Simplifies to previous result in limit qRc 1
Quantum Effects: theories for microwave-induced
resistance oscillations
Large ωcτ 1 =⇒ Landau level quantization =⇒ modulated density of states
Durst et al. (2003): disorder-induced scattering
n+3
−∆x
+∆x
n+2
ω
n+1
ωc
n
εn = nωc + eEdcx
V
• Qualitative agreement with experiments
Dmitriev et al. (2003): oscillatory structure in distribution function
• Stationary kinetic equation:
2σ
D
(ω) X
f () − fT ()
E
γ̃(
±
ω)
[f
(
±
ω)
−
f
()]
=
2ω 2 γ 2 ±
τin
• Assume weak DOS modulations
γ̃() =
2π
1 − Γ cos
~ωc
Γ = 2e
− ωcπτq
γ
1
• Implies oscillations in elastic
scattering rate
τ −1 () = τ −1
γ̃()
γ
• However, there are microwave-induced corrections to dc conductivity
Larger than Durst mechanism by τin/τq 1
Andreev et al. (2003): Negative resistance → zero resistance
ρd (j 2)
j2
Ex
j2
0
j
−j0
0
jx
Instability – only time independent
state with negative linear response
dissipative resistivity ρd is characterized by a current with magnitude j0 almost everywhere, with ρd(j02) = 0
Explains how zero resistance arises from negative resistance
Current domains
Vx
I
Ly
E0
j0
Vy
d
E0
j0
V2
Quantum Effects from SAW
• For low frequencies ω ωc (i.e. high B fields) balance equation is
2
0
γ̃
σ
f00
2 ωq
0
|Eωq |
∂ (f + fT ) =
2 00
γ
γ̃
γ
τin
±ωq
X
σωq is longitudinal component of classical non-local conductivity and τ in−1 τ −1
0
0
• Solution of the equation has smooth and oscillatory parts f 00
= f00
+δf 0 ().
Pick out oscillatory part and average over energy:
Isotropic magneto-oscillations for 1/τ . ω ωc
3
2.5
2
ρxx/ρ0
1.5
1
Geometric commensurability
oscillations superposed on
frequency oscillations
0.5
0
-0.5
ωτ = 7, vF/s = 63
ωτ = 60, vF/s = 84
-1
-1.5
0
0.2 0.4 0.6 0.8
1 1.2 1.4 1.6 1.8
ωc/ω
2
Comparison of classical and quantum contributions
• Quantum correction to resistance is isotropic , classical is anisotropic
• Quantum correction =⇒ possibility of zero resistance states
• Final results for τ −1 . ω ωc are
2
δρxx
v
δσyy
τ
in
≈
= 2J02 (qRc ) E 2 F2 −
(2πΓν)2 ,
ρ0
σ0
s
τ
δρyy
τin
δσxx
≈
= −2J02 (qRc) E 2 (2πΓν)2 ,
ρ0
σ0
τ
• Control parameter is
η=
2πΓνs
vF
r
τin
τ
• Regimes
η 1: Correction in direction of SAW propagation, anisotropic, positive
η ' 1: Correction in direction perpendicular to SAW, anisotropic, negative
η 1: Correction is isotropic, negative
Long-range disorder – Classical Resistance
Correction
• Disorder potential in high quality 2DEGs is from coulomb potentials of donors,
hence long-range
• Leads to small-angle scattering and diffusive motion of electrons, instead of
isotropic scattering off a short-range potential
– Magnetic field dependent scattering time τ∗ = 2τ /(qRc)2
Classical resistivity correction
Long-range potential
Short-range potential
1.4
1.4
ωτ = 60, vF/s = 84
1.35
1.3
1.3
*
ωcτ* < 1
ωc τ > 1
1.25
ρxx/ρ0
1.25
ρxx/ρ0
ωτ = 60, vF/s = 84
1.35
1.2
1.15
1.2
1.15
1.1
1.1
1.05
1.05
1
0
2
4
6
ωc/ω
8
10
1
0
0.2
0.4
0.6
0.8
ωc/ω
1
1.2
1.4
Long range disorder – Zero resistance states
• Long-range disorder does not change:
1) Prediction of Zero Resistance States , isotropic magnetoresistance
2) Regime of η where quantum effects dominate classical effects
Quantum correction to resistivity
5
ωτ = 7, vF/s = 63
ωτ = 60, vF/s = 84
4
Geometrically commensurate
and temporally commensurate
ZRS occupy separate magnetic
field ranges
qRc/ωτ << 1 for ωτ = 7, vF/s = 63
3
ρxx/ρ0
2
1
0
-1
qRc/ωτ << 1 for ωτ = 60, vF/s = 84
-2
-3
0
2
4
6
ωc/ω
8
10
12
Microwaves again
• Theory: Dmitriev et al.
ZRS depend on the sense of
circular polarization
Other mechanisms ?????
• Experiment: Smet et al.
Sense of circular polarization
of microwaves doesn’t really
affect ZRS
Work still required to understand the microwave induced ZRS
Future Directions
Probing with SAW + microwaves
Microwaves
(Ω)
SAW modulation
(ω, q )
Electrons move in
cyclotron orbits in
response to
magnetic field
Two dimensional electron gas
• Simultaneous probe of energy and wavenumber dependence of collective
excitations
• Means to probe higher energy collective excitations at ν = 12
– Theory and experiment indicate Fermi Liquid at low energies,
– What about higher energies?
– Experimentally most information comes from inelastic light
scattering, doesn’t allow for systematic tuning of ω and k
• Need theoretical results to compare with experiments
Summary
• New experimental regimes of SAW frequency and 2DEG quality
require new theoretical investigations
• Classical results can be understood in terms of guiding centre
drift. Combined geometrical and temporal resonances (qRc and
ω/ωc)
• Found quantum corrections arising from oscillatory non-equilibrium
distribution function (leading effect for τin τ )
• Predict SAW-induced zero resistance states when ω ωc,
which show geometric oscillations as a function of qRc
• Prediction of ZRS is robust for different types of disorder, but
nature of states changes
• Future Work – probing collective excitations of 2DEGs with both
SAWs and microwaves