domains in zero resistance state - oist groups

Nonequilibrium phenomena in complex quantum systems,

Okinawa Institute of Science & Technology, April 26, 2012

Domains in zero resistance state

Ivan Dmitriev

I. A. Dmitriev, M. Khodas, A. D. Mirlin, D. G. Polyakov,work in progress

I. A. Dmitriev, S. I. Dorozhkin, A. D. Mirlin ’09, ’11,related photovoltaic phenomena

Ivan Dmitriev (KIT& Ioffe Institute) Domains in zero resistance state 26.04.2012 1 / 30

j · E < 0 ⇒ Domains

• Maxwell equations:

Negative absolute conductivity of homogeneous state, j · E < 0

⇒ Electrical instability: Translational symmetry spontaneously broken

⇒ Electric domains ≡ Spontaneously formed inhomogeneous state

ZRS ?=? Static domains Hall bar Corbino

Andreev,Aleiner,Millis’03


Microwave Induced Resistance Oscillations (MIRO)and associated Zero Resistance States (ZRS)

Experiments: MIRO & ZRS

MIRO: Zudov,Du,Simmons,Reno’97-’01,Ye,Tsui,Simmons,Wendt,Vawter,Reno’01,..

ZRS: Mani,Smet,von Klitzing,Narayanamurti,Johnson,Umansky’02,Zudov,Du,Pfeiffer,West’03, ..

Theory: MIRO & ZRS

MIRO: δR ∝ − sin(2πω/ωc) → ZRS

Degenerate 2DEG, classically strong B,Landau quantization, suppressed SdHO

τ−1 � ωc, τ−1q < T � εF

ZRS: electric instability → domains


j · E < 0 ⇒ Domains

• Maxwell equations:

Negative absolute conductivity of homogeneous state, j · E < 0

⇒ Electrical instability: Translational symmetry spontaneously broken

⇒ Electric domains ≡ Spontaneously formed inhomogeneous state

ZRS ?=? Static domains Hall bar Corbino



Outline

Model: Key ingredients

• Nonlinear conductivity

• Violated Einstein relation & nonequilibrium screening length

• Nonlocal electrostatics

Stable field & density distribution in the domain state

• Domain state and its stability, Lyapunov functional

• Finite-size effects:Negative conductance and size-dependent instability threshold

• Relation to other models: power-law vs. exponential decay

Photovoltaics near transition to the domain state

• Spontaneous vs. explicit symmetry breaking

• Divergence of the linear response at the instability threshold


Overview of the model: Key ingredients

Domain state ≡ A stable solution of coupled Continuity and Poisson

equations for the 2D charge density ρ and the in-plane electric field E

compatible with the boundary conditions.

Continuity equation: ∂tρ+∇ · j = 0 (Static domains ⇒ ∂t = 0)

j = σ(E)E−D∇ρ → conductivity σ(0) < 0, diffusion coefficient D > 0

Poisson equation: −ε∆φ = 4πρδ(z) ⇒ ρ(x) = −

∫εdx ′

2π2

E(x ′)

x− x ′

Resulting self-consistent nonlinear integral equation

∂xj=0 ⇒ j=σ(E)E(x) +D∂x

∫εdx ′

2π2

E(x ′)

x− x ′


Microscopic mechanisms of MIRO

Displacement mechanism

Average 〈∆X〉 ⇔ 〈ν(ε)ν(ε+ω+ eE∆X)〉

⇒ The sign of 〈∆X〉 oscillates with ω/ωc

⇒ jph ∝ P∂ω〈ν(ε)ν(ε+ω)〉E at E→ 0

Ryzhii’69, Ryzhii,Suris,Shchamkhalova’86, Vavilov,Aleiner’04,

Durst,Sachdev,Read,Girvin’03, ID,Mirlin,Polyakov’07,

Khodas,Vavilov’08, ID,Khodas,Mirlin,Polyakov,Vavilov’09,..

Inelastic mechanism

j ∝ −E∫dεν2(ε)∂εf(ε) - The sign of ∂εf(ε)

at DOS maxima oscillates with ω/ωc

⇒ jph ∝ (τin/τ)P∂ω〈ν2(ε)ν(ε+ω)〉E

ID,Mirlin,Polyakov’03,’04,’07, Dorozhkin’03

ID,Vavilov,Aleiner,Mirlin,Polyakov’05, Khodas,Vavilov’08,

ID,Khodas,Mirlin,Polyakov,Vavilov’09

At sufficiently strong P both mechanisms yield negative σ, j · E < 0

In both mechanisms D is positive and nearly P-independent


Nonlinear conductivity of homogeneous state

Homogeneous 2DEG: I-V in minima

I-V characteristics j = Eσ(E):

σ(E<Ec) < 0, σ(E>Ec) > 0 cf. ↑↑

ID,Vavilov,Aleiner,Mirlin,Polyakov’05

Microscopic theory: σ(E)&D

• σ(0)<0, σ(E) grows with E

Critical field Ec: σ(Ec)=0

• D=const(E)>0

• Dimensionless σ ≡ σ/|σ(0)|changes between -1 and 0


Violated Einstein relation & Nonequilibrium screening length

λ =εD

2π|σ(0)|- nonequilibrium screening length (e2χD 6= σ)

• Equilibrium ⇒ Einstein relation e2χD = σ

=⇒ λ→ λTF =ε h2

2me2(Thomas–Fermi) & j = σ∂xη/e (single term!)

η - electrochemical potential, χ =m/π h2 - static compressibility

• Shine microwaves =⇒ e2χD 6= σ & j = σE−D∂xρ (irreducible to 1 term)

j = σ(E)E(x) +εD

2π∂x

∫dx ′

π

E(x ′)

x− x ′→ j

|σ(0)|=σ(E)E(x)+λ∂x

∫dx′

π

E(x′)

x− x′

Diverges when |σ(0)|→ 0 !


Electrostatics: Nonlocal 2D and auxiliary 3D model

Quasicorbino geometry: ∞ stripe of width L

2D charge density: ρ(x) = ene(x) − en+

Boundary conditions: ρ(0) = ρ(L) = 0

Hall current jy(x) = E(x)ρ−1H

Plain contacts: (i) Do not create nonuniform ρ(x) ⇒ Spontaneous symmetry breaking

(ii) Image charges ⇒ Periodicity: ρ(x) = ρ(x+ 2L) ⇒ Analytic solution feasible

Alternative 3D model

3D media with σ<0&D>0 and 1D modulation, E(x) and ρ(x)

The Poisson equation ∂xE = (4π/ε)ρ ⇒ Local relation utilized in all existing works on ZRS

A. Auerbach et al., Phys. Rev. Lett. 94, 196801 (2005)I. G. Finkler and B. I. Halperin, Phys. Rev. B 79, 085315 (2009)J. Alicea et al., Phys. Rev. B 71, 235322 (2005)A. F. Volkov and V. V. Pavlovskii, Phys. Rev. B 69, 125305 (2004)


Electrostatics: Nonlocal 2D and auxiliary 3D model

Quasicorbino geometry: ∞ stripe of width L

2D charge density: ρ(x) = ene(x) − en+


Hall current jy(x) = E(x)ρ−1H

2D electrostatics & Hilbert transform

• The Poisson equation −ε∆φ = 4πρδ(z) ⇒ 2πρ(x) = εEz|z=+0

• ∆φ(x,z > 0)=0 ⇒ Cauchy-Riemann relations between Ex(z > 0) and Ez(z > 0)

=⇒ E(x) ≡ Ex(z = 0) and ρ(x) are harmonic conjugates (a Hilbert pair):

2πρ(x) = −H[εE(x)], εE(x) = H[2πρ(x)], H[E(s)] = 1π

∫ds ′ E(s

′)s−s′


Overview of the model: Key ingredients

Domain state ≡ A stable solution of coupled Continuity and Poisson

equations for the 2D charge density ρ and the in-plane electric field E

compatible with the boundary conditions.

Continuity equation: ∂tρ+∇ · j = 0 (Static domains ⇒ ∂t = 0)

j = σ(E)E−D∇ρ → conductivity σ(0) < 0, diffusion coefficient D > 0

Poisson equation: −ε∆φ = 4πρδ(z) ⇒ ρ(x) = −

∫εdx ′

2π2

E(x ′)

x− x ′

Resulting self-consistent nonlinear integral equation

∂xj=0 ⇒ j=σ(E)E(x) +D∂x

∫εdx ′

2π2

E(x ′)

x− x ′


Solution for L� λ: Preliminary remarks

Solve “j = 0”

σ(E)E+ λ∂xH[E(x)] = 0

with BCs E(±∞) = ±Ec

Hint: Consider an infinitely sharp domain wall, i.e. E(x) = Ecsgn(x)

⇒ ∂xE = 2Ecδ(s), H[∂xE] =2Ecπx

, ρ = −εEc

π2ln |x| captures behavior at x� λ

Put into a nicer form: Introduce holomorphic Ψ ≡ π

2EcE+ i

π2

εEcρ =

π

2Ec(E− iH[E])

⇒ ∂xΨ = (γ+ ix)−1|γ→0 , Ψ = −i ln(γ+ ix)|γ→0 = π2 sgn(x) − i ln |x|

Suggests a trial solution ∂xΨ = (λ+ ix)−1 (same behavior at x� λ)

⇒ Ψ = −i ln(1 + ix/λ) = arctanx/λ− i ln√x2/λ2 − 1


Solution for L� λ

.. holomorphic Ψ ≡ π

2EcE+ i

π2

εEcρ =

π

2Ec(E− iH[E])

.. trial ∂xΨ = (λ+ ix)−1 ⇒ Ψ = −i ln(1 + ix/λ) = arctanx/λ− i ln√x2/λ2 − 1

This works! ∂xΨ = (λ+ ix)−1 solves σ(E)E+ λ∂xH[E(x)] = 0

provided σ(E) = −sin(πE/Ec)

πE/Ec(⇒ sin Re2Ψ+ λ∂xIm2Ψ=0)


Single domain wall in 2D

• Solved j=0 for L� λ ⇒ L→∞ with BC: E(x→±∞) = ±Ec

2D: σs(E) = −Ec

πEsinπE

Ec⇒ E(x) =

2Ecπ

arctanx

λ; ρ(x) =

εEc

2π2lnL2/4 + λ2

x2 + λ2

σ(E)=−E2

c +E2

E2c+E

2 |σ(0)|σdark

σ2: |σ(0)|σdark

� 1;

σs: |σ(0)|σdark'0.77;

−−: |σ(0)|σdark

=3

• Model conductivity σs meets all requirements

• Nonequilibrium screening length λ ≡ width of the domain wall

• Specifics of 2 dimensions: E(x→±∞) = ±Ec requires infinite charge density

• Power-law/logarithmic behavior@x→±∞ compared to exp(−x/λ3D) in 3D → next


Single domain wall in 2D & 3D

SDW: Solve j=0 for L� λ ⇒ L→∞ with BC: E(x→±∞) = ±Ec

2D: σ(E)E(x) + λ∂x

∫dx′

π

E(x′)

x− x′= 0 3D: σ(E)E(x) − λ2

3D∂2xE(x) = 0

λ =εD

2π|σ(0)|, λ3D =

(εD

4π|σ(0)|

)1/2

SDW in 3D: Bergeret,Huckestein,Volkov‘03

2D: σs(E) = −Ec

πEsinπE

Ec⇒ E(x) =

2Ecπ

arctanx

λ; ρ(x) =

εEc

2π2lnL2/4 + λ2

x2 + λ2

3D: σs(E) = −Ec

πEsinπE

Ec⇒ E(x) =

2Ecπ

sn

(x√

2λ3D

, 1

); ρ(x) =

ε

4π∂xE(x)

3D: σ2(E) = −1 +E2

E2c

⇒ E(x) = Ectanhx√

2λ3D

; ρ(x) =εEc

4√

2πλ3D

sech2 x√2λ3D

A side note: ∂xE(x)∝X≡(1 + s2)−1 solves the T=0 overdamped Caldeira-Leggett model,

∂XV(X) + ∂s

∫ds ′

π

X(s ′)

s− s ′= 0, V = X2/2 −X3


Periodic solution, L > πλ

Solve sin Re2Ψ+ λ∂xIm2Ψ=0

Periodic Ψ ≡ π

2EcE+ i

π2

εEcρ : Ψ(x) = Ψ(x+ 2L)


Before - SDW: ∂xΨ = (λ+ ix)−1, L� λ Now: 2L-periodicity ⇒ Image charges..

iL

π∂xΨ =

∞∑−∞

(−1)n

ξ−nπ≡ 1

sinξ≡ 2i

∞∑k=0

e−i(2k+1)ξ , ξ =πx

L−π

2− i ln

√L+πλ

L−πλ

E(x)

Ec= −

2

πarctan

(√l2 − 1 cos

πx

L

) 2πρ(x)

εEc=

2

πartanh

(√1−l−2 sin

πx

L

)l=

L

πλ

l = 30

l = 10

l = 2

l = 1.1

l = 1.01


Size-dependent instability threshold: σc = −εD/2L

Homogeneous phase ρ(x) = 0: Linear stability analysis

Stability with respect to small&slow spatio-temporal fluctuations δρ(x,y, t);

∂tρ+ ∂xj = 0, j = σ(E)E(x) −D∂xρ(x), ρ(0) = ρ(L) = 0

• L→∞: ρ(x) ≡ 0 stable for σ > 0 & ∂E(Eσ) > 0

• Finite L: ρ(x) ≡ 0 & E(x) ≡ 0 stable for σ > −εD/2L ⇔ l = L/πλ > 1

• Phenomena at −εD/2L < σ < 0 (0<l<1) → Dorozhkin,Dmitriev,Mirlin,PRB’11

π

23/2EcE(x) = −

√l− 1 cos

πx

Lπ2

21/2εEcρ(x)=

√l− 1 sin

πx

Ll=

L

πλ

l = 30

l = 10

l = 2

l = 1.1

l = 1.01


Instanton (SDW) vs Periodic domain solutions (PDS)

3D: σ(E)E− λ23D∂

2xE = 0

−→ λ23D(∂xE)

2/2 + g(E) = g(EM)

g = −E∫0

σ(E′)E′dE′→ |σ(0)|2E2c

π2sin2 πE

2Ec

• EM=Ec ⇒ instanton≡SDW;

• EM<Ec ⇒ PDS: E(x+2NL)=E(x), N=1, 2 . . .

E(x)

Ec= −

2

πarctan

(√l2 − 1 cos

πx

L

) 2πρ(x)

εEc=

2

πartanh

(√1−l−2 sin

πx

L

)l=

L

πλ

l = 30

l = 10

l = 2

l = 1.1

l = 1.01


Single- vs Multiple-domain wall solutions

3D: σ(E)E− λ23D∂

2xE = 0

−→ λ23D(∂xE)

2/2 + g(E) = g(EM)

g = −E∫0

σ(E′)E′dE′→ |σ(0)|2E2c

π2sin2 πE

2Ec

• EM=Ec ⇒ instanton≡SDW;

• EM<Ec ⇒ PDS: E(x+2NL)=E(x), N=1, 2 . . .

E(x)

Ec=−

2

πarctan

(√l2n−1 cos

πnx

L

)ρ(x)

ρ0=

2

πartanh

(√1−l−2

n sinπnx

L

)ln=

L

nπλ

l1 = 7.1l2 = 3.55l3 = 2.37l4 = 1.77l5 = 1.42l6 = 1.18l7 = 1.01


Stability: Lyapunov functional

Lyapunov functional Φ = −G+ K: Stable solution ≡ Global minimum of Φ

• bulk gain G=

∫dxg[E(x)] = −

∫dx

E(x)∫0

σ(E ′)E ′dE ′≷ 0

• inhomogeneous domain walls contribution K =D

2

∫dxE(x) CE(x)> 0

• capacitance C: ρ(x) = Cφ(x), [C,∂x] = 0 ⇒ ∂xρ = −CE

• Allow E(x, t): Φ =

∫dx [σ(E)E E+DE C E] =

∫dx [σ(E)E−D∂xρ] E =

∫dx j E

• Poisson E = −C−1∂xρ & Continuity ρ = −∂xj ⇒ E = C−1 ∂2xj

=⇒ Φ =

∫dx j E =

∫dx j C−1 ∂2

xj = −

∫dx (∂xj) C

−1 (∂xj)6 0

Application to ZRS (3D, limit λ� L): A. Auerbach, I. Finkler, B. I. Halperin, A. Yacoby’05


Lyapunov functional in 2D & 3D: Sine model

Constructed Φ =∫dx[D2 ECE− g(E)], g(E) = −

E(x)∫0

σ(E′)E′dE′ Proved: ∂tΦ 6 0

Apply to the sine model: σ = σ(0)sin 2θ

2θ, θ =

πE

2Ec, σ(0) < 0; Φ≡ π2Φ

2E2c|σ(0)|

3D: Local C3DE = −ε

4π∂2xE

Φ3D=

∫dx[λ2

3D(∂xθ)2−sin2 θ]

2D: Nonlocal C2DE =

∫εdx ′

2π2

E(x) −E(x ′)

(x− x ′)2

Φ2D=λ

∫dxdx ′

2π

[θ(x) − θ(x ′)

x− x ′

]2

−

∫dx sin2 θ

l1 =L

πλ= 7.1

l2 = 3.55l3 = 2.37l4 = 1.77l5 = 1.42l6 = 1.18l7 = 1.01


Evolution with system size

0 1 2 3 4 5 6 7 8 9 10 11 120

-0.2

-0.4

-0.6

-0.8

-1L � Π Λ

F�

F0

E(x)

Ec= −

2

πarctan

(√l2 − 1 cos

πx

L

) 2πρ(x)

εEc=

2

πartanh

(√1−l−2 sin

πx

L

)l=

L

πλ

l = 30

l = 10

l = 2

l = 1.1

l = 1.01


Multidomain solutions

0 1 2 3 4 5 6 7 8 9 10 11 120

-0.2

-0.4

-0.6

-0.8

-1L � Π Λ

F�

F0

Multidomain solution can be stabilized in the presence of long-range potential fluctuations.

A. Auerbach, I. Finkler, B. I. Halperin, A. Yacoby’05

l1 = 7.1

l2 = 3.55l3 = 2.37l4 = 1.77l5 = 1.42l6 = 1.18l7 = 1.01


Resistance in zero-resistance states

Next: Current response of the domain state.


Instanton: Zero resistance in zero-resistance states

Hall bar Corbino


Static domains

Both Corbino and Hall bar geometry:

Domains with jc⊥Ec = ρHjc,

where σd(Ec)=0 and ρH=B/enc.

External bias moves the domain wall

Hall bar – zero resistance state:

V=0 whatever I=jc(L1 − L2) is.

Corbino – zero conductance state:

I = 0 whatever V = Ec(L1 − L2) is


Residual conductance in the domain state: L ∼ λ

Solve σ(E)E−D∂xρ(E) = j to 1st order: δE(x),δρ(x) ∝ j

2D case, σ(E)E ∝ − sinπE/Ec =⇒ Linear response of the domain state:

δE(x) +iε

2πδρ(x) =

j

G0L

[2L

πλ− 1 −

4L

πλ

(1 +

L+πλ

L−πλe2πix/L

)−1]

Here G0 = σ(0)/L < 0 is conductance at L < πλ [ where E = δE = j/σ(0)]

⇒ Residual conductance G =j

< E > L=

G0

2L/πλ− 1in the domain state in 2D case

in contrast to G ∝ e−L/λ at L� λ in 3D case /Bergeret,Huckestein,Volkov‘03/

G2D

G0

�

Π Λ

2 L - Π Λ

G3D µ e-L� Λ,

L p Λ

G=G0, L< Π Λ

0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

1.2

L�Π Λ

G�G

0

0.2 0.4 0.6 0.8 1.0

-0.4

-0.2

0.2

0.4

0.6

0.2 0.4 0.6 0.8 1.0

0.1

0.2

0.3

0.4

0.5

0.6

0.2 0.4 0.6 0.8 1.0

-0.5

0.5

0.2 0.4 0.6 0.8 1.0

0.5

1.0

1.5

2.0

Shift of domain wall:

E/Ec (left)& ρ/ρ0 (right)

at j 6= 0 (red)& j = 0 (blue)

forL

πλ= 1.5 (top)

&L

πλ= 9 (bottom)


Spontaneous vs explicit symmetry breaking

• ρ(0)=ρ(L)=0 ⇒ Homogeneous state ρ ≡ E ≡ 0 isstable for L < πλ, or σ > σc = −σdark(πλTF/L):

Spontaneous symmetry breaking at σ = σc

• Explicit symmetry breaking: ρ(0) = −Uc(me/π h2)

Regions with E > 0 and E < 0 already at σ < σc

Experiment: Dorozhkin,Pechenezhskiy,Pfeiffer,West,Umansky,vonKlitzing,Smet’09; Theory: Dmitriev,Dorozhkin,Mirlin’09


Summary and Outlook

Summary: Found stable density/field distribution and the linear response of thedomain state in a finite-size 2D system with negative linear conductivityI. A. Dmitriev, M. Khodas, A. D. Mirlin, D. G. Polyakov, work in progress

Outlook• Mean field: Nonlinear response to external bias, contact potentials, smoothdisorder, spatial modulation etc.• Critical behavior at the transition to ZRS: Influence of noise, nature oftransition, dynamics at finite temperature• Domain structure in zero differential resistance states in Hall and Corbinogeomeries

• Experimental evidence of domains: S. I. Dorozhkin et al., Nature Phys. 7, 336 (2011)• Review: I. A. Dmitriev, A. D. Mirlin, D. G. Polyakov, M. A. Zudov, arXiv:1111.2176, toappear in Rev. Mod. Phys.• Domains in “3D”: A. Auerbach et al., Phys. Rev. Lett. 94, 196801 (2005); I. G. Finkler andB. I. Halperin, Phys. Rev. B 79, 085315 (2009); J. Alicea et al., Phys. Rev. B 71, 235322(2005); A. F. Volkov and V. V. Pavlovskii, Phys. Rev. B 69, 125305 (2004); A. F. Volkov andS. M. Kogan, Sov. Phys. Usp. 11, 881 (1969).• Negative conductance: S. I. Dorozhkin, I. A. Dmitriev, A. D. Mirlin, Phys. Rev. B 84,125448 (2011).


Remarks

Denote ϑ=πE

2Ec, r=

π2ρ

εEc, and s=

x

λ; Hilbert transform H[f] =

∫dx

π

f(x ′)

x− x ′

2D stripe of width L: ∆φ(x, z)=0 & 2πρ(x)=εEz(x, z→+0) → Ψ = ϑ+ ir

• ϑ & r are harmonic conjugates: Cauchy-Riemann ϑ = H[ρ], r = −H[ϑ]

• Model of contacts ⇒ 2L–periodicity ⇒ Ψ =

∞∑0

fne−iqns, qn = n

πλ

L

• σ =sin 2ϑ

2ϑ⇒ sin Re2Ψ+ ∂sIm2Ψ = 0 ⇒ ∂sΨ =

2i

cos(qns+ i artanhqn)

Limit L→∞: Single domain wall ↔ Caldeira-Legett model

• L→∞ ⇒ ∂sΨ = (1 + is)−1 ↔ ϑ = arctan s

• X ≡ ∂sϑ = (1 + s2)−1 solves the T=0 overdamped Caldeira-Leggett model:

∂XV(X) +

∫ds ′

π

X(s) − X(s ′)

(s− s ′)2= 0, V = X2/2 − X3


domains in zero resistance state - oist groups

Documents