Coulomb Drag of Electrons for Bicyclists.
A. G. Yashenkin
PNPI and SPbSU
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Outline.
•
Constituents of the Theory.
•
The Essence of the Coulomb Drag Effect.
•
Detailing the Effect.
•
Extensions and Exotics.
•
Conclusion.
1. Constituents of the Theory: Diagramatics.
(r, t)
(r’, t’)
(k, ω )
Electron Propagator
ψH (r, t) = eiHt ψ(r) e−iHt ,
(Green Function)
k+q
q
k
2πe2
vq = q
Coulomb Interaction
Impurities
k
q
H = H0 + Hint
k’
k’−q
k+q
†
G(r − r 0, t − t0) = −i h T̂ ( ψH (r, t)ψH
(r 0, t0) ) i
→
Uq (ω) ∝ (q + κ)−1 ∝ e−κq /r
∝ (ω−ωpl)−1
ω > vF q
Gaussian averaging over random configurations
k’
k’−q
1. Constituents of the Theory: Diffusion and Interference.
Bloch Theorem:
ε0(k) → ε(k)
Classical Drude formula:
n e2 τ
σD = m
Why it works?
l – Mean Free Path
Einstein relation:
σD ∝ e 2 D ν
λF l
Ordinary Path →
h |A1 + A2|2 itraj ≈ 2|A1|2
Classical Drude formula:
D – Diffusion coefficient
Closed Path →
h |A1 + A2|2 itraj ≈ 4|A1|2
Quantum Interference
1. Constituents of the Theory (Cont’d)
Diffuson (diffusion)
D ∝ Dq21−iω
Cooperon (interference)
C∝
Diffusive vs Ballistic:
1
DQ2−iω+τφ−1
τϕ (lϕ) – Dephasing Time (Length)
l lϕ
FTD: connecting susceptibilities and correlators. Kubo formula.
m = χ̂ h,
Ĉmm = hm, mi :
J = σ̂ E,
χ̂ ∝ Ĉmm
ĈJJ = hJx, Jxi :
J
J
σ̂ ∝ ĈJJ
1. Constituents of the Theory (Cont’d)
Diffuson (diffusion)
Cooperon (interference)
D ∝ Dq21−iω
C∝
1
DQ2−iω+τφ−1
τϕ (lϕ) – Dephasing Time (Length)
Diffusive vs Ballistic:
l lϕ
FTD: connecting susceptibilities and correlators. Kubo formula.
m = χ̂ h,
Ĉmm = hm, mi :
J = σ̂ E,
χ̂ ∝ Ĉmm
ĈJJ = hJx, Jxi :
ε+ω
J
p
ε
p
J
σ̂ ∝ ĈJJ
1. Constituents of the Theory (Cont’d)
Diffuson (diffusion)
D ∝ Dq21−iω
Cooperon (interference)
C∝
Diffusive vs Ballistic:
1
DQ2−iω+τφ−1
τϕ (lϕ) – Dephasing Time (Length)
l lϕ
FTD: connecting susceptibilities and correlators. Kubo formula.
m = χ̂ h,
Ĉmm = hm, mi :
J = σ̂ E,
χ̂ ∝ Ĉmm
ĈJJ = hJx, Jxi :
J
J
σ̂ ∝ ĈJJ
2. The Essence of the Coulomb Drag Effect.
Pogrebinskii ’77, Gramila et al. ’91, Eisenstein ’92
ρ21 = (V2/J1)J2=0
Transresistivity
Conventional theory
T=0
n
F
r
F
F
sliding
roughnesses
friction
ρ21 = −σ21/detσ̂
dragging
n
LT
T>0
r
thermal density fluctuations
probabilities
ρ 21
T / EF &
independent
( T / EF )2
2. The Essence of the CDE (Cont’d).
ε k = vF ( k − kF ) +
n
holes
+ (2m)−1 ( k − kF )2 + ...
Particle−hole
electrons
J2
asymmetry
J1
σ 21 = i ω −1 < J2 , J1 >ω −>0
2. The Essence of the CDE (Cont’d).
ε k = vF ( k − kF ) +
n
holes
+ (2m)−1 ( k − kF )2 + ...
Particle−hole
electrons
J2
asymmetry
J1
σ 21 = i ω −1 < J2 , J1 >ω −>0
2. The Essence of the CDE (Cont’d).
ε k = vF ( k − kF ) +
n
holes
+ (2m)−1 ( k − kF )2 + ...
Particle−hole
electrons
J2
asymmetry
J1
σ 21 = i ω −1 < J2 , J1 >ω −>0
2. The Essence of the CDE (Cont’d).
ε k = vF ( k − kF ) +
n
holes
+ (2m)−1 ( k − kF )2 + ...
Particle−hole
electrons
asymmetry
J2
J1
2
σ 21 = i ω −1 < J2 , J1 >ω −>0
1
U21(q, ω) =
πe2
q
κ2
sinh qd
κd 1
2
2
1 π ζ(3) T
1
ρ21 = 2
2 d4
e
16
EF
κ 2 kF
2. The Essence of the CDE: Diff ’s and Sim’s.
•
Phonon Drag: phonons drag electrons when the temperature gradient
is applied. FD manifests itself in thermopower.
•
”Mixed” transport coefficients. Examples:
Hall coefficient < JxJy > is small in (ωcτ );
Thermopower < JkJ > is small in (T /EF ) due to PHA;
Transconductivity < J2J1 > is small in (T /EF )2 due to PHA.
•
Interaction induced corrections to < JxJx > .
Aslamasov-Larkin contribution is smaller than Altshuler-Aronov one.
However, now it is not a correction but the entire effect!
3. Detailing the Effect: Diffusion.
Typically, (ωτ < 1, ql < 1, lϕ < l) → T τ < 1. For CDE qd < 1.
A
A
R
A
R
U21(q, ω) =
ρ21 =
πe2 q
κ 2 sinh qd
1 π2
e2 12
T
EF
2
R
−iω + Dq 2
Dq 2
1
(κd kF l)
ln
2
T0
T
2
.
Thus, diffision dominates the drag only at vanishingly small T .
3. Detailing ...: Spin CDE and Correlated Impurities.
spin
Coulomb drag
interlayer
transresistivity
m
ρ ud= ( Eu / Jd )J =0
u
t
Dc
ρ 21= ( E2 / J1 )J =0
1
double quantum wells
no spin flip!
Ds ( 1 − ρ ud σ D )
spin ( charge )
diffusion
Gornyi et al. ’99: experimental set up of correlated impurities in DQW
Impurity potential correlations. Quantum coherence.
the same impurity
potential for spin
up and spin down
different potentials
in layers
3. Detailing ...: Spin CDE and Correlated ... (Cont’d).
(a)
R
R
A
A
(b)
Anomalous Maki-Thomson processes.
ρ21 =
ρ21
1 π4
ln(T τg )
e2 24 (kF d)4 (κl)
,
2
Low-T enhancement.
τg−1 < T < τtr
1 π4
(T τg )2
= 2
,
e 6 (kF d)4 (κl)2
T < τg−1
τg measures the mismatch of random potentials in two layers.
3. Detailing the Effect: Interaction and Temperature.
•
The interlayer interaction is effectively static and screened. Fermiliquid result T 2.
•
The interaction is retarded (dynamic) but overdamped. It excites
the particle-hole pairs from the continuum. Linear-in-T behavior.
•
T 3 temperature dependence is due to interlayer plasmon modes.
•
High-temperature regime wherein the density fluctuations could be
treated hydrodynamically. Transresisitvity decays as T −1.
4. Extensions and Exotics.
CDE and SCDE between nanowires (including chiral LL’s )
CDE of massless Dirac fermions (graphene and TI)
Interlayer macroscopic coherence (excitonic insulator, QH states)
CDE of composite fermions (FQHE)
Giant mesoscopic fluctuations of transconductance
DE of disordered bosons
Residual CDE (third-order)
CDE in low and intermediate magnetic fields
SCDE in presence of polarization and density mismatch
Phonon-mediated CDE
CDE between different fermionic systems
CDE in presence of tunneling briges
Conclusions.
•
We overview the Coulomb Drag Effect in 2DEG.
•
Over the past two decades, the Coulomb Drag Effect became
a powerful tool for studying the intrinsic properties of interacting disordered low-dimensional electron systems.
Narozhny, Levchenko, Rev. Mod. Phys. ’16
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