Out-of-Equilibrium Admittance of Single Electron Box Under Strong Coulomb Blockade
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Out-of-Equilibrium Admittance of Single Electron Box Under StrongCoulomb Blockade
Ya. I. Rodionov and I. S. Burmistrov L.D. Landau Institute for Theoretical Physics RAS, 119334 Moscow, Russia
Submitted October 17, 2018
We study admittance and energy dissipation in an out-of-equlibrium single electron box. The systemconsists of a small metallic island coupled to a massive reservoir via single tunneling junction. The potentialof electrons in the island is controlled by an additional gate electrode. The energy dissipation is caused byan AC gate voltage. The case of a strong Coulomb blockade is considered. We focus on the regime whenelectron coherence can be neglected but quantum fluctuations of charge are strong due to Coulomb interac-tion. We obtain the admittance under the specified conditions. It turns out that the energy dissipation ratecan be expressed via charge relaxation resistance and renormalized gate capacitance even out of equilibrium.We suggest the admittance as a tool for a measurement of the bosonic distribution corresponding collectiveexcitations in the system.PACS: 73.23.Hk, 73.43.Nq
It is well-known that the phenomenon of Coulombblockade is an excellent tool for observation of interac-tion effects in single electron devices [1–5]. Recently,due to the progress in the field of thermoelectricity theCoulomb blockade under out-of-equilibrium conditionshas come into the focus of the theoretical [6–11] and ex-perimental research [12–14]. The simplest mesoscopicsystem displaying Coulomb blockade is a single electronbox (SEB). The properties of such a system are essen-tially affected by electron coherence and interaction.The set-up is as follows (see Fig.1). Metallic island iscoupled to an equilibrium electron reservoir (tempera-ture T r ) via tunneling junction. The island is also cou-pled capacitively to the gate electrode. The potentialof the island is controlled by the voltage U g of the gateelectrode. The distribution function of electrons in thereservoir is assumed to be equilibrium (Fermi distribu-tion) while the one inside the island is arbitrary.The physics of the system is governed by the Thou-less energy of an island E Th , its charging energy E c ,and the mean single-particle level spacing δ . Through-out the paper the Thouless energy is considered to bethe largest scale in the problem. This allows us to treatthe metallic island as a zero dimensional object withvanishing internal resistance. The characteristic energy( ε d ) of electrons inside the island obeys the condition δ ≪ ε d ≪ E c , E Th . This implies that characteristicenergy is high enough to render the system incoherentand low enough to keep it strongly correlated due toCoulomb interaction [15]. The dimensionless conduc-tance of a tunneling junction is small g ≪ e-mail: [email protected] [email protected]. A single electron box does not allow for conductancemeasurements since there is no DC-transport. This wayan essential dynamic characteristic becomes the set-upadmittance, which is a current response to an AC-gatevoltage U g ( t ) = U + U Ω cos Ω t .Paper [16] sparked both theoretical and experimentalattention to the admittance of such a set-up [17–22]. Asit is well-known, the real part of admittance determinesenergy dissipation in an electric circuit. Classically, theaverage energy dissipation rate of a single electron boxis given as follows W Ω = Ω C g R | U Ω | , R = he g , ~ Ω ≪ gE c , (1)where C g denotes the gate capacitance, e - the electroncharge, and h = 2 π ~ - the Planck constant. Expres-sion (1) presents us with a natural way of extractingthe resistance of a system from its dissipation rate. C g U + U ω cos ωt C g C , g g IslandLead U g ( t ) ε d T r Fig. 1. The set-up. A SEB is subjected to a constantgate voltage U . The current through the tunnelingjunction is caused by a weak AC voltage U g ( t ). T ≪ δ ) coherent regime was pioneered in Ref. [16]. Itwas shown that the energy dissipation rate W Ω can befactorized in accordance with its classical appearance(1) but the definition of physical quantities comprisingit becomes different. Geometrical capacitance C g shouldbe substituted by a new observable: mesoscopic capac-itance C µ . This leads to the establishment of anotherobservable: charge relaxation resistance R q such that R → R q in Eq. (1). Charge relaxation resistance ofa coherent system differs drastically from its classicalcounterpart. In particular, the charge relaxation resis-tance of a single channel junction was predicted to beindependent of its transmission and equal to h/ (2 e ) atzero temperature [16]. However Coulomb interaction inRef. [16] and subsequent works [18] was accounted for onthe level of classical equations of motion only. Recentlythe result for quantization of the charge relaxation resis-tance in SEB at T ≪ δ has been rigorously derived [23].The admittance in this low temperature regime was in-vestigated experimentally by Gabelli et al. [19].The knowledge of the charge relaxation resistance hasbeen extended to a SEB at the transient temperatureswhen thermal fluctuations smear out electron coherencebut electron-electron interaction is strong. The expres-sion for the energy dissipation at this transient temper-atures keeps its classical appearance if one substitutesthe renormalized gate capacitance C g and the charge re-laxation resistance R q for C g and R respectively [24].Unlike the latter, C g and R q have strong temperatureand gate voltage U g dependance.The recent experiment by Persson et al [22] exploredthe energy dissipation rate at these transient tempera-tures. The admittance of SEB was measured at fixedfrequency as a function of pumping amplitude U Ω andthe DC part of gate voltage U in a wide range. Thetheoretical analysis of the data in Ref. [22] was carriedout under assumption of linear response to the AC gatevoltage: the electrons inside the island were assumed tobe in the equilibrium with the reservoir. However it hasnot been verified experimentally. It is natural to expectthat this assumption is violated for the set of data withhigh values of the amplitude U Ω .Motivated by the experiment [22] we study the ad-mittance of a single electron box under the out-of-equilibrium conditions. We consider the linear responseof a SEB with arbitrary electron distribution functionin the island to the AC gate voltage.A single electron box is described by the Hamiltonian H = H + H c + H t , (2) where H = X k,i ε k a † k a k + X α ε ( d ) α d † α d α (3)describes free electrons in the lead and the island, H c describes Coulomb interaction of carriers in the island,and H t describes the tunneling. Here operators a † k ( d † α )create a carrier in the lead (island). Then the tunnelingHamiltonian is H t = X + X † , X = X k,α t kα a † k d α . (4)The charging Hamiltonian of electrons in the box istaken in the capacitive form: H c = E c (cid:0) ˆ n d − q (cid:1) . (5)Here E c = e / (2 C ) denotes the charging energy, q = C g U g /e the gate charge, and ˆ n d is an operator of a par-ticle number in the island ˆ n d = P α d † α d α . To char-acterize the tunneling it is convenient to introduce theHermitean matrix:ˇ g αα ′ = (2 π ) h δ ( ε ( d ) α ) δ ( ε ( d ) α ′ ) i / X k t † αk δ ( ε k ) t kα ′ . (6)The energies ε k , ε ( d ) α are accounted from the Fermi level,and the delta-functions should be smoothed on the scale δE , such that δ ≪ δE ≪ T r , ε d . The classical di-mensionless conductance (in units e /h ) of the junc-tion between a reservoir and the island can be ex-pressed as follows g = P α ˇ g αα . Therefore, each non-zero eigenvalue of ˇ g corresponds to the transmittance ofsome ‘transport’ channel between a reservoir and the is-land [25]. The effective dimensionless conductance ( g ch )of a ‘transport’ channel and their effective number ( N ch )are given by g ch = P αα ′ ˇ g αα ′ ˇ g α ′ α P α ˇ g αα , N ch = (cid:18)P α ˇ g αα (cid:19) P αα ′ ˇ g αα ′ ˇ g α ′ α . (7)The dimensionless conductance becomes g = g ch N ch . Inwhat follows we will always assume g ch ≪ , N ch ≫ , g ≪ . (8)Throughout the paper we keep the units such that ~ = e = k B = 1 except for the final results.In the presence of time dependent gate voltage thegate charge q in Eq. (5) is changed as q → C g U g ( t ) /e .The gate voltage is coupled to the operator of particle ut-of-Equilibrium Admittance of Single Electron Box Under Strong Coulomb Blockade E ch ( n , q ) k + 1 q k + 2 k ∆ Fig. 2. Charging energy E ch = E c ( n − q ) as a functionof gate charge q . number inside the island only. Therefore the admit-tance of the system (the response of the charge in theisland to AC part of the gate voltage) is determinedby autocorrelation function of fluctuating particle num-ber: iθ ( t ) h [ˆ n d ( t ) , ˆ n d (0)] i , where θ ( t ) is Heaviside step-function. Due to the presence of strong Coulomb in-teraction the behavior of the autocorrelation function isnon-trivial. It corresponds to collective bosonic modessimilar to the case of Fermi liquid where the density-density correlator is governed by the electron-hole ex-citations [26–28]. The latter determines the behaviorof the autocorrelation function in the absence of theCoulomb interaction. In an out-of-equilibrium regimewe generally expect the collective mode distribution tobe different from the distribution of the electron-holeexcitations. As shown in Ref. [11], the collective modedistribution coincides with the one for the electron-holeexcitations even out of equilibrium: B ω ( τ ) = R (cid:2) − F dε ( τ ) F rε − ω ( τ ) (cid:3) dε R (cid:2) F dε ( τ ) − F rε − ω ( τ ) (cid:3) dε . (9)Here function F d,rε ( τ ) is given in terms of the Wignertransform f d,rε ( τ ) of the electron distribution function f d,r ( t, t ′ ) inside the island/reservoir: F d,rε ( τ ) = 1 − f d,rε ( τ ), where a slow time τ = ( t + t ′ ) /
2. In the equi-librium F d,rε = tanh( ε/ T r ) and B ω = coth( ω/ T r ). Results. – We focus on the most interesting case ofCoulomb peak: the vicinity of a degeneracy point q = k + 1 / k is an integer. In this parametric regimethe transport is dominated by the two closest chargingstates [29] (see Fig. 2) which in the case of g = 0 areseparated by the Coulomb gap ∆ = 2 E c ( k + 1 / − q ).Due to the presence of the tunneling (finite g ) all the ob-servables, e.g., ∆, undergo strong renormalization nearthe Coulomb peak.For not very high frequencies Ω ≪ max {| ¯∆ | , T r , ε d } we obtained the following expression for admittance ofthe SEB G Ω = C g C Z ¯ g π ¯∆ ∂ ¯∆ B − ¯∆ B − ¯∆ i Ω − i Ω − ¯ g ¯∆ B − ¯∆ π . (10) Here the scaling parameter Z is defined as Z ( λ ) = (cid:16) g π λ (cid:17) − / , λ = Z B ω ω dω, (11)and ¯ g, ¯∆ are renormalized tunneling conductance andCoulomb gap respectively:¯ g = gZ ( λ ) , ¯∆ = ∆ Z ( λ ) . (12)The integral in Eq. (11) runs over frequencies E c ≫| ω | ≫ ω = max { T r , ε d , | ¯∆ |} . The energy scale ω de-termines the natural scale at which the RG procedurehas to be stopped [11]. Within logarithmic accuracy wefind λ = ln E c /ω .We stress that our result (10) is valid for an arbitraryelectron distribution. To make predictions more con-crete we consider the case of quasi-equilibrium F dε =tanh ε/ T d , T d > T r as an example. This regime is typ-ical for a SEB with the metallic island. It is achievedwhen the energy relaxation rate due to electron-electroninteraction in the island 1 /τ ee ≫ gδ (see e.g., [11]).The real part of admittance (10) at fixed Ω as a func-tion of q is shown in Fig. 3 for the out-of-equilibriumregime with T d > T r . At fixed C g , C and g the height ofthe maximum is controled by the effective temperatureof electron-hole excitations T eh = lim ¯∆ → ( ¯∆ / B ¯∆ [30].As it was shown, T r T eh T d and T eh ≈ T d ln 2for T d ≫ T r [11]. Therefore, out-of-equilibrium ad-mittance is confined within the boundaries Re G Ω ,T d < Re G Ω < Re G Ω ,T r , where G Ω ,T d ( G Ω ,T r ) are equilibriumadmittances at temperatures T d ( T r ).The dissipative part of the admittance in a SEB hasbeen addressed experimentally via radio-frequency re-flectometry measurements. The device was exposed toa continuous rf-signal [22]. In the experiment the tun-neling conductance was estimated to be equal g = 0 . q T r = 0.08 E c , T d = 0.12 E c k k + 1/2 k + 1Re G [ a . u ] T r = T d = 0.08 E c T r = T d = 0.12 E c Fig. 3 The real part of admittance of the SEB at fixedΩ as a function of q . We use g = 0 .
5, Ω = 0 . E c and C g /C = 0 .
24. See text. q k + 1 k + 1/2 k Re G [ a . u ] C g / C = 0.24, T r = T d = 0.08 E c C g / C = 0.32, T r = T d = 0.067 E c T r = 0.054 E c , T d = 0.08 E c C g / C = 0.31, Fig. 4 The dissipative part of admittance of the SEB atfixed Ω as a function of q . Three curves correspondingto three different formulae are presented. Dashed linecorresponds to Eq. (10) with Z = 1, ¯∆ = ∆, ¯ g = g and B ¯∆ = coth ∆ / T r . Dotted line is plotted according toEq. (10) with B ¯∆ = coth ¯∆ / T r . Solid line correspondsto Eq. (10) with non-equilibrium B ¯∆ given by Eq. (9).We use g = 0 . . E c . See text. such that the SEB was in the strong Coulomb blockaderegime. We plot the real part of the admittance (10) atfixed Ω as a function of q in Fig. 4. There, for a sake ofcomparison, we present Re G Ω computed i) in the equi-librium without taking into account the renormalizationeffects, i.e., with Z = 1 and B ¯∆ = coth ∆ / T r (dashedline); ii) in the equilibrium and with the renormaliza-tion effects, i.e., with B ¯∆ = coth ¯∆ / T r (dotted line);iii) out of equilibrium with T d > T r and B ¯∆ determinedby Eq. (9) (solid line). In all three cases, we use thesame values of g , E c and Ω corresponding to the experi-ment [22]. As one can see from Fig. 4, a slight variationof ratios C g /C and T r,d /E c allows us to make curvesfor cases i), ii) and iii) indistinguishable. In Ref. [22]it is assumed that the electrons inside the island arein the equilibrium with the reservoir and the renormal-ization effects are not important (case i) above). Val-ues of C g /C and T r /E c are used as fitting parameters.The curves presented in Fig. 4 however demonstrate amore subtle picture. As one can see, the successful fit-ting of the experimental data by a ‘theoretical’ curvegives yet no confidence that these assumptions are sat-isfied. Therefore, more careful analysis of the experi-mental data of Ref. [22] is needed.The electron-hole distribution B ω enters admittancein a twofold way. The analytical structure of admit-tance as a function of external frequency Ω is entirelydetermined by B ω at ω = − ¯∆. The scaling parameter Z arising from the renormalization however contains in-formation on B ω in wide domain ω < | ω | < E c . Ad-mittance (10) can serve as a tool for direct experimental measurement of B ω . As such can be the measurement ofa real part of admittance Re G Ω at two different drivingfrequencies. Other possibility would be the simultane-ous measurement of the real and imaginary parts of G Ω at a given frequency [19]. Then one can read out ¯∆ B − ¯∆ in the entire span of ¯∆ by tuning the DC gate voltage U . Measurements of frequency dependence of Re G Ω atthe Coulomb peak ( ¯∆ = 0) provide an access to the ef-fective bosonic temperature T eh . Thus the admittanceof a SEB under AC gate voltage can be used as the ther-mometer for the electron-hole excitations similar to theCoulomb blockade thermometer based on measurementsof the differential DC conductance in a single electrontransistor (SET) [12].The real part of admittance determines energy dissi-pation rate: W Ω = ( C g / C )Re G Ω | U Ω | . At quasi-staticregime Ω → W Ω = Ω C g R q | U Ω | , R q = he g ′ , C g = ∂q ′ ∂U . (13)The charge relaxation resistance R q and the renormal-ized gate capacitance C g are related to physical observ-ables formally defined as [31] g ′ = 4 π Im ∂K R ( ω ) ∂ω , q ′ = Q + Re ∂K R ( ω ) ∂ω . (14)Here Q = h ˆ n d i is the average charge in the island, thecorrelation function K R ( t ) = iθ ( t ) h [ X ( t ) , X † (0)] i andthe limit ω → g ′ thencoincides with the SET conductance [32,33]. The quan-tity q ′ is specific to Coulomb blockade physics and canbe addressed as the quasi-particle charge [31].With the help of definitions (14) we obtained thefollowing results in the out-of-equilibrium regime (for g ≪ g ′ = −
12 ¯ g ¯∆ ∂ ¯∆ ln B − ¯∆ , q ′ = k + 12 + 12 1 B − ¯∆ . (15)Equations (15) generalize the results for g ′ [32, 33] and q ′ [31] derived under the equilibrium conditions. Derivation. – Below we describe the main steps of thederivation. Further details will be given in [34]. Follow-ing Ref. [29], we write the Hamiltonian (2) in the trun-cated Hilbert space of electrons on the island account-ing for two charging states: with Q = k and Q = k + 1 ut-of-Equilibrium Admittance of Single Electron Box Under Strong Coulomb Blockade × H = H + H t + ∆ S z + ∆ / E c (16)where H t = X k,α t kα a † k d α S − + H.c. (17)and S z , S ± = S x ± iS y are ordinary (iso)spin 1 / Rs ( t ) = iθ ( t ) h [ S z ( t ) , S z (0)] i [24]: G Ω = − i Ω C g Π Rs (Ω) /C. (18)To deal with spin operators out of equilibrium the gen-eralization of Abrikosov’s pseudo-fermions (PF) ψ † α , ψ α is used [35, 36]. Integrating out electrons in the limit N ch ≫
1, we arrive at the following effective action [11] S = Z dt ¯ ψ (cid:16) i∂ t − σ z ∆2 + η (cid:17) ψ + g Z ¯ ψ ( t ) γ i σ − ψ ( t ) × Π ij ( t, t ′ ) ¯ ψ ( t ′ ) γ j σ + ψ ( t ′ ) dtdt ′ . (19)Here the pseudo-fermion fields ψ , ¯ ψ are understood asvectors in the tensor product of isospin and Keldyshspaces. We inserted the factor exp( η ¯ ψψ ) with η → −∞ into the density matrix in order to fulfill the constraint¯ ψ ( t ) ψ ( t ) = 1. The matrices σ z , σ ± = ( σ x ± iσ y ) /
2, and γ ≡ τ x , γ ≡ τ are the Pauli matrices in (iso)spin andKeldysh spaces respectively. Π ij stands for the matrix:Π = A Π R Π K ! , (20)Π R,A,K ( t, t ′ ) = Z dω π Π R,A,Kω ( τ ) e − iω ( t − t ′ ) , (21)Π R,Aω ( τ ) = ∓ i Z (cid:2) F dε ( τ ) − F rε − ω ( τ ) (cid:3) dε π , (22)Π Kω ( τ ) = 2 i Z (1 − F dε ( τ ) F rε − ω ( τ )) dε π . (23)The PF dynamical spin susceptibility is given as [34]:Π Rs,pf ( ω ) = Z X σ Z ( Γ RKRσ ( ε + ω, ε, ω ) G Rσ,ε + ω G Rσ,ε + Γ RARσ ( ε + ω, ε, ω ) h G Rσ,ε + ω G Kσ,ε + G Kσ,ε + ω G Aσ,ε i + Γ KARσ ( ε + ω, ε, ω ) G Aσ,ε + ω G Aσ,ε ) dε πi , (24)where the renormalized Green’s function [24] G R,Aσ,ε = Z ( λ ) ε − ¯ ξ σ ± i ¯ g Γ σ ( ε ) , ¯ ξ σ = − η + σ ¯∆ / , Γ σ ( ε ) = 18 π ( ε − ¯ ξ − σ )[ F − σ ¯ ξ − σ + B ε − ¯ ξ − σ ] . (25) The pseudo-fermion distribution F σε is not known a pri-ori. It is to be determined self-consistently from corre-sponding kinetic equation. It obeys [11]: F σε = B − σ ( ε + ∆ σ + η ) F − σ ¯ ξ − σ − σB − σ ( ε + ∆ σ + η ) − σ F − σ ¯ ξ − σ . (26)As was shown in [24] all terms of G R G R and G A G A type are controlled by renormalization scheme and canbe discarded. Then, Eq. (24) becomes simplified:Π Rs,pf ( ω ) = Z X σ ∂ ¯ ξ σ F σ ¯ ξ σ h − ω Γ RARσ ( ¯ ξ σ + ω, ¯ ξ σ , ω ) ω + 2 i ¯ g Γ σ i (27)where Γ σ = Γ σ ( ¯ ξ σ ). The vertex function Γ RAR solvesthe following Dyson equation Γ RARσ ( ε + ω, ε, ω ) = 1 + ig Z dx π G R − σ,ε + ω + x G A − σ,ε + x × Im Π Rx ( B x − σ ) Γ RAR − σ ( ε + ω + x, ε + x, ω ) . (28)By using Eqs (26)-(27) and the solution of Eq. (28): Γ RARσ ( ¯ ξ σ + ω, ¯ ξ σ , ω ) ω + 2 i ¯ g Γ σ = 1 ω ω + 2 i ¯ g (Γ − σ − Γ σ ) ω + 2 i ¯ g (Γ − σ + Γ σ ) , (29)we obtain expression (10) for the admittance.The computation of q ′ and g ′ is entangled with thecomputation of K Rω (see Eq. (14)). Using the definitionof K R ( t ) in terms of the operators X ( t ), one can obtainthe following expression [34]: K Rω = − g π Z dω ′ π h i Im D ω ′ ( B ω ′ − B ω ′ − ω )+Re D ω ′ B ω ′ − ω i Z ( F dε + ω ′ − ω − F rε ) dε. (30)Here we introduce the transverse spin susceptibility D R ( t ) = iθ ( t ) h [ S − ( t ) , S + (0)] i . Following Eq. (14) onestraight forwardly establishes: g ′ = g Z dω π Im D Rω ω∂ ω B ω , (31) q ′ = Q + g π Z dω π Re D Rω ∂ ω ( ωB ω ) . (32)The average charge in the island is given in terms of theaverage isospin as: Q = k + 1 / − h S z i . Using the resultfor the transverse spin susceptibility [11]: D Rω = 1 B − ¯∆ Z ( λ ) ω + ¯∆ + i + , (33)we obtain results (15) for g ′ and q ′ . In summary, the paper addresses the admittance andenergy dissipation in an out-of-equlibrium single elec-tron box under strong Coulomb blockade ( g ≪ ≪ max { T r , ε d , | ¯∆ |} .We found that the energy dissipation rate retains itsuniversal appearance in the quasi-stationary limit evenout of equilibrium. It is achieved in terms of speciallychosen physical observables: the charge relaxation resis-tance and the renormalized gate capacitance. We pro-pose the admittance as a tool for a measurement of theeffective bosonic distribution corresponding to electron-hole excitations in the system.The authors are grateful to A. Ioselevich, Yu.Makhlin, and J. Pekola for stimulating discussions. Theresearch was funded in part by the Russian Ministry ofEducation and Science under Contract No. P926, theCouncil for Grant of the President of Russian FederationGrant No. MK-125.2009.2, RFBR Grants No. 09-02-92474-MHKC and RAS Programs “Quantum Physicsof Condensed Matter” and “Fundamentals of nanotech-nology and nanomaterials”.
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