zero resistance state - gatech.edu

ICCMSE 2014, 04-07 April 2014, Athens, Greece

Transport in zero resistance state

Ivan Dmitriev, M. Khodas, A. D. Mirlin, D. G. Polyakov

Phys. Rev. Lett. 111, 206801 (2013)

Ivan Dmitriev (MPI FKF) Zero resistance state 05.04.2014 1 / 23

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

Nonequilibrium phenomenain high Landau levels

2001-Present:

• Integer and fractional MIRO and ZRSin semiconductor quantum Hall systems• similar microwave-inducedoscillations and dissipationless statesin 2D electron system on liquid He• magnetooscillations induced bystrong dc current• magnetooscillations induced byresonant interaction with acousticphonons• photovoltaic effects. . .

Review:Dmitriev, Mirlin, Polyakov, Zudov,Rev. Mod. Phys. 84, 1709 (2012)

Experiments: MIRO & ZRS

ZRS: Mani,Smet,von Klitzing,Narayanamurti,Johnson,Umansky’02,Zudov,Du,Pfeiffer,West’03,Dorozhkin’03, Willett,Pfeiffer,West’04 ..MIRO: Zudov,Du,Simmons,Reno’01,Ye,Tsui,Simmons,Wendt,Vawter,Reno’01,..


Microscopic mechanisms of MIRO ⇒ σ(E) and D

MIRO: linear DC conductivity under MWs oscillates with ω/ωc,(δσ)mw ∝ −P sin(2πω/ωc), and can become negative at high P.

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


Nonlinear conductivity of homogeneous state

Homogeneous 2DEG: I-V in minima

Linear σ(E→ 0): σ < 0 in minima

Nonlinear σ(E): σ(E=Ec) = 0 (inset)

ID,Vavilov,Aleiner,Mirlin,Polyakov’05

Microscopic theory: σ(E)&D

Static σ(E) under radiation:σ(0) < 0;

Critical field Ec: σ(Ec)=0

-Ec Ec0

0 0

D

s(0)

s(E )

D(E )

E

Microscopic theory:

Ec ∼ωc/e

max{lin, Rc}. 1 V/cm

- agrees with experiments

ID, Mirlin, Polyakov, Zudov,Rev. Mod. Phys. 84, 1709 (2012)


j · E < 0 ⇒ Domains

Maxwell equations:

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

⇒ Electrical instability: Uniform state is unstable

⇒ Domain state ≡ Resulting inhomogeneous state

σ < 0

←→

ZRS ?=? Static domains


Uniform state with j · E < 0 ⇒ Domains ⇒ j · E = 0

• Uniform j = 0 & E = 0 state - unstable

• Two domains with |E| = Ec are formed:

σ(Ec) = 0, σd(Ec) > 0

=⇒ For any V =∫

dxE(x), the current j = 0

=⇒ zero conductance j/V = 0 in the domain state


Related phenomena

• Instability of the negative conductivity: Zakharov (1960), Ridley(1963), Bonch-Bruevich

and Kogan(1965), Elesin and Manykin(1967), Volkov and Kogan(1967)..

• Gunn and similar effects:

Differential conductivity σd = ∂E[Eσ(E)] < 0 at finite 0 < E1 < |E| < E2:

⇒ Electrical instability: Moving domains and high-frequency generation

review Volkov, Kogan (1969),books Bonch-Bruevich,Zvyagin,Mironov (1975), Pozhela (1981), Scholl (2001)

In our case:

• σd = σ < 0 at E→ 0 ⇒ Static domains

• Isotropic: uniform state with σ(E) < 0 and σd(E) > 0 - unstable

• Nonlocal 2D electrostatics: exponential vs. power-law behavior

• Magnetic field: duality of S– and N–shape CVC


Zero conductance ↔ Zero resistance

Hall bar Corbino

Andreev,Aleiner,Millis’03

Domains with jc⊥Ec = ρHjc,

where σ(Ec)=0 and ρH=B/enc.

External biasmoves the domain wall

Hall bar – zero resistance state:

V=0 for any I= jc(L1 − L2)

Corbino – zero conductance state:

I = 0 for any V = Ec(L1 − L2)


1D model of the domain state

• Static continuity equation ∂xj = 0, j=σ(E)E(x) −D∂xρ(x)

• Poisson ∂xρ = −∂x

∫εdx ′

2π2

E(x ′)

x− x ′— non-local (in 3D ∂xρ =

ε

4π∂2xE)

L

microwavesz

y

x

B

V

-Ec Ec0

0 0

D

s(0)

s(E )

D(E )

E

E E

jH

jH

j=σ(E)E(x) +D∂x

∫εdx ′

2π2

E(x ′)

x− x ′+ boundary conditions


Step 1: Domain solution in infinite system

• Solve j = 0 with boundary conditions E|x→±∞=±Ec

• Take σ(E) = σ(0)sinπE/EcπE/Ec

, σ(0) < 0

-Ec Ec0

0 0

D

s(0)

s(E )

D(E )

E

σ(E)E(x) +D∂x

∫εdx ′

2π2

E(x ′)

x− x ′= 0 ⇒ − sin

πE

Ec+ λ ∂x

∫dx ′

x− x ′E(x ′)

Ec= 0

Solution:E

Ec=

2

πarctan

x

λ, λ =

εD

2π|σ(0)|

Nonequilibrium screening length λ:

• The only spatial scale, width of domain wall

• Diverges at σ(0)→ 0 ⇒ critical parameter

• reduces to λTF =ε

2me2in equilibrium

-5 0 5E

E/

c

0

-1

1

x / l

-1 1


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 !


Step 2: Domain solution in unbiased finite system

• Solve j = 0 in 2D stripe of width L → parameter l ≡ L/πλ

2DEG

x

y

z

0

• Plane contacts do not create nonuniform ρ(x)

⇒ Spontaneous symmetry breaking

• Image charges → periodicity: E(x) = E(x+ 2L)

• Boundary conditions: ρ(0) = ρ(L) = 0

l=L/πλ

l = 30

l = 10

l = 2

l = 1.1

l = 1.01

E(x)

Ec= −

2

πarctan

(√l2 − 1 cos

πx

L

) 2πρ(x)

εEc=

2

πartanh

(√1−l−2 sin

πx

L

)


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

Fluctuations e−ηqteiqyy sin(qxx) Stability: ηq = 2πεq

(σd(Ex)q

2x +σ(Ex)q

2y

)+Dq2 > 0

Consider linear stability of the uniform ρ ≡ 0 unbiased E ≡ 0 state:

• L→∞: long-wavelength (q→ 0) instability at σc = 0

• Finite L: q > π/L ⇒ instability shifts to σc = −εD/2L ⇔ l ≡ L/πλ = 1

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


Step 3: Domain solution in a biased 2D stripe

• Solve sinπE

Ec− λ ∂x

∫dx ′

Ec

E(x ′)

x− x ′= j; (Anti)dissipative current j =

π j

σ(0)Ec

0.0 0.2 0.4 0.6 0.8 1.0

0.0

-0.5

-1.0

0.5

1.0

L / pl = 3

x L/

EE

/c

j = {0, 0.25, 0.5, 0.75, 0.9, 8/9}

0.0 0.2 0.4 0.6 0.8 1.0

0.0

-0.5

-1.0

0.5

1.0

x L/

EE

/c

j = {0, 0.03, 0.1,

0.5, 0.9, 899/900}

L / pl = 30

Finite bias j: Shift w(j) and broadening β(j) of the domain wall

Full solution: Ψ ≡ π

2EcE+ i

π2

εEcρ = i ln

cosh(ξ− iw)

sinhξ−

1

2arcsin j, |j| 6

√1 − l−2,

where 2ξ = iπ(x/L− 1/2) + iw+β, w = arctan(jl), β = arcoth(l

√1 − j2)


Domains ⇒ zero resistance/conductance

Hall bar Corbino

Andreev,Aleiner,Millis’03

Domains with jc⊥Ec = ρHjc,

where σ(Ec)=0 and ρH=B/enc.

External biasmoves the domain wall

Hall bar – zero resistance state:

V=0 for any I= jc(L1 − L2)

Corbino – zero conductance state:

I = 0 for any V = Ec(L1 − L2)


Negative conductance: Linear response

Small V: j =<σ> V/L

Conductance G =<σ> /L

• L < π/λ : <σ>= σ(0)

• L� λ :<σ>

σ(0)=πλ

2L

<s

>

s(0)

01

2L / pl -1

s(0)

e-L / l3D

L / pl

3D electrostatics, ∂xE = 4πρ/ε −→ <σ>

σ(0)∼ e−L/λ3D , λ3D=

√εD

4πσ(0)

Appropriate description for 3D ruby crystals Dyakonov’84, Dyakonov,Furman’84

Also utilized to describe ZRS in 2DA. F. Volkov and V. V. Pavlovskii, Phys. Rev. B 69, 125305 (2004)A. Auerbach et al., Phys. Rev. Lett. 94, 196801 (2005)I. G. Finkler and B. I. Halperin, Phys. Rev. B 79, 085315 (2009)


General approach to stability: Lyapunov functional

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

Application to ZRS (3D, limit λ� L) Auerbach,Finkler,Halperin,Yacoby’05

• Bulk gain G=−

∫dx

E(x)∫0

σ(E ′)E ′dE ′ : σ < 0 ⇒ Maximized for |E(x)|=Ec

• Domain walls K =D

2

∫dxE(x) CE(x)> 0 ⇒ Minimized for ∂xE=0

capacitance C: ρ(x) = Cφ(x)

Sine model: Φ=πλ

8E2c

∫dxdx ′

[E(x) − E(x ′)

x− x ′

]2

−

∫dx sin2 πE

2Ec> −1


Full CVC, phase diagram, and current discontinuity

2.5

10

100

1000

2

0.0 0.5 1.0

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

V / E Lc

- j

= p j

/ |s(0)|

Ex

c

L = 1.2pl

0.25 0.75

Vc3 Vmin Vc1 VmaxVc2

<s>

s(0)

0

1L / pl

3

0.1 1 10 210 310 410L / pl

stablehomogeneous state

0.25

0.0

1.0

0.5

0.75

V /

EL

c

hysteresis

2L / pl - 1

s(0)

L -l3De

0 2

1st and 2nd order phase transitions:

• l 6√

2: Kink in j(V)

• l >√

2: Discontinuity in j(V)

• Frozen transverse fluctuations ⇒ hysteresis

• Transverse fluctuations ⇒ instabilityat large V and L

0.1 1 102

103

104

10

0.25

0.0

1.0

0.5

0.75

V /

EL

c

L / pl

stablehomogeneous state

in state

homogeneous

breakdown of 1D approach

+


Possible 2D domain structure at low and high bias

• Finkler,Halperin’09: DWs at π/4 to E in domains, ρ∝∂xE (3D)

=⇒ � structure is stable, Φ� ≶ Φ1D, for ∂2Eσ↘ with E

• High bias structure in 1D (L↑ � L) vs 2D (Double domain walls)

• Realistic 2D electrostatics (ρ∝x−1 ◦ E) ⇒ Φ� > Φ1D

=⇒ 1D domain structure is preferable ⇒ different 2D structure at high V


Summary

Analytic model for the domain state in a biased 2D stripe

Step 1: Domain state in unbiased large system, L→∞, V → 0

Step 2: Finite-size effects in unbiased system

Step 3: Domain state in biased finite system

Outcome:

• Nontrivial dependence of domain structure on system size L and bias V

• V − L phase diagram including continuous and discontinuous phase transitions

• Transverse instability at large V and L: transition to a 2D domain state


Direct evidence for static domains in ZRS?

Random telegraphphotovoltage signals betweeninternal probes undercontinuous microwaveillumination

Interpreted as spontaneousswitching between two nearlydegenerate configurations ofdomains

Dorozhkin,Pfeiffer, West, von Klitzing, Smet’11

configuration I configuration II

UB4, B5

UB3, B4

UB2, B3

UB1, B2

Ui, j

Time t (s)-1.0 -1.5 0.0

- 0

- 0

- 0

- 0

- 0

B

1 mV

B BA AC C

SSD D

1

2

34

5

I I


Wigner electron liquid on surface of liquid He

SSE

liquid 3He

sinter

VB VG1

VF VG2

+ +

++-

- -

-

G2 C2 C1

G1 B

F �y�y�y�y�y�y�y�y�y�y�y�y�y�y�y�y

I1

I2

0 1 2 3

I2

t (s)

2x105

4x105

6x105

Q (i

n un

its o

f e)

I1

-0.3

0.0

0.3

(b)

off

on

I (pA

)

0

on

off

0.52 T

0.62 T

0.78 T

(a)

0

Konstantinov and Kono’09,10,Konstantinov,Chepelianskii,Kono’12

• Magnetooscillations and zero dissipationin the minima under microwave illumination

• Time-resolved capacitive measurementfor radiation switched on and off:

Charge transfer due to domain formation!

0.0 0.2 0.4 0.6 0.8

0.0

0.4

0.8 (b)

I (pA

)

Magnetic field B (T)

(a)

0.0

0.1

0.2

0.3

0.4

l = 7l = 6 l = 5

xx (n

S)

/2 =90.9 GHzns=1.4x106 cm-2

T=0.2 Kl = 4

0.60 0.62 0.64 0.660.0

0.1

0.2

xx (n

S)

B (T)

0.0

0.2

0.4

I (pA

)


Importance of transverse fluctuations by example of ruby

• Laser-illuminated Ruby Al2O3 : Cr3+

→ Spontaneous field E ∼ ±106 V/cm

along or against the trigonal axis C3

detected in the luminescence ofStark-shifted Cr3+ levels

No transverse fluctuations are possible!

⇒ Homogeneous state with σd > 0 isstable whatever σ is (σ > 0 or σ < 0)

⇒ Hysteresis with respect to directionof electric field sweep

Experiment: Liao,Glass,Humphrey’80Basun,Kaplyanskii,Feofilov’83,’84

Theory of domains: Dyakonov’84Dyakonov,Furman’84

j · E < 0 is stable!


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