Harvesting fluctuations in electrical hot spots

Rafael Sanchez

Instituto de Ciencia de Materiales de Madrid (ICMM-CSIC)

In collaboration with: Markus Buttiker, Bjorn Sothmann (Geneve)

Rosa Lopez, David Sanchez (Mallorca)

Andrew N. Jordan (Rochester)

Cargese, 23 October, 2012

Relative stability controlled by hot spots

ρ(A)ρ(B−)

= e−(UA−UB)/kTL

ρ(B−)

ρ(B+)= f(TL, TH)

ρ(B+)

ρ(C−)= e−(UB−UC)/kTH

ρ(C−)

ρ(C+)= 1/f(TL, TH)

ρ(C+)

ρ(D)= e−(UC−UD)/kTL

ρ(A)

ρ(D)= e

−UA−UD

kTL e−(UB−UC)

(

1kTH

−1

kTL

)

Noise in rarely occupied states must be considered.

R. Landauer, J. Stat. Phys. 53, 233 (1988)

Noise induced transport

State-dependent diffusion:

Boltzman factor

e−V (q)/kT −→ e−ψ(q)

Nonequilibrium potential:

ψ(q) = −∫ q dp

v(p)D(p)

, v(q) = −µ dVdq

M. Buttiker, Z. Phys. B 68, 161 (1987)N.G. van Kampen, IBM J. Res. Dev. 32, 107 (1988)Ya. M. Blanter, and M. Buttiker, Phys. Rev. Lett. 81, 4040 (1998)

P. Olbrich, et al., Phys. Rev. Lett. 103, 090603 (2009)

Coulomb coupled conductors

Harvard Stuttgart

Motivation

Noise induced transport

Capacitive coupled systems

Four terminals: Drag current and current correlations

Nonequilibrium fluctuation relations

Three terminals: Quantum dot heat engines

Coulomb blockade quantum dots

Detection of heat transfer statistics

Interacting open conductors

Conclusions

Time resolved charge detection

◮ Quantum point contact weakly coupled to a quantum dot

◮ Coulomb blockade regime

◮ Single electron tunneling detection

=⇒✞✝

☎✆Full counting statistics, PN (t)

Cumulant generating function: F(iχl) = ln∑

NlP (Nl)e

iNlχi

S. Gustavsson et al., Phys. Rev. Lett. 96, 076605 (2006)T. Fujisawa et al., Science 312, 1634 (2006)

State dependent counting and the fluctuation theorem

P (Nl) = eqNlVlkT P (−Nl)

F(iχl) = F (−iχl +Al) , Al =qVl

kT

J. Tobiska and Yu.V. Nazarov, Phys. Rev. B 72, 235328 (2005)K. Saito and Y. Utsumi, Phys. Rev. B 78, 115429 (2008)H. Forster and M. Buttiker, Phys. Rev. Lett. 101, 136805 (2008)

Effect of gate voltage!

B. Kung et al., Phys. Rev. X 2, 011001 (2012)

Cross-correlations in coupled quantum dots

✄✂

�✁Interactions-induced positive crosscorrelations

Understood in terms of sequential

tunneling.

D.T. McClure et al., Phys. Rev. Lett. 98, 056801 (2007)

◦ Small intradot interaction

◦ Non interacting electrons in each

conductor

M.C. Goorden, M. Buttiker, Phys. Rev. B 77, 205323 (2008)

Coulomb drag in quantum circuits

ID(V )=

∫dω

4πω2Tr[Z(ω)Sdrive(ω, V )Z(−ω)Γdrag(ω)]

Γdrag(ω): rectification

Sdrive(ω, V ): noise

∆i: curvature of the barrier

Linear drag (T ≪ ∆1,2):☛✡

✟✠ID = V

RQ

α+(0)π2

6T2

∆1∆2

1cosh2(eVg/2∆1)

⇒ near-equilibrium thermal excitations

* Very different with localized statesN.A. Mortensen et al., Phys. Rev. Lett. 86, 1841 (2001)

Nonlinear drag (T ≪ eV ≪ ∆1):✓

✒

✏

✑

ID = eV 2

∆2RQα−(0)

∑

n

|tn|2(1− |tn|

2)

︸ ︷︷ ︸

Shot noise

✄✂

�✁The mesoscopic regions do not interact!

Localized states?

A. Levchenko, A. Kamenev, Phys. Rev. Lett. 101, 216806 (2008)

Capacitively coupled quantum dots

V1 V2

V3 V4

Γ+1Γ−

1 Γ+2Γ−

2

γ+1γ−

1 γ+2γ−

2

Γ−3

Γ−3

γ+3

γ−3Γ−

4

Γ−4 γ+

4

γ−4

C1 C2

C3 C4

V1 V2

V3 V4

C

Tunneling rates:

Γ−

l = Γlf(∆l)

Γ+l = Γl[1− f(∆l)]

γ−l = γlf(∆l + EC)

γ+l = γl[1− f(∆l + EC)]

EC =2q2C

CΣuCΣd − C2

Drag current, V1 = V2

✞✝

☎✆Idrag = q (γ1Γ2 − γ2Γ1) sinh

qVdrive2kT

G({Vl})

1

2

3

4

∼Γ1γ2

EC

C

1

2

3

4

∼γ1Γ2

EC

C

◦ Asymmetry due to energy dependent tunneling is required.

◦ Broken detailed balance.

◦ Possible negative Coulomb drag

Drag current and current-current correlations V1 = V2

◦ Backaction as a gate effect

◦ Equilibrium larger that drag fluctuations

◦ Positive cross correlations!

◦ Drag current ∼ cross correlations

R. Sanchez, R. Lopez, D. Sanchez, and M. Buttiker, Phys. Rev. Lett. 104, 076801 (2010)

Nonequilibrium fluctuation relations

F(iχα) = F(−iχα + qVα/kT )

-0.002

0

0

0.001

0 2 4 6

0

0.002

0 2 4 6γ1/Γγ1/Γ

[q3Γ/kT]

[q3Γ/kT]

[q3Γ/kT]

kTG2,12

kTG2,22

kT (2G2,12+G2,22)

kTG2,44

kTG4,24

kT (2G4,24+G2,44)

kTG2,14

kTG2,24

kTG4,12

kT (G4,12+G2,14+G2,24)

S22,2

S22,1

S24,4

S44,22S24,4+S44,2

S12,4

S24,1

S24,2S24,1−S24,2+S12,4

(a) (b)

(c) (d)

(e) (f)

Ii =∑

j

Gi,jVj +1

2

∑

j,k

Gi,jkVjVk + . . .

Sij = S(0)ij +

∑

k

Sij,kVk + . . .

Fluctuation-dissipation theorem:

S(0)ij = 2kTGi,j

Nonequilibrium relations:

Sαα,α = kTGα,αα

Sαα,α = −kTGα,αα = (2Gα,αα+Gα,αα),

2Sαβ,β+Sββ,α = kT (Gα,ββ+2Gβ,βα)

Sαβ,α−Sαβ,α + Sαα,β = kT (Gβ,αα+Gα,αβ−Gα,αβ)

No drag ⇒ Sαα,β = 2kTGα,αβ

H. Forster and M. Buttiker, Phys. Rev. Lett. 101, 136805 (2008)R. Sanchez, R. Lopez, D. Sanchez, and M. Buttiker, Phys. Rev. Lett. 104, 076801 (2010)

Capacitively coupled quantum dots

V1,T1 V2,T2

Vg ,Tg

I

Jg

gate quantum dot

charge conductingquantum dot

C1 C2

Cg

V1 V2

Vg

C

Three terminals −→ Uncoupled directions of charge and heat currrents

q EC =2q2C

CΣuCΣd − C2

R. Sanchez, and M. Buttiker, Phys. Rev. B 83, 085428 (2011)

Related model (bosons): O. Entin-Wohlman, Y. Imry, and A. Aharony, Phys. Rev. B 82, 115314 (2010)

Heat into electric motion

Ts = Tg : V1 = V2 :

0

0.2

0.4

0.6

0.8

-1 0 1 2 3 4-2

0

2

-0.4

-0.2

0

0.2

0.4

-100 -50 0 50 100 150 200 0

2

4

6

0

0

I 2/qΓ

I 2/qΓ

[×10−

2]

Jl/E

CΓ

Jl/E

CΓ

[×10−

2]

J1

J2

J3

q(V1 − V2)/hΓ (T3 − T )/T

∆01 =0∆0

2 =0

◦ Negative differential conductance

◦ Current changes sign with

temperature

◦ Heat is not conserved

◦ Heat and charge currents correlated

◦ Heat rectification ⇒ Heat diodeT. Ruokola, T. Ojanen, Phys. Rev. B 83, 241404 (2011)

Cooling by transportq

q(V1 − V2)/hΓ

(Tg−

Ts)/

Ts−

0.20.2

−100 −50

0

0 50 100 150−1

1

2

Jg/ECΓ

Jg > 0

Jg < 0

Unbiased transport: V1 = V2

✞✝

☎✆I = q (γ1Γ2 − γ2Γ1) sinh

[EC2

(1kTg

− 1kTs

)]

F ({Vl}, {Tl})✞✝

☎✆Jg = EC(Γ1 + Γ2)(γ1 + γ2) sinh

[EC2

(1kTg

− 1kTs

)]

F ({Vl}, {Tl})

Entropy produced after charge transfer: ∆S± = ±EC

(

1Ts

− 1Tg

)

11 11

22

3 333

44

EC EC

C C

Heat to charge conversion:

I = qEC

γ1Γ2−Γ1γ2(Γ1+Γ2)(γ1+γ2)

JgIf γ1 = Γ2 = 0 ⇒

✞✝

☎✆

Iq= −

JgEC

Energy quanta to charge conversion!

Efficiency

Quantum dots as 0D contactsT. Bryllert et al, Appl. Phys. Lett. 80, 2681 (2002)

Total heat into charge current conversion.

ΓcΓcΓcΓc

Γ

ΓΓΓΓ

Γ

I+ I−

J+3J−

3

ECEC

C C∆S < 0

Power against the potential: P = I(V1 − V2)

Efficiency: η = P−Jg

=q(V1−V2)

−EC

Stall potential: I+ = I− ⇔ V∗ = −ECqηC

⇒

✞✝

☎✆ηmax = qV∗

−EC= ηC (reversibility)

0 0.5 10

0.5

1

η/η C

(V1 − V2)/V∗

Efficiency at maximum power

Carnot efficiency: ηc = 1− TsTg

Curzon-Albhorn efficiency: ηca = 1−√

TsTg

0 0.2 0.4 0.6 0.8 10

2

4

6

0

0.25

0.5

0.75

1

0 0.25 0.5 0.75 1

ηc = 1

ηc = 0.25

ηc = 0.75ηc = 0.5 ηc

ηca

ηm

efficiency

q∆V/ECηc

P/E

CΓ

[×10

−3]

ηc

Efficiency at maximum power: ηm ≈ 12ηc

R. Sanchez, and M. Buttiker, Phys. Rev. B 83, 085428 (2011)

Fluctuation relations for charge currents

I QPC

IQPC

I(0, 0)

I(1, 0)I(0, 1)

I(1, 1)

time

s

g

1’

1’2’

2’

3’

3’

4’

4’

1

1

2

2

3

3

4

4

γ+=Γ−10Γ

−31Γ

+21Γ

+30 γ−=Γ−

30Γ−21Γ

+31Γ

+10

ECC

V1, Ts V2, Ts

V3, Tg

Γ±1n Γ±

2n

Γ±3n

Transferred heat:

Eg = EC(Ng1 −Ng0)

Level resolved counting statistics:

F(iχln) = F(−iχln +Aln)

Aln = (Eαn − qVl)βl

◦ Valid for voltage and temperature gradients

◦ Configuration dependent

R. Sanchez and M. Buttiker, arXiv:1207.2587

T. Krause, G. Schaller, T. Brandes, Phys. Rev. B 84, 195113 (2012)O.-P. Saira, Y. Yoon, T. Tanttu, M. Mottonen, D.V. Averin, J.P. Pekola, arXiv:1206.7049

(Incomplete) fluctuation relations for charge currents

Iln = Nln/t

Jg = EC(Ig1 − Ig0)

For charge flows in the conductor:

1

tln

P ({Isn})

P ({−Isn})=IC(V1−V2)βs−

1

2Jg(βg−βs) = ξ

Gate dependent

Optimal converter:

1

tln

P (IC)

P (−IC)= IC[q(V1−V2)βs − Ec(βg−βs)]

q(V1−V2)/hΓ

q(V1−V2)/hΓ

〈ξ〉/〈I

C〉

〈ξ(I

C,I

11,I

20)〉/hΓ

-5

1086

5

4

2

2

1

0.1

0.02

0.04

0.06

0.08

00

00

Ts=Tg:Ts 6=Tg:

asymmetricsymmetric

optimal

(Incomplete) fluctuation relations for charge currents

Iln = Nln/t

Jg = EC(Ig1 − Ig0)

For charge flows in the conductor:

1

tln

P ({Isn})

P ({−Isn})=IC(V1−V2)βs−

1

2Jg(βg−βs) = ξ

Gate dependent

Optimal converter:

1

tln

P (IC)

P (−IC)= IC[q(V1−V2)βs − Ec(βg−βs)]

q(V1−V2)/hΓ

q(V1−V2)/hΓ

〈ξ〉/〈I

C〉

〈ξ(I

C,I

11,I

20)〉/hΓ

-5

1086

5

4

2

2

1

0.1

0.02

0.04

0.06

0.08

00

00

Ts=Tg:Ts 6=Tg:

asymmetricsymmetric

optimal

Two terminals:

IQPC

Vs, Ts Vg, Tg

Cs g

F(iχsn)=F(−iχsn+EC (βg−βs))

Universal!

R. Sanchez and M. Buttiker, arXiv:1207.2587

Hot spots in interacting chaotic cavities

C1 C2

Cg

C

UV1, T1 V2, T2Tl(E) Tr(E)

gateV g, T g

Semiclassic kinetic equation:

eνiFdfi

dt= eνiF

∂fi

∂UiUi +

e

h

∑

r

Tir(fir − fi) + δiΣ

Fluctuations:

fi =

∑

r Tirfir∑

r Tir+ δfi

Tir = T 0ir − qT ′

irδUi

δfi, δUi related by self consistency

→ nonlinear Langevin equations for δUi (Noise: δIir)

〈δIr(t)δIr(0)〉 = Dr(δUi)δrr′δ(t)

Current:

I1r =e

h

∫

dET1r(f1r − f1) + δIr

B. Sothmann, R. Sanchez, A.N. Jordan, M. Buttiker, Phys. Rev. B 85, 205301 (2012)

Hot spots in interacting chaotic cavities

τRC =Ceff

GeffΛ = e

T 01RT

′

1L − T 01LT

′

1R

T 21Σ

∼ N−1

Generated currents:

〈I1L〉 =Λ

τRCkB(Θ1 −Θ2)

〈JH〉 =1

τRCkB(Θ2 −Θ1)

Maximal power and efficiency:

Pmax =Λ2

4G1τ2RC(kB(Θ1 −Θ2))

2 ∼ N−1

ηmax =Λ2

4G1τRCkB(Θ2 −Θ1) ∼ N−2

→ Larger currents (∼ 0.1nA), but lower efficiency

B. Sothmann, R. Sanchez, A.N. Jordan, M. Buttiker, Phys. Rev. B 85, 205301 (2012)

Conclusions

◦ Current driven by fluctuations (electric or thermal non equilibrium)

◦ Asymmetry by energy dependent tunneling processes.

◦ Energy is transferred in quanta EC

◦ Negative drag

◦ Heat to charge conversion at the maximal (Carnot) efficiency

◦ High efficiency at maximum power

◦ Direction of heat current uncoupled from direction of electric motion

◦ Fluctuation theorems for charge flows modified in the presence of a hot spot

◦ Detection of heat transfer statistics by electron counting

◦ Open systems: Larger currents, lower efficiency

◦ More details in:R. Sanchez, R. Lopez, D. Sanchez, and M. Buttiker, Phys. Rev. Lett. 104, 076801 (2010)

R. Sanchez and M. Buttiker, Phys. Rev. B 83, 085428 (2011)

B. Sothmann, R. Sanchez, A.N. Jordan, M. Buttiker, Phys. Rev. B 85, 205301 (2012)

R. Sanchez and M. Buttiker, arXiv:1207.2587.

Current-current correlations

Probability of ni events in a time t: P (ni, t)

Generating function: F(iχi) = ln∑

niP (ni)e

iniχi

Cumulants:

Mean value: 〈ni〉 −→ Stationary current: Ii = q ddt〈ni〉

Variance: 〈n2i 〉 − 〈ni〉

2 −→ Noise: Sii = q2 ddt

(〈n2i 〉 − 〈ni〉

2)

=⇒ Correlations: Sij = q2 ddt

(〈ninj〉 − 〈ni〉〈nj〉)

Fluctuation relations:

At equilibrium: Fluctuation-dissipation theorem: Seq = 2kT ddVI

Out of equilibrium?

Current-current correlations

Probability of ni events in a time t: P (ni, t)

Generating function: F(iχi) = ln∑

niP (ni)e

iniχi

Cumulants:

Mean value: 〈ni〉 −→ Stationary current: Ii = q ddt〈ni〉

Variance: 〈n2i 〉 − 〈ni〉

2 −→ Noise: Sii = q2 ddt

(〈n2i 〉 − 〈ni〉

2)

=⇒ Correlations: Sij = q2 ddt

(〈ninj〉 − 〈ni〉〈nj〉)

Fluctuation theorem:

P (ni) = eqniVikT P (−ni)

F(iχi) = F (−iχi +A1) , Ai =qVi

kTi

J. Tobiska and Yu.V. Nazarov, Phys. Rev. B 72, 235328 (2005)K. Saito and Y. Utsumi, Phys. Rev. B 78, 115429 (2008)H. Forster and M. Buttiker, Phys. Rev. Lett. 101, 136805 (2008)

Linear regime

Idrag =q2Vdrive

kT(γ1Γ2 − γ2Γ1)(γ3Γ4 − γ4Γ3)Geq

Onsager symmetry preserved

Breaking of detailed balance in both conductors

Non trivial temperature dependence

0 5 10 15 200

0.001

0.002

0.003

kT

G2,4

∼ 1kT

∼ e−(ε′

u0+ε′

d0+EC)/kT

kT

Equilibrium crosscorrelations: S(0)24 = 2q2(γ1Γ2 − γ2Γ1)(γ4Γ3 − γ3Γ4)Geq

⇒

✞✝

☎✆S

(0)24 = 2kTG2,4

Importance of gate voltages

Γ1 Γ2

qV

Γ−

l = Γlf((∆U − qVl)/kTl)

Γ+l = Γl[1− f((∆U − qVl)/kTl)]

∆U = ε+ q2

2CΣ+ qCΣ

(C1V1+C2V2+CgVg)

ρ1 =(

Γ+1 +Γ+

2

)

ρ0 −(

Γ−

1 +Γ−

2

)

ρ1⇓

F(iχb) = F(−iχb +Ab) Independent of gate voltage!

χb = χ1 − χ2

Ab = A1 −A2 = qV1kT1

− qV2kT2

◮ Nonlinear transport coefficients: G2,11 6= G2,22

◮ Inhomogeneous temperature: F(iχb) = F(−iχb +Ab − (ε+∆U︸ ︷︷ ︸

V1,V2,Vg

)(

1T1

− 1T2

)

)

Detection of noise induced transport

Fano factor: F = SqI

⇒ Diverges at equilibrium

Symmetric coupling

(no conversion)

Asymmetric coupling

(heat conversion)

Selective coupling

(optimal conversion)

Thermal motor

hot bath cold bath

system

load

W

Q1 Q2

η =W

Q1= ηC −

TcSprod

Q1

ηC = 1−Tc

Th, Carnot efficiency

A single cavity

U

C1 C2

Cg

V1 V2

Vg

UV1, T1 V2, T2Tl(E) Tr(E)

gateV g, T g

Semiclassic kinetic equation:

qνFdf

dt= qνF

∂f

∂UU +

q

h

∑

l

Tl(fl − f) + δiΣ

Fluctuations:

f =

∑

l Tlfl∑

l Tl+ δf

Tl = T 0l − qT ′

l δU

δf , δU related by self consistency

→ nonlinear Langevin equations for δU (Noise: δIl)

〈δIl(t)δIl′ (0)〉 = Dl(δU)δll′δ(t)

Current:

Il =q

h

∫

dE Tl(fl − f) + δIl

B. Sothmann, R. Sanchez, A.N. Jordan, M. Buttiker, Phys. Rev. B 85, 205301 (2012)

Charge fluctuations:

Qc = qνF∑

l

∫

dETl

TΣfl − q2νFU

=∑

l=1,2

Cl(U − Vl) + Cg(U − Vg)

⇒

∫

dEδf = q

(CΣ

Cµ+

χ

q2νF

)

δU +T ′

Σχ

νFT0Σ

(δU)2

χ = q3νF

1

2

∑

l

Λll(Vl − Vl) Λlm =T ′lT

0m − T 0

l T′m

(T 0Σ)2

Langevin equation:✞✝

☎✆CΣ

˙δU = − q2

hT 0Σ

(CΣCµ

+ χq2νF

)

δU + q3

hT ′

ΣCΣCµ

(δU)2 + δIΣ

Diffusion coefficients:

〈δIl(t)δIl(0)〉 =2q2

h

∫

dETl[fl(1− fl) + f(1− f) + (fl − f)2(1− Tl)

]δ(t)

= (D0l +D1lδU +D2lδU2

︸ ︷︷ ︸

Dl(δU)

)δ(t)

In equillibrium: Deq0l = 4q2

h T 0l kBΘ, Deq

1l = −4q3

h T ′l kBΘ, Deq

2l = 0 (Einstein relation)

Non linear Langevin equation: x = f(x) +√

2D(x)l(t)

x = CΣδU 〈l(t)l(0)〉 = δ(t)

Fokker-Planck equation:

∂P

∂t=

∂

∂x

(

−f(x)P + α∂D(x)

∂xP + D(x)

∂P

∂x

)

.

kinetic: α = 0, Stratonovich: α = 1/2, Ito: α = 1

d

dt〈x〉 = 〈f(x)〉 − (α−1)〈

∂D

∂x〉

d

dt〈x2〉 = 2〈xf(x)〉 − 2(α−1)〈x

∂D

∂x〉+ 2〈D〉.

Kinetic prescription (α = 0):

〈δU〉eq = 0

〈δU2〉eq =2CµkBΘ

C2Σ

(Equipartition theorem)

Stratonovich and Ito do not describe equilibrium properties!Y.L. Klimontovich, Phys. Usp. 37, 737 (1994).

The current

Il =q

h

∫

dE Tl(fl−f) + δIl

Equilibrium:

〈Il〉eq = 0. X (only for α=0)

Linear response:

〈Il〉lin =q2

h

[T 0l T

0l

T 0Σ

−2q2kBΘCµT 0

Σ3Λ2ll

4q2CµkBΘT ′

Σ2+C2

ΣT0Σ2

]

(Vl − Vl).

Interaction correction:

◦ proportional to asymmetry Λll

◦ Scales as N−1