Mechanism for large thermoelectric power in negative-U molecular quantum dots
aa r X i v : . [ c ond - m a t . m e s - h a ll ] D ec Mechanism for large thermoelectric power in molecular quantum dots described bythe negative- U Anderson model
S. Andergassen , T. A. Costi , and V. Zlati´c , , Institut f¨ur Theorie der Statistischen Physik,RWTH Aachen University and JARA-Fundamentals of Information Technology,D-52056 Aachen, Germany Peter Gr¨unberg Institut and Institute for Advanced Simulation,Research Centre J¨ulich, D-52425 J¨ulich, Germany Institute of Physics, HR-10001 Zagreb, Croatia J. Stefan Institute, SI-1000 Ljubljana, Slovenia (Dated: August 30, 2018)We investigate with the aid of numerical renormalization group techniques the thermoelectricproperties of a molecular quantum dot described by the negative- U Anderson model. We show thatthe charge Kondo effect provides a mechanism for enhanced thermoelectric power via a correlation-induced asymmetry in the spectral function close to the Fermi level. We show that this effect resultsin a dramatic enhancement of the Kondo-induced peak in the thermopower of negative- U systemswith Seebeck coefficients exceeding 50 µV /K over a wide range of gate voltages. PACS numbers: 71.27.+a,72.15.Jf,72.15.Qm,73.63.Kv
Introduction.—
Thermoelectric devices currently usebulk materials, e.g., Si-Ge, PbTe or Bi Te . In the fu-ture, devices made of nanoscale objects, such as quantumdots or molecules, could offer alternatives, particularlyfor low-temperature applications, such as on-chip cool-ing of microprocessors or low-temperature refrigeration.Nanoscale objects have some potential advantages overtheir bulk counterparts, for example, in scalability or intheir high degree of tunability (e.g., via a gate voltage),allowing them to be operated at optimal thermoelectricefficiency. Molecular quantum dots, in particular, couldbe interesting to study, since a large variety of such sys-tems could be fabricated and investigated for interestingthermoelectric properties .The description of electrical and thermal transportthrough quantum dots is, however, a challenging theo-retical task. Electrons tunneling from the leads throughthe quasi-localized levels of the dot typically experiencea large Coulomb repulsion on the dot, giving rise to thespin Kondo effect . The latter profoundly affects trans-port, resulting, for example, in the lifting of Coulombblockade at low temperatures for a wide range of gatevoltages and an enhanced conductance close to the uni-tary limit, G ≈ G = 2 e /h , for symmetric coupling tothe leads . Recent experimental and theoretical workhas also addressed the effects of Kondo correlations onthe thermoelectric properties of such quantum dots .However, the Kondo-induced enhancement of the ther-mopower at the Kondo temperature T K was found to bevery small , suggesting that the spin Kondo effect, in itssimplest manifestation, is ineffective for realizing efficientthermoelectric devices.In this Rapid Communication we consider a molecu-lar quantum dot with an attractive onsite Coulomb in-teraction, U <
0, described by a negative- U Andersonimpurity model, Eq. (1) below. Such a model has beenused to explain the dielectric properties of amorphous semiconductors , to describe highly polarized heavyfermion states and to investigate the noise and non-equilibrium transport through negative- U molecules .For a molecular quantum dot, several mechanisms couldresult in U <
0, for example, screening by electronsin metallic leads can reduce an initially repulsive localCoulomb interaction to negative values , or, a vibratingmolecule with a local electron-phonon interaction couldresult in a net attractive Coulomb interaction . Fortypically used metallic electrodes, such as gold, screen-ing is expected to ensure the locality of the attractiveinteraction in Eq. (1).A negative- U quantum dot supports a charge Kondoeffect in which the role of spin-up and spin-down statesin the conventional spin Kondo effect are played by thenon-magnetic empty and doubly occupied states of thedot . As in the usual spin Kondo effect, this chargeKondo effect results in a renormalized Fermi liquid at lowtemperatures which has important consequences for elec-trical and thermal transport. It is also believed to be theorigin of superconductivity in PbTe doped with Tl, wherethe valence skipper Tl acts as a negative- U center .While some aspects of the electrical transport through anegative- U molecule have been investigated , the mostinteresting feature of such a system, elucidated below, liesin its remarkable low-temperature Kondo-induced ther-moelectric response which, to the best of our knowledge,has not been previously addressed. Model and calculations.—
Specifically, we consider aquantum dot described by the following two-lead Ander-son impurity model H = X σ ε d n dσ + U n d ↑ n d ↓ + X kασ ǫ kα c † kασ c kασ + X kασ ( t α c † kασ d σ + h.c. ) , (1)where, ε d is the energy of the molecular level, U < σ labels the spin, and α = L, R labels left and right electron lead states withkinetic energies ǫ kα . The couplings of the dot to the leadsare denoted by Γ α ( ω ) = 2 πρ α ( ω ) | t α | , where ρ α ( ω ) = P k δ ( ω − ǫ kα ) is the density of states of lead α .The linear response transport properties can be cal-culated from the single-particle spectral function of thedot A σ ( ω ) = − Im[ G dσ ( ω + iδ )] /π, where G dσ ( ω + iδ ) = hh d σ ; d † σ ii is the Fourier transform of the retarded single-particle Green function of (1). The thermopower is givenby S = − | e | T R dω ω T ( ω ) ( − ∂f /∂ω ) R dω T ( ω ) ( − ∂f /∂ω ) , (2)where f is the Fermi function, e is the electronic charge,and T ( ω ) = 2 π Γ( ω ) P σ A σ ( ω ) is the transmission func-tion of the dot with Γ( ω ) = Γ L ( ω )Γ R ( ω )Γ L ( ω )+Γ R ( ω ) . At low tem-perature, a Sommerfeld expansion leads to S ( T ) = − π k B | e | k B T (cid:18) Γ ′ ( ǫ F )Γ( ǫ F ) + P σ A ′ σ ( ǫ F ) P σ A σ ( ǫ F ) (cid:19) (3)where ǫ F = 0 is the Fermi level of the leads. In theabsence of a magnetic field A ↑ ( ω ) = A ↓ ( ω ) = A ( ω ) is spinindependent. A large thermopower at low temperaturecan be achieved by either tailoring the band structure ofthe leads to give a highly asymmetric Γ( ω ) at ǫ F witha large slope Γ ′ ( ǫ F ) (Ref. 23) or tailoring correlations toyield a highly asymmetric A ( ω ) at ǫ F with a large slope A ′ ( ǫ F ), or both. We concentrate on the latter which isrobust to details of the lead density of states, and assumea smooth Γ( ω ) around ǫ F , i.e., we take Γ( ω ) = Γ = 0 . A ( ω, T )is calculated by using the numerical renormalizationgroup (NRG) method . Results for U/ Γ = − −| e | V g = ( ε d + U/
2) in the range | V g | ≤
8Γ (setting e = 1). In addition, for T = 0, wehave compared results for occupation numbers n d withthose from functional renormalization group (fRG) andBethe ansatz techniques (see Fig. 3 below). In the fol-lowing, T K = p | U | Γ / e − π | U | / (Ref. 5) denotes the rel-evant low energy charge Kondo scale of (1). Due to theexponential dependence on U and Γ, T K can vary by or-ders of magnitude, e.g. for positive- U systems from 1 to200 K . For U = − T K = 2 . × − Γ ≪ Γ. Results.—
Figure 1 shows the T = 0 spectral func-tion for several gate voltages. At V g = 0 the pseudo-spinstates n d = 0 and n d = 2 are degenerate, and the spec-tral function is symmetric, with a Kondo resonance ofwidth O ( T K ) at ω = 0 and two Hubbard satellite peaksat ω = ε d > ω = ε d + U <
0. A finite gate voltage V g induces a splitting ∆ E = − V g of the pseudo-spinstates which is analogous to a magnetic field in the con-ventional spin Kondo effect, i.e. the spectral functionbecomes highly asymmetric due to the polarizing effectof V g , with n d changing substantially from its “perfectlyscreened” value of n d = 1 . This asymmetry in the -4 -2 0 2 4 ω / Γ A ( ω , T = ) -20 0 20 ω /T K A ( ω , T = ) -4 0 4 ω / Γ A ( ω , T ) V g /T K =V g /T K = 1 T/T K =(a) (b) FIG. 1: (Color online)
Main panel: T = 0 spectral functionfor U/ Γ = − V g /T K . Inset (a): A ( ω, T = 0) near ω = 0. Inset (b): Temperature dependenceof A ( ω, T ) for V g /T K = 1. -2 -1 T/T K S ( µ V / K ) V g /T K =0.1V g /T K =1V g /T K =10V g /T K =10010 T/T K -50050 S ( µ V / K ) U/ Γ =-8, V g /T K =10U/ Γ =+8, V g /T K =10x 100 FIG. 2: (Color online)
Thermopower S vs temperature atdifferent gate voltages V g /T K and U/ Γ = −
8. Inset: Com-parison with
U > V g /T K = 10. single-particle spectral function with a large slope at ǫ F ,for both spin components, is the origin of the large ther-mopowers to be discussed below. The analogy to the spinKondo effect in a magnetic field, can be made precise forthe case of particle-hole symmetric bands which we con-sider: A particle-hole transformation on the down spinsallows the negative- U Anderson model in the absence ofa local magnetic field to be mapped onto the positive- U symmetric Anderson model in a finite local magnetic field B = 2 ǫ d + | U | = − V g , thereby explaining the highlyasymmetric spectral function of (1) shown in Fig. 1. Thepolarizing effect of finite V g ∼ B is strongest at T = 0and diminishes for T ≫ T K [see Fig. 1(b)]. In terms ofthe above analogy, this corresponds to the quenching ofthe magnetization M = ( n d ↑ − n d ↓ ) / U model in a field B .Figure 2 shows the main result of this Rapid Commu-nication: a dramatic enhancement of the Seebeck coeffi-cient induced by a finite gate voltage V g & T K exceed-ing 50 µV /K for V g & T K . The maximum in the ther-mopower occurs on a temperature scale which correlateswith V g and is therefore highly tunable. CorrespondingSeebeck coefficients for U > S is due to the correlation-induced asymmetryin the spectral function at finite V g . At low tempera-tures, explicit calculations, within Fermi-liquid theory ,also shed light on this enhancement. In this limit, thethermopower may be expressed in terms of the occupancy n d of the dot as S ( T ) = − πγT | e | cot( πn d /
2) (4)with γT ≪
1, and γ being the linear coefficient of spe-cific heat of the dot (with γ ∼ /T K for V g = 0). Afinite V g ∼ T K polarizes the charge Kondo state, lead-ing to n d ∼ V g >
0. This enhances a nomi-nally small ( ≪ k B / | e | ) thermopower by the large factorcot( πn d / ≫
1. Note also, that while a finite mag-netic field for
U > A ↑ ( ω ) and A ↓ ( ω ) around ǫ F , the asymmetryin the Kondo regime is opposite for spin up and spindown. Consequently, it largely cancels in the combina-tion P σ A σ entering (2) and the thermopower is not en-hanced. [Furthermore, for n d ≃
1, the factor cot( πn d / S ( T ) of the negative- U model in Fig. 2 does not exhibit a sign change with in-creasing temperature for any finite V g , in contrast to thecase of U > : The sign change of the latter is due toa change in slope of the spectral function at the Fermilevel, induced by a collapse of the Kondo resonance withincreasing temperature. This cannot occur in the U < V g at all relevant temperatures.The gate voltage dependence of the thermopower andelectrical conductance is shown in Fig. 3(a) and 3(b) atseveral temperatures. Except at T . T K , a large Seebeckcoefficient exceeding 50 µV /K can always be realized bya suitable choice of gate voltage. By tuning the gatevoltage to positive or negative values about the chargeKondo state at V g = 0 one can realize the p -type or n -type legs of a thermoelectric device. Note the absenceof a Kondo plateau in G ( V g ) at T ≪ T K in Fig. 3(b),which contrasts with the U > V g ∼ T K of G ( V g ) due to the suppression ofthe Kondo state by the finite gate voltage acting as amagnetic field in the conventional Kondo effect .NRG results for n d ( T = 0) vs V g compare very well withfRG calculations at U/ Γ = − , − U/ Γ = − G ( T ) exhibits the typical Kondoscaling behavior at small gate voltages V g . T K . -60-202060 S ( µ V / K ) -20 -15 -10 -5 0 5 10 15 20 V g /T K G ( / h ) -2 T/T K G ( / h ) -4 -2 -V g / Γ n d V g /T K =(d)(c)(a)(b) T/T k = -U/ Γ = FIG. 3: (Color online)
Gate voltage dependence of ther-mopower S (a) and conductance G (b) at typical temperatures T /T K . Inset (c): Dot occupation number n d vs gate voltagefor U/ Γ = − , − , −
8. fRG results for U/ Γ = − , − U/ Γ = − G at selected gate voltages V g /T K . -2 -1 T/T K S G / G ( k B / e ) -1 T/T K -3 -2 -1 ZT T K Γ V g /T K = FIG. 4: (Color online)
The power factor S G divided by G =2 e /h vs temperature and for a range of gate voltages V g /T K for U/ Γ = −
8. The locations of T K and Γ are indicated byarrows. Inset: Upper and lower bounds for ZT , as defined inthe text, at several gate voltages. The thermoelectric efficiency of a nanoscale device isrelated to its dimensionless figure of merit defined by ZT = P T /K , where P = S G is the power factor, and K = K e + K ph the thermal conductance due to elec-trons (e) and phonons (ph). For metallic leads, K e willgive the dominant contribution to K , while for semi-conducting leads, K ph will also be important. Since, tothe best of our knowledge, no calculation of K ph in thepresence of Kondo correlations is available, we discussthe efficiency of our system in terms of the power factor P V g ( T ), shown in Fig. 4, and give upper and lower boundestimates for ZT below. The power factor is largely inde-pendent of details of the leads, making it a useful quan-tity for future comparison with experiments. It is also arelevant quantity for on-chip cooling of a hot source inmicroelectronics . For each V g the power factor exhibitsa maximum at a temperature which is related to V g . Theenvelope of these curves has two maxima, one at T ≈ T K for V g ≈ T K and another at high temperatures T ≈ V g ∼ ≫ T K . In contrast, for U >
0, the power fac-tor is vanishingly small in the Kondo regime, with largervalues being obtained only at the border between mixedvalence and Kondo regimes . Turning to ZT , an up-per bound estimate is obtained by setting K ph = 0. Alower bound estimate is obtained by assuming that themolecule is transparent to phonons. In this case, eachphonon mode contributes the maximum ballistic ther-mal conductance of κ = π k B T / h . For three phononmodes we have K ph = 3 κ resulting in a lower bound es-timate for ZT . Both bounds (see the inset of Fig. 4) showa maximum at a temperature T that correlates with V g ,with the upper bound exceeding 1 for V g /T K ≫ T /T K ≫
1. In a real device, phonons will be inelasti-cally scattered, e.g. by vibrational modes of the molecule,thereby reducing K ph below its ballistic value, especiallyat higher temperatures where anharmonic effects becomeimportant. Hence, our lower bound for ZT is likely toostringent so that a suitable choice of gate voltage couldallow interesting values of ZT ∼ . − T ∼ T K . Conclusions.—
In summary, we investigated the ther-moelectric properties of a negative- U molecular quantumdot exhibiting the charge Kondo effect. A small gatevoltage V g & T K is found to polarize the charge on thedot creating a single-particle spectral function which ishighly asymmetric about the Fermi level. This yieldsa large enhancement of the Seebeck coefficient exceed-ing 50 µV /K on a temperature scale comparable to V g .The device is highly tunable and allows large power fac-tors to be achieved at virtually any temperature by asuitable choice of gate voltage. In addition to the above-mentioned possible realizations of such devices, molecularcomplexes similar to those in Ref. 27, but with valenceskipping ions such as Bi, Tl or In, acting as negative- U centers, and attached to gold leads, could be promisingsystems to look into in the future. Reducing the dimen-sionality of the leads, e.g. by using carbon nanotubes ,could further enhance the power factor . Acknowledgments
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