Thermopower for a molecule with vibrational degrees of freedom
TThermopower for a molecule with vibrationaldegrees of freedom
Sergei Kruchinin and Thomas Pruschke E-mail: [email protected] Bogolyubov Institute for Theoretical Physics,NASU, Kiev 03143, Ukraine Institut f¨ur Theoretische Physik, Georg-August-Universit¨at G¨ottingen,Friedrich-Hund-Platz 1, 37077 G¨ottingen, Germany
Abstract.
We propose a simple model to study resonant tunneling through anorganic molecule between to conducting leads, taking into account the vibrationalmodes of the molecule. We solve the model approximately analytically in the weakcoupling limit and give explicit expressions for the thermopower and Seebeckcoefficient. The behavior of these two quantities is studied as function of modelparameters and temperature. For a certain regime of parameters a rather peculiarvariation of the thermopower and Seebeck coefficient is observed.Although the model is very simple, we expect it to give some nontrivial insightinto thermal transport properties through nan-devices. Furthermore, because wecan provide an analytical solution, it may eventually serve as benchmark for moreadvanced analytical or computational methods.PACS numbers:
Keywords : Nano devices, thermopower, bose-fermi model
1. Introduction
In a recent experiment, Reddy et al. [1] studied transport through an organic molecule(benzenedithiol (BDT)) attached to a scanning-tunneling-microscop (STM) tip anda substrate, generated by a break junction using the STM tip. The authors wereable to measure different transport properties, including the thermopower respectivelySeebeck coefficient of this setup, which is is shown schematically in Fig. 1. Quitegenerally, the idea to contact organic molecules to leads in a controlled fashion is ratherfashionable and - if successful - can lead to a variety of interesting applications, rangingfrom nanoscale computing to quantum computing [2–4]. First progress of reproduciblyand controlled contacting organic molecules has been made using carbon nanotubes[5–8], but similar attempts with smaller molecules usually suffer from several problems,among them the actual question of how the contact is established and how one cancontrol its quality [9–11].It is well known, that the physics of nanoobjects coupled to conducting leads ishighly nontrivial (for an early review see for example [12], more recent aspects arediscussed in [13]), and already concentrating on electronic degrees of freedom onlypresents an extreme challenge to both experiment and especially theory. The latter isdue to the fact that correlations on these nanostructures can usually not be ignored a r X i v : . [ c ond - m a t . m e s - h a ll ] A ug hermopower for a molecule with vibrational degrees of freedom BDT Molecule
SH SH M e t a l s ub s t r a t e S T M T i p Figure 1.
Schematic experimental setup according to [1]. The STM tip isconnected to a reservior which maintains a constant tip temperature. Thesubstrate, on the other hand, can be heated, thus creating a temperature gradientacross the molecule. and hence one is plagued with complicated quantum impurity problems [14, 15], whichcan be solved only numerically and only in equilibrium and for rather restricted modelsetups. The situation becomes even worse if one wants to take into account vibrationalmodes present in complex molecules [16–19]. Here, even advanced numerical toolsrather quickly meet their limits [19].And last, but by no mean least, the conventional experimental probes used innano-systems are transport properties, in particular by applying a bias voltage or, asin the above-mentioned experiment, a temperature difference across the nano-object.In this situation, the assumption of thermal equilibrium is quickly questionable, andreliable tools to study correlated systems out of thermal equilibrium – even only thestationary state – are not yet existing.Therefore, setting up simple models which preferably are solvable analytically andwhich allow to study the physical properties also off equilibrium in the stationary state,are still a highly relevant approach to obtain an at least qualitative feeling how variousdegrees of freedom influence the transport properties of nano-objects. Particularlythermal transport and here thermopower or Seebeck coefficients are an interestingarea because possible applications in the area of energy conversion or storage havegained strong renewed interest over the past decade. However, in this area relativelylittle theoretical work on thermopower for mesoscopic systems exists to our knowledgeso far [20].The situation in an experimental setup like the one shown in Fig. 1 has theadvantage, that one can, for example by stretching the bond through removing thetip, control the coupling between the leads and the molecule very sensitively andactually shift the system into the weak coupling regime. Here, theoretical calculationsbecome much easier. For our purpose, we will therefore assume that the system isin this regime, and suggest a very simple model which will allow us to see how theinterplay between electronic and vibrational degrees of freedom influence transportproperties.The model setup and general expressions for thermopower and possible situationswill be presented in the next section. The central result, viz analytical expressions forthe thermopower and Seebeck coefficient will be derivedin section 3, and a summarywill conclude the paper. hermopower for a molecule with vibrational degrees of freedom
2. Setup and model
We study an organic molecule attached to two metallic leads. The leads are assumed tobe in thermal equilibrium but can have different temperatures and chemical potentials.We will take the temperature and chemical potential of the right leads as referenceand write T R = T , T L = T + ∆ T respectively µ R = µ , µ L = µ + ∆ µ . In the following,we use µ = 0 as reference of energy.Using first-principle approaches, such a setup has been studied before in a quitegeneral way [21], concentrating however on the local properties of the molecule.Concentrating on a subset of degrees of freedom allows the use of more advancedanalytical and numerical tools and has been applied to several model situations relatedto experiments mentioned in the introduction [18, 19] (see also further references in[19]).Here, we intend to investigate transport properties, and how local vibration modesinfluence them. Calculating transport for a general situation becomes rather quickly acumbersome task. Therefore, in order to be able to gain some insight into the physicsof the problem, while at the same time having as little parameters as possible, weconcentrate her on a very simple model. Furthermore, restricting the study to a bare-bone model also allows for an explicit analytical solution. Such a solution is rathervaluable in several respect, for example as possible benchmark for more advancedmethods.To this end we assume that the HOMO of the molecule has a certain energy (cid:15) below the chemical potential of the leads, but the LUMO well above it, so that we ε BDT Molecule S ub s t r a t e S T M t i pL e ft B a rr i e r R i gh t B a rr i e r t L t R T+ Δ T T
Figure 2.
Schematic setup of the model. We consider only one relevant orbitalof the BDT molecule, which we assume to be the HOMO. The coupling betweenthe substrate respectively the STM tip are modeled as tunneling barriers. can neglect it. Also, all other electronic states of the molecule are assumed to be wellseparated, too. The model motivated above is shown schematically in Fig. 2, wherewe also explicitly included the various model parameters.Up to now the model is the standard setup to describe resonant transport througha mesoscopic system like molecules or quantum dots. Here, however, we also allow for hermopower for a molecule with vibrational degrees of freedom ω . This is for example permissible whenone studies low temperature, where only the lowest-lying modes play a role anyway.Finally, the molecule is coupled to the leads via tunneling barriers.Thus, the Hamiltonian for our model to describe the experimental setup fromFig. 1 reads H = H L + H M + H L − M (1) H L = (cid:88) k ,α = L,R ε α, k c † α, k c α, k (2) H M = (cid:15) d † d + ω b † b + λω d † d (cid:0) b † + b (cid:1) (3) H L − M = (cid:88) α = L,R (cid:16) t α c † α,B d + h . c . (cid:17) (4)where we use standard second quantized notation, λ denotes the dimensionlesscoupling between the electronic and phononic modes, and t α the tunneling matrixelements to the leads. The operator c ( † ) α,B denotes the projection of the band states tothe barrier.Note that we did not include the spin degrees of freedom of the electrons in themodel (1). Although this is at first glance a rather crude approximation, we want toemploy it nevertheless for two reasons. First, including the spin makes already themolecular problem highly nontrivial and would prevent the semi-analytical solutionwe will discuss in the section about the results. Second, we are presently aiming at amore qualitative insight into the influence of vibrational modes on transport effects.To this end we believe that the additional complications due to additional electronicdegrees of freedom will surely alter the results quantitatively, but possibly not stronglyom a qualitative level. Nevertheless, calculations based on more realistic models arein progress.For our simple setup we can use the Meir-Wingreen formula [22] to calculate theelectric current as J = J (cid:90) dω [ f L ( ω ) − f R ( ω )] ρ M ( ω, ∆ µ, ∆ T ) (5)where ρ M ( ω, ∆ µ, ∆ T ) denotes the electronic density of states on the molecule in thepresence of a potential difference ∆ V and a temperature difference ∆ T between theleads, and f α ( ω ) = 11 + exp (cid:16) ω − µ α k B T α (cid:17) the Fermi functions of the left and right lead, respectively. The quantity J finallycollects the coupling parameters and the natural constants like charge and Planck’sconstant.Calculating the DOS in the presence of the leads for arbitrary coupling t α , ∆ µ and∆ T is not possible. However, in the case t α → ρ M ( ω, ∆ µ, ∆ T ) by the result for the isolated molecule, which reads [23] ρ M ( ω, ∆ µ, ∆ T ) = 2 πe λ ∞ (cid:88) l = −∞ λ | l | | l | ! δ ( ω − (cid:15) − ( l + λ ) ω ) (6) hermopower for a molecule with vibrational degrees of freedom µ and ∆ T explicitely. When we insert expression (6) intothe Meir-Wingreen formula (5), the expression for the current becomes J = 2 πJ e λ ∞ (cid:88) l = −∞ g | l | | l | ! [ f L ( (cid:15) + ( l + λ ) ω ) − f R ( (cid:15) + ( l + λ ) ω )]Since we are interested in the thermopower, the current has to be zero, i.e. for a giventemperature difference ∆ T between left and right lead we have to adjust ∆ µ such that ∞ (cid:88) l = −∞ λ | l | | l | ! [ f L ( (cid:15) + ( l + λ ) ω ) − f R ( (cid:15) + ( l + λ ) ω )] = 0 (7)In general, we are interested in low temperatures, i.e. typically we will have | (cid:15) | , ω (cid:29) k B T . In this case, only a certain range of values of l ∈ [ l min , l max ], which havethe property that (cid:15) + ( l + λ ) ω = O ( k B T ), will actually be important, because forthe other terms the Fermi functions are either both zero or both one, i.e. cancel in thedifference. This situation is schematically depicted in Fig. 3a. m m Dm w (a) m m Dm w (b) Figure 3.
Schematic representation of the different situations arising fromEq. (7). The left and right boxes denote the left and right leads, with the Fermifunctions as full lines. In the left lead, we included some ∆ µ and ∆ T . In (a)there are several polaronic modes which lie in the Fermi window, while in (b) thetemperature is much smaller than the level splitting, resulting in effectively onlytwo modes within the Fermi window. Let us take this idea to the extreme and assume, that T is low enough suchthat only one vibrational mode with index l will contribute, see the schematicrepresentation in Fig. 3b. The necessary conditions for transport in this case are (cid:15) + ( l + λ ) ω < ≤ ˜ (cid:15) := (cid:15) + ( l + 1 + λ ) ω < ω from which l can be determined. The equation (7) then simplifies to11 + exp (cid:16) ˜ (cid:15) − ω − ∆ µk B ( T +∆ T ) (cid:17) −
11 + exp (cid:16) ˜ (cid:15) − ω k B T (cid:17) + λl + 1
11 + exp (cid:16) ˜ (cid:15) − ∆ µk B ( T +∆ T ) (cid:17) −
11 + exp (cid:16) ˜ (cid:15) k B T (cid:17) = 0 (8) hermopower for a molecule with vibrational degrees of freedom µ for given model parameters and ∆ T .Although this is strictly speaking not the case for the experimental setup in Fig. 1, wewant to note that in nano devices (cid:15) can often be controlled through a gate voltage, (cid:15) = − eV G , and consequently the active polaronic mode respectively l is controllableexperimentally.
3. Results
Without vibrational modes, the thermoelectric response of the model (1) has beenstudied before by Ermakov et al. [20] Within their model, the authors observed a linearrelation between the temperature difference and thermovoltage. In the following wewant to try to understand what modifications arise through the presence of polaronicmodes in the molecule.In the present paper we want to concentrate on the situation shown in Fig. 3b.Here we can use the fact that ω (cid:29) k B T and try to solve equation (8) approximatelyanalytically and provide explicit expressions for ∆ µ and the Seebeck coefficient asfunction of model parameters and temperature.The limiting cases ˜ (cid:15) → (cid:15) → ω turn out to be uninteresting, because inthe first case the combination f L (˜ (cid:15) − ω ) − f R (˜ (cid:15) − ω ) vanishes, while in the second f L (˜ (cid:15) ) − f R (˜ (cid:15) ) →
0. In both cases one thus recovers the solution by Ermakov etal.[20] in the limit when the level position is close to the chemical potential.The more interesting case therefore is ˜ (cid:15) finite and reasonably far away fromthe chemical potential. Then we can furthermore assume ˜ (cid:15) (cid:29) k B T as well as | ˜ (cid:15) − ω | (cid:29) k B T , and replace the Fermi functions by f ( x ) ≈ e − βx , if x >
0, respectively f ( x ) ≈ − e βx , if x <
0. Inserting these approximations into Eq. (8) one obtains aquadratic equation for e ∆ µk B( T +∆ T ) . From the two solutions one must pick the one thathas the correct limit ∆ µ = ( ω − (cid:15) ) ∆ TT as λ →
0. The solution fulfilling this requirement reads∆ µ ≈ ( T + ∆ T ) ln e − ˜ (cid:15) k B T λl +1 (cid:34) e ˜ (cid:15) k B( T +∆ T ) (cid:18) λl + 1 − e (cid:15) − ω k B T (cid:19) + (cid:115) e (cid:15) k B( T +∆ T ) (cid:18) e (cid:15) − ω k B T − λl + 1 (cid:19) + 4 λl + 1 e (cid:15) − ω k B( T +∆ T ) + (cid:15) k B T (cid:35) (9)Since this expression is rather hard to interpret, we inspect it behavior for small λ .Taylor expanding it to first order results in∆ µ ≈ ( ω − ˜ (cid:15) ) ∆ TT − λ ( T + ∆ T ) l + 1 e ω − (cid:15) k B T (cid:104) e ω Tk B T ( T +∆ T ) − (cid:105) The first term is identical to the result by Ermakov et al. [20]. The correction isnegative, i.e. the vibrations tend to initially reduce the thermopower. Note that thestrenght of this effect depends on the relation between ˜ (cid:15) and ω : If ω − (cid:15) <
0, it willbe exponentially suppressed, while it is actually stronlgy enhanced for ω − (cid:15) > µ . As generic parameter set we chose ˜ (cid:15) = ω / hermopower for a molecule with vibrational degrees of freedom λ/ ( l + 1) = 1 /
40. Due to the exponential structure a numerical solution fails fortoo small T . Using multi-precision arithmetic, one can extend the range of solvabilityslightly. We checked the stability of the root finding algorithm by comparing theresults for ∆ µ at λ → T ≥ · − ω using a precision of 100 digits. The resultingfunction ∆ µ (∆ T ) for T = 5 · − ω , 10 − ω and ω / T we observe a linear -1 D T-0.100.10.20.30.4 Dm / w T=10 -1 w T=5×10 -1 w T=5×10 -2 w Figure 4. ∆ µ vs. T − ∆ T for various values of T . Other parameters see text.The dashed curves are the results from the approximate formula (9). dependence with negative coefficient, while at high T the sign of the thermopowerreverses. For intermediate T , however, one can see a rather peculiar sign changeof ∆ µ as function of ∆ T . Changing the model parameter does not influence thisqualitative behavior as long as ω − (cid:15) >
0. For example, decreasing ˜ (cid:15) moves thezero in ∆ µ towards smaller values of ∆ T , but also decreases the magnitude of theeffect. Increasing λ produces a similar behavior. However, when ω − (cid:15) < λ →
0, i.e. linear with slope (˜ (cid:15) − ω ) /T > µ (∆ T → S for the junction. The resultas function of temperature for the same model parameters used in Fig. 4 is shown in hermopower for a molecule with vibrational degrees of freedom e denotes the electric charge) S ≈ k B e k B T ˜ (cid:15) − ω λl +1 e ω − (cid:15) k B T (10)derived from the approximate formula (9) as dashed curve in Fig. 5. It apparentlydescribes the overall behavior quite well, becoming asymptotically exact for low T . B T/ w -505 S [ m V / K ] ~ e =w /8 ~ e =3w /4 Figure 5.
Seebeck coefficient S = − ∆ V ∆ T (cid:12)(cid:12)(cid:12) ∆ T → as function of T for twocharaceristic values of ˜ (cid:15) . Other parameters as in Fig. 4. The dashed curvesare the result from the approximate formula (10). Interestingly it also provides the correct asymptotic for high T . Note that the unit ofthe thermopower is, as usual, determined by the universal ratio k B /e and is hencenot model or parameter dependent. As already noted in the case of ∆ µ , we here, too,obtain a sensible dependence of the Seebeck coefficient on ˜ (cid:15) . When ω − (cid:15) >
0, itshows strong variation with T due to the denominator in Eq. (10), while for ω − (cid:15) < (cid:15) and hence ˜ (cid:15) experimentally, at least forelectrically gated molecules, the described effects and their switching could in principlebe observable.
4. Summary
A simple model for an organic molecule attached to two conducting leads via tunnelbarriers was introduced. Main emphasis was laid on the inclusion of vibration modesfor this molecule and to study their impact on transport properties, here especiallythe thermopower. We derived an analytical approximate formula for the thermopower hermopower for a molecule with vibrational degrees of freedom µ (∆ T ) and the Seebeck coefficient and showed that for a rather broad range of modelparameters the influence of the molecular oscillations can strongly influence these twoquantities. In particular, the thermopower can develop a pronounced extremum andsign changes as function of temperature gradient across the molecule, and the Seebeckcoefficient showed pronounced nonmomotonic behavior in this regime. When oneshifted the local electronic level of the molecule to larger values, one can observe atransition into a regime where the vibrational modes have negligible influence andthe system behaves similar to the case without any elastic degrees of freedom. Sincein some experimental setups the local parameters are accurately controllable via e.g.gate voltage or the like, we are certain that one can test these theoretical predictionsto some extent.Of course, the simplifications introduced to allow for an analytical solution arerather severe: Electron spin as well as additional molecular levels were neglected, andthe vibration modes were restricted to a single mode. Furthermore, for the calculationof the transport only two of the polaron levels were taken into account. The latterapproximation can more easily be relaxed, even within the proposed model (see Fig.3a). Similarly, including several vibration modes can be done straightforwardly (see forexample [23]), too, while taking into account more realistic electronic level structureswill actually require more refined analytical approaces [16] or a numerical solutionof the resulting model for the molecule [19]; and possibly also a reinspection of the.validity of the Meir-Wingreen formula for the current.In any case, adding these additional and physically important features to themodel of the molecule makes the problem harder to treat. At least as long as we stayin the weak-coupling limit, we can expect that one can handle them reasonably withinexisting numerical approaches, or even, up to the solution of the transport equation,to a large extent analytically. The actual problematic part is hidden in the solutionof the transport equation, even in weak coupling. Here, the exponential nature of theFermi functions renders the numeric root searching extremely awkward and severelylimits the accessible parameter and especially temperature regime (see e.g. discussionof Fig. 4). Finally, relaxing the condition of weak coupling to the leads leads to afull-scale many-body problem out of equilibrium, which at present cannot be reallyattacked with reliable analytical or numerical tools.Thus, to conclude, we believe that even such an extremely simple model will beof some importance to increase our understanding of properties of nano-systems, andcan eventually also serve as a benchmark for more advanced methods to be developedin the future. Acknowledgments
We would like to acknowledge financial support by the Deutsche Forschungsgemein-schaft through project PR 298/12 (T.P. and S.K.) and S.K. the SASIIU of the projectUkraine – Republic of Korea (S.K.). S.K. thanks the Institute for Theoretical Physicsof the University of G¨ottingen and T.P. the Bogolyubov Institute for TheoreticalPhysics of the NASU in Kiev for their support and hospitality during the respectivevisits.We also thank Dr. Ermakov and Dr. Zolotovsky for the fruitful discussions.
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