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