Gate-modulated thermopower in disordered nanowires: I. Low temperature coherent regime

Using a one-dimensional tight-binding Anderson model, we study a disordered nanowire in the presence of an external gate which can be used for depleting its carrier density (field effect transistor device configuration). In this first paper, we consider the low temperature coherent regime where the electron transmission through the nanowire remains elastic. In the limit where the nanowire length exceeds the electron localization length, we derive three analytical expressions for the typical value of the thermopower as a function of the gate potential, in the cases where the electron transport takes place (i) inside the impurity band of the nanowire, (ii) around its band edges and eventually (iii) outside its band. We obtain a very large enhancement of the typical thermopower at the band edges, while the sample to sample fluctuations around the typical value exhibit a sharp crossover from a Lorentzian distribution inside the impurity band towards a Gaussian distribution as the band edges are approached.

figure of merit σ κ = ZT S T 2 , which governs the efficiency of thermoelectric conversion at a given temperature T. Indeed, they allow one to reduce the phonon contribution κ ph to thermal conductivity κ [2][3][4]. On the other hand, through their highly peaked density of states they offer the large electron-hole asymmetry required for the enhancement of the thermopower S [5,6]. This now ranks them among the best thermoelectrics in terms of achievable values of ZT, with other nanostructured materials. Yet, maximizing the figure of merit is not the ultimate requirement on the quest for improved thermoelectrics. The actual electric power that can be extracted from a heat engine (or conversely the actual cooling power that can be obtained from a Peltier refrigerator) is also of importance when thinking of practical applications. From that point of view, nanowire-based thermoelectric devices are also promising: they offer the scalability needed for increasing the output power, insofar as they can be arranged in arrays of nanowires in parallel.
The main issue of this and the subsequent paper [7] is the determination of the dopant density optimizing the thermopower in a single semiconductor nanowire. From the theory side, this question has mainly been discussed at room temperature when the semi-classical Boltzmann theory can be used [8][9][10], or in the ballistic regime [11], when the presence of the disorder is completely neglected. The goal was to describe the thermoelectric properties of nanowires at room temperature where the quantum effects become negligible, and in particular to probe the role of their geometry (diameter, aspect ratio, orientation etc). From the experimental side, investigations have been carried out by varying the carrier density in the nanowire with an external gate electrode [6,[12][13][14][15][16]. Different field effect transistor device configurations can be used: either the nanowire and its metallic contacts are deposited on one side of an insulating layer, while a metallic back-gate is put on the other side (see for instance [14,17]), or one can take a top-gate covering only the nanowire (see for instance [18]). Recently, Brovman et al have measured at room temperature the thermopower of silicon and silicon-germanium nanowires and observed a strong increase when the nanowires become almost depleted under the application of a gate voltage [17]. Interestingly, this work points out the importance of understanding thermoelectric transport near the band edges of semiconductor nanowires. It also reveals a lack of theoretical framework to this field that we aim at filling.
For that purpose, we shall first identify, as a function of the temperature T and the applied gate voltage V g , the dominant mechanism of electronic transport through a given nanowire. At low temperature < T T x , transport is dominated by elastic tunneling processes and quantum effects must be properly handled. Due to the intrinsic disorder characterizing doped semiconductors, the electronic transport is greatly affected by Anderson localization while electron-phonon coupling can be neglected inside the nanowire. Above the activation temperature T x , electron-phonon coupling inside the nanowire starts to become relevant. One enters the inelastic variable range hopping (VRH) regime [19] where phonons help electrons to jump from one localized state to another, far away in space but quite close in energy. At temperatures higher than the Mott temperature T M , the VRH regime ceases and one has simple thermal activation between nearest-neighbor localized states. The different regimes are sketched in figure 1 for a nanowire modeled by a one-dimensional (1D) tight-binding Anderson model. Note that they are highly dependent on the gate voltage V g . The inelastic VRH regime will be addressed in a subsequent paper [7].
In this work, we focus our study on the low temperature elastic regime or more precisely, to a subregion < T T s inside the elastic regime in which the thermopower can be evaluated using the Landauer-Büttiker scattering formalism and Sommerfeld expansions. An experimental study of the gate dependence of the electrical conductance of Si-doped GaAs nanowire in this elastic coherent regime can be found in [18]. We will mainly consider nanowires of size N larger than their localization length ξ, characterized by exponentially small values of the electrical conductance. Obviously, this drastically reduces the output power associated with the thermoelectric conversion. Nevertheless, the advantage of considering the limit ξ ≫ N is twofold: first, the typical transmission at an energy E is simply given by ξ − N exp [ 2 ], in this limit, and second, at weak disorder, ξ E ( ) is analytically known. This makes possible to derive analytical expressions describing the typical behavior of the thermopower. To avoid the exponential reduction of the conductance at large ξ N , one should take shorter lengths ( ξ ≈ N ). To study thermoelectric conversion in this crossover regime would require one to use the scaling theory discussed in [20,21]. Furthermore, another reason to consider ξ ≫ N is that the delay time distribution (which probes how the scattering matrix depends on energy) has been shown to have a universal form [22] in this limit. We expect that this should also be the case for the fluctuations of the thermopower (which probes how the transmission depends on energy). This gives the theoretical reasons for focusing our study on the limit ξ ≫ N . , W = t and N = 1000. T s is given for ϵ = 0.01% (see section 6). Transport exhibits the bulk behavior of the nanowire impurity band as far as V g does not exceed a value of order t 1.5 and its edge behavior in the interval is necessary for completely depleting the nanowire in the limit → ∞ N . This paper is restricted to the study of region (I), corresponding to low temperatures < T T s at which the Sommerfeld expansion can be applied for the calculation of the thermoelectric coefficients. The VRH region (II) will be studied in [7].
The outline of the manuscript is as follows. Section 2 is a reminder about the Landauer-Büttiker formalism which allows one to calculate thermoelectric coefficients in the coherent regime. In section 3, we introduce the model and outline the numerical method used in this work, which is based on a standard recursive Greenʼs function algorithm. Our results are presented in sections 4, 5 and 6. Section 4 is devoted to the study of the typical behavior of the thermopower as the carrier density in the nanowire is modified with the gate voltage. We show that the thermopower is drastically enhanced when the nanowire is being depleted and we provide an analytical description of this behavior in the localized limit. In section 5, we extend the study to the distribution of the thermopower. We show that the thermopower is always Lorentzian distributed, as long as the nanowire is not completely depleted by the applied gate voltage and provided it is long enough with respect to the localization length. Interestingly, the mesoscopic fluctuations appear to be basically larger and larger as the carrier density in the nanowire is lowered and the typical thermopower increases. As a matter of course, this ceases to be true when the gate voltage is so large that the nanowire, almost emptied of carriers, behaves eventually as a (disordered) tunnel barrier. In that case, the thermopower distribution is found to be Gaussian with tiny fluctuations. The evaluation of the 'crossover temperature' T s (see figure 1) is the subject of section 6. Finally, we draw our conclusions in section 7.

Thermoelectric transport coefficients in the Landauer-Büttiker formalism
We consider a conductor connected via reflectionless leads to two reservoirs L (left) and R (right) in equilibrium at temperatures T L and T R , and chemical potentials μ L and μ R . To describe the thermoelectric transport across the conductor, we use the Landauer-Büttiker formalism [23]. The heat and charge transport are supposed to be mediated only by electrons and the phase coherence of electrons during their propagation through the conductor is supposed to be preserved. In this approach, the dissipation of energy takes place exclusively in the reservoirs while the electronic transport across the conductor remains fully elastic. The method is valid as long as the phase-breaking length (mainly associated to electron-electron and electron-phonon interactions) exceeds the sample size. From a theoretical point of view, it can be applied to (effective) non-interacting models. In this framework, the electric (I e ) and heat (I Q ) currents flowing through the system are given by [24,25] 1 is the Fermi distribution of the lead α and  E ( ) is the transmission probability for an electron to tunnel from the left to the right terminal. k B is the Boltzmann constant, < e 0 the electron charge and h the Planck constant. The above expressions are given for spinless electrons and shall be doubled in case of spin degeneracy.
We now assume that the differences Δμ μ μ = − where the linear response coefficients L i are given by The electrical conductance G, the electronic contribution K e to the thermal conductance K, the Seebeck coefficient  (or thermopower) and the Peltier coefficient Π can all be expressed in terms of the Onsager coefficients L i as The Seebeck and Peltier coefficients turn out to be related by the Kelvin-Onsager relation [26,27]  Π = T (9) as a consequence of the symmetry of the Onsager matrix. Note that, by virtue of equation (4), in the presence of particle-hole symmetry we have  Π = = 0. Furthermore, the link between the electrical and thermal conductances is quantified by the Lorenz number  = K GT e . In the zero temperature limit → T 0, the Sommerfeld expansion [28] can be used to estimate the integrals (4). To the first order in k T E B F , the electrical conductance reduces to (ignoring spin degeneracy) while the thermopower simplifies to The Lorenz number  takes in this limit a constant value, . This reflects the fact that the electrical and thermal conductances are proportional and hence cannot be manipulated independently, an important although constraining property known as the Wiedemann-Franz (WF) law. This law is known to be valid for non-interacting systems if the low temperature Sommerfeld expansion is valid [29,30], when Fermi liquid (FL) theory holds [28,31] and for metals at room temperatures [28], while it could be largely violated in interacting systems due to non FL behaviors [32,33].

Model and method
The system under consideration is sketched in figure 2(a). It is made of a 1D disordered nanowire coupled via perfect leads to two reservoirs L (left) and R (right) of non-interacting electrons, in equilibrium at temperature The nanowire is modeled as a 1D Anderson chain of N sites, with lattice spacing a = 1. Its Hamiltonian reads, where † c i and c i are the creation and annihilation operators of one electron on site i and t is the hopping energy. The disorder potentials ϵ i are (uncorrelated) random numbers uniformly distributed in the interval −W W [ 2,2]. The two sites at the ends of the nanowire are connected with hopping term t to the leads which can be 1D semi-infinite chains or 2D semi-infinite square lattices, with zero on-site potentials and the same hopping term t. The simpler case of the wide band limit (WBL) approximation, where the energy dependence of the self-energies of the leads is neglected, is also considered. Finally, an extra term (a) Sketch of the system: a 1D nanowire made of N sites is connected to two leads at its extremities. An external gate voltage V g is applied. (b) Band diagram. The impurity band of the nanowire (in blue) can be shifted by the application of V g in order to probe either the bulk, the edges or the outside of the impurity band at Fermi energy E F . Here, the leads are bidimensional (conduction band of the leads in red) and hence, ∈ − E t t [ 4 , 4 ] F . is added in the Hamiltonian (12) to mimic the presence of an external metallic gate. It allows to shift the whole impurity band of the nanowire.

Recursive Greenʼs function calculation of the transport coefficients
In the Greenʼs function formalism, the transmission  E ( ) of the system at an energy E is given by the Fisher-Lee formula [23]  L R in terms of the retarded single particle Greenʼs function i ( ) describe the coupling between the conductor and the lead α = L or R. A standard recursive Greenʼs function algorithm [34] allows us to compute the transmission  E ( ). The logarithmic derivative  E dln d can be calculated as well with the recursive procedure, without need for a discrete evaluation of the derivative. It yields the thermopower  in the Mott-Sommerfeld approximation (10). Hereafter, we will refer to a dimensionless thermopower We now discuss the expressions of the self-energies Σ E ( ) of the left and right leads which are to be given as input parameters in the recursive Greenʼs function algorithm. The nanowire of length N sites is supposed to be connected on one site at its extremities to two identical leads, which are taken 1D, 2D or in the WBL approximation. Hence, the self-energies Σ α (as well as the operator Γ α ) are N×N matrices with only one non-zero component (identical for both leads) that we denote with Σ (or Γ). When the wide-band limit is assumed for the leads, Σ is taken equal to a small constant imaginary number independent of the energy E. When the leads are two 1D semi-infinite chains or two 2D semi-infinite square lattices, Σ is given by the retarded Greenʼs function G lead of the lead under consideration evaluated at the site X (in the lead) coupled to the nanowire, . Knowing the expressions of the retarded Green functions of the infinite 1D chain and the infinite 2D square lattice [35], it is easy to deduce G lead for the semi-infinite counterparts by using the method of mirror images. For 1D leads, one finds Σ cos and k is the electron wavevector [23]. For 2D leads, the expression of Σ E ( ) is more complicated (see appendix A). As far as the Fermi energy E F is not taken near the edges of the conduction band of the leads, the thermopower behaviors using 1D and 2D leads coincide with those obtained using the WBL approximation (see section 4). This shows us that the dimensionality D becomes irrelevant in that limit, and we expect that taking 3D leads will not change the results.

Scanning the impurity band of the Anderson model
The density of states per site ν E ( ) of the Anderson model, obtained by numerical diagonalization of the Hamiltonian (12), is plotted in figure 3 c are the edges of the impurity band. In the bulk of the impurity band (i.e. for energies | | ≲ E t 1.5 ), the density of states is given with good precision by the formula derived for a clean 1D chain (red dashed line in figure 3(a)),  (17) and (18), expected in the bulk and at the edges of the nanowire conduction band for → ∞ N . (b) Localization length ξ of the 1D Anderson model (12) (with = W t 1) as a function of energy E. The circles correspond to numerical data (obtained with equation (21)). The red dashed line and the blue line are the theoretical predictions (22) and (23) As one approaches the edges ± E c , the disorder effect cannot be neglected anymore. The density of states is then well described by the analytical formula obtained by Derrida and Gardner around ± E c , in the limit of weak disorder and large N (see [36]), In this paper, we study the behavior of the thermoelectric coefficients as one probes at the Fermi energy E F electron transport either inside or outside the nanowire impurity band, and more particularly in the vicinity of its band edges. Such a scan of the impurity band can be done in two ways. One possibility is to vary the position of the Fermi energy E F in the leads. In doing so, we modify the distance between E F and the band edges ± E c but also the one between E F and the band edges of the leads. This can complicate the analysis of the data, the dimensionality of the leads becoming relevant when . To avoid this complication, we can keep E F fixed far from ± E c and vary the gate voltage V g (see figure 2(b)).

Localization length of the Anderson model
In the disordered 1D model (12) we consider, all eigenstates are exponentially localized, with a localization length ξ. As a consequence, the typical transmission of the nanowire drops off exponentially with its length N. More precisely, when ξ ≫ N (localized limit), the distribution of  ln is a Gaussian [37,38] centered around the value 0 as long as the energy E of the incoming electron is inside the impurity band of the nanowire. The inverse localization length ξ 1 can be analytically obtained as a series of integer powers of W when → W 0. To the leading order (see e.g. [39]), this gives The formula is known to be valid in the weak disorder limit inside the bulk of the impurity band (hence the index b). Strictly speaking, it fails in the vicinity of the band center E = 0 where the perturbation theory does not converge [40] but it gives nevertheless a good approximation. As one approaches one edge of the impurity band, the coefficients characterizing the expansion of ξ 1 in integer powers of W diverge and the series has to be reordered. As shown by Derrida and Gardner [36], this gives (to leading order in W) the non analytical behavior ξ ∝ W 1 2 3 as one edge is approached instead the analytical behavior ξ ∝ W 1 2 valid in the bulk of the impurity band. More precisely, one finds in the limit as E approaches the band edges ± t 2 . The integrals  i and the parameter X have been defined in equations (19) and (20). As shown in figure 3(b), both formula (22) and (23) are found to be in very good agreement with our numerical evaluation of ξ E ( ), in the respective range of energy that they describe, even outside a strictly weak disorder limit (W = t in figure 3(b)).

Typical thermopower
We compute numerically the thermopower S for many realizations of the disorder potentials ϵ i in equation (12), and we define the typical value S 0 as the median of the resulting distribution P(S). As will be shown in section 5, P(S) is typically a smooth symmetric function (Lorentzian or Gaussian), and thus its median coincides with its most probable value. We study the behavior of S 0 as one scans the energy spectrum of the nanowire by varying the position of the Fermi energy E F in the leads or the gate voltage V g .
In figure 4(a), the typical thermopower S 0 of a long nanowire in the localized regime ). Indeed, the random potentials being symmetrically distributed around a zero value, one has a statistical particle-hole symmetry at the band center and the thermopower can only be a statistical fluctuation around a zero typical value. As E F is increased, the statistical particle-hole symmetry breaks down and S 0 becomes finite. Here > S 0 0 because charge transport is dominated by holes for > E 0 F . When the wide band limit is assumed for both leads (triangles in figure 4(a)), we find that the typical thermopower S 0 increases with E F and reaches a maximum just before = + F , charge transfer being impossible outside the conduction band of the leads). This singularity at the band edge of the leads can be easily understood using equations (14) and (15) and noticing that for 1D [2D] leads, . This is obvious in the case of 1D leads where Γ = − E t E t ( ) 2 1 ( 2 ) 2 and it can also be shown for 2D leads. We will see in section 6 that this apparent divergence of the thermopower is actually only valid in an infinitesimally small range of temperatures above 0 K.
With the gate voltage V g , we can explore the impurity band of the nanowire while keeping E F fixed. The behavior of S 0 as a function of V g is shown in figure 4(b) figure 4(b) and we have no doubt that it also remains true with 3D leads. Moreover, the results are unchanged if E F is fixed to any other value, as long as it does not approach too closely one edge of the conduction band of the leads (but it can be chosen close enough to one band edge to recover the continuum limit of the leads). Our main observation is that the typical thermopower S 0 increases importantly when the Fermi energy probes the region around the edges of the impurity band of the nanowire. Qualitatively, this is due to the fact that the typical transmission of the nanowire drops down when the edges are approached: this huge decrease results in an enhancement of the typical thermopower, the thermopower being somehow a measure of the energy dependence of the transmission. A quantitative description of this behavior can also be obtained. Indeed, since the distribution of the transmission  is log-normal in the localized regime [37,38] and the ln Gaussian distribution (which in this case coincides with the most probable value). Moreover, according to equation (21), the energy dependence of  [ln ] 0 is given by the energy dependence of the localization length, i.e. by equations (22) and (23). This allows us to derive the following expressions for the typical thermopower in the bulk and at the edges: where now X is modified to in order to take into account the effect of the gate voltage V g . When the outside of the impurity band, rather than the inside, is probed at E F (i.e. when the wire is completely depleted), no more states are available in the nanowire to tunnel through. Electrons coming from one lead have to tunnel directly to the other lead through the disordered barrier of length N. We have also calculated the typical thermopower of the nanowire in that case, assuming that the disorder effect is negligible (see appendix B). We find Figure 4(b) shows a very good agreement between the numerical results (symbols) and the expected behaviors (equations (25), (26) and (28)). One consequence of these analytical predictions is that the peak in the thermopower curves gets higher and narrower as the disorder amplitude is decreased (and viceversa).

Thermopower distributions
In the coherent elastic regime we consider, the sample-to-sample fluctuations of the thermopower around its typical value are expected to be large. The most striking illustration occurs at the center of the impurity band of the nanowire ( = E V F g ), when the typical thermopower is zero due to statistical particle-hole symmetry but the mesoscopic fluctuations allow for large thermopower anyway. Van   ) and remains a good approximation as far as one stays in the bulk of the impurity band. But the distribution P(S) is no more centered around zero as one approaches the band edge.
We propose here to investigate how the thermopower distribution P(S) is modified when this is not only the bulk, but the edges (or even the outside) of the impurity band which are probed at the Fermi energy E F . To fix the ideas, we set the Fermi energy to = E 0 F and the disorder amplitude to W = t (so that the band edges are . First, we check in figure 5(a) that at = V 0 g and in the localized regime, the thermopower distribution is indeed a Lorentzian with a width Λ ∝ N . We note that very long chains of length ξ ≈ N 50 (ξ ≈ 100 here) are necessary to converge to the Lorentzian (29). Moreover, we have checked that this it is also in this limit that the delay time distribution converges towards the universal form predicted in [22].
Then we increase the gate potential up to = V t 2 g to approach the edge − E c of the impurity band and find that the thermopower distribution remains a Lorentzian in the localized regime ( ξ ≳ N 50 ) with a width Λ ∝ N , as shown in figure 5(b). It turns out actually that the fit of the thermopower distribution with a Lorentzian (in the large N limit) is satisfactory in a broad range of gate potentials ≲ V t 2.25 g , as long as the Fermi energy = E 0 F probes the impurity band without approaching too closely its edges + ± V E g c . In figure 5(c), we show in addition that in this regime, the widths Λ of the Lorentzian fits to the thermopower distributions P(S) obey Λ π ν = Nt (2 ) F , i.e. equation (30). Therefore (figure 5(d)), we can use this parameter to rescale all the distributions obtained in a broad range of parameters, on the same Lorentzian function (30) is that the mesoscopic fluctuations of the thermopower are maximal for When the gate voltage V g is increased further, the number of states available at E F in the nanowire decreases exponentially and eventually vanishes: one eventually approaches a regime where the nanowire becomes a long tunnel barrier and where the thermopower fluctuations are expected to be smaller and smaller. In this limit, we find that the thermopower distribution is no more a Lorentzian but becomes a Gaussian, ⎡ for two values of V g . The Gaussian thermopower distribution is centered around a typical value S 0 given by equation (28) and its width λ is found with great precision to increase linearly with N and W.
To be more precise, we find that the dependency of λ on the various parameters is mainly captured by the following formula  at least for 35 6 F g and ≳ N 100 (see figure 6(c)). We stress out that equation (32) is merely a compact way of describing our numerical data. In particular, the apparent divergence of λ when − → + outside the range of validity of the fit. To double-check the validity of equation (32), we have rescaled with the parameter λ given by equation (32), a set of thermopower distributions obtained in the disordered tunnel barrier regime, for various W and V g . All the resulting curves (plotted in figure 6(d)) are superimposed on the unit Gaussian distribution, except the one for the smallest disorder value = W t 0.5 for which the fit (32) to λ is satisfactory but not perfect.
To identify precisely the position of the crossover between the Lorentzian regime and the Gaussian regime, we introduce now the parameter η, G L G which measures, for a given thermopower distribution P(S) obtained numerically, how closed it is from its best Gaussian fit P S ( ) G and from its best Lorentzian fit P S ( ).  figure 7 that η converges at large N for any V g (inset). The asymptotic values of η (given with a precision of the order of 0.05 in the main panel) undergo a transition from η ≈ 1 to η ≈ 0 when V g is increased from 0 to t 4 . This reflects the crossover from the Lorentzian to the Gaussian thermopower distribution already observed in the top panels of . Upon shifting the spectrum of the nanowire with V g , the thermopower distribution moves from a Lorentzian distribution for . 1 One could be tempted to compare an arbitrary thermopower distribution P(S) to the Lorentzian and Gaussian distributions given in equations (29), (30) and (31), (32) respectively. However, to define η for any set of parameters, one should extend to the outside of the spectrum the formula (30) for the width Λ of the Lorentzian, and to the inside of the spectrum the formula (32) for the width λ of the Gaussian. We avoid this problem by taking instead the best Lorentzian and Gaussian fits to P(S) in the definition of η. It allows us to distinguish whether P(S) is a Lorentzian or a Gaussian (or none of both) but of course, the precise form of P(S) is not probed by η as defined.
figures 5 and 6. We see in addition that the crossover is very sharp around the value ≈ V t 2.3 g , indicating a crossover which remains inside the impurity band of the infinite nanowire, since the band is not shifted enough when ≈ V t 2.3 g to make the Fermi energy coincide with the band edge We have obtained the same results for other values of the disorder amplitude. After checking the convergence of η at large N, we observe the same behavior of the asymptotic values of η as a function of V g , for any W. Only the position of the crossover is disorder-dependent. Those results are summarized in the right panel of figure 7 where one clearly sees the crossover (in white) between the Lorentzian regime (in blue) and the Gaussian regime (in red). It occurs around ≈ in a region where the number of states available at E F in the nanowire becomes extremely small. To be precise, we point out that the values of η in the 2D colorplot are given with a precision of the order of 0.1. Hence, one cannot exclude that the white region corresponding to the crossover actually reduces into a single line V W ( ) g c . One could also conjecture the existence of a third kind of thermopower distribution (neither Lorentzian, nor Gaussian) associated to this critical value V g c . Our present numerical results do not allow one to favor one scenario (sharp crossover) over the other (existence of a critical edge distribution).

Temperature range of validity of the Sommerfeld expansion
All the results discussed in this paper have been obtained in the low temperature limit, after expanding the thermoelectric coefficients to the lowest order in k T E B F . To evaluate the temperature range of validity of this study, we have calculated the Lorenz number  = K GT e beyond the Sommerfeld expansion, and looked at its deviations from the WF law   = 0 (see equation (11)): we have computed numerically the integrals (4) enterings equations (5) and (6), deduced  T ( ) for increasing values of temperature, and then recorded the temperature T s above which  T ( ) differs from  0 by a percentage ϵ, . We did it sample by sample and deduced the temperature T s averaged over disorder configurations. Our results are summarized in figure 8.
In panel (a), we analyze how sensitive T s is to the precision ϵ on the Lorenz number . We find that T s increases linearly with ϵ , ϵ ϵ = * T T ( ) , T s is found to decrease as V g is increased. More precisely, we find in the large N limit ( ξ ≳ N 10 ) that ν ∝ − Nk T B s F 1 with a proportionality factor depending on ϵ (solid line in figure 8(b)). The temperature T s is hence given by (a fraction) of the mean level spacing at E F in this region of the spectrum ( spectrum, this increase of T s with V g is well understood as follows: since in the tunnel barrier regime, the transmission behaves (upon neglecting the disorder effect) as  ζ , the Sommerfeld expansion of integrals (4) holds for temperatures Our numerical results are in perfect agreement with this prediction (dashed line in figure 8(b)).
In figure 8(c), we investigate the behavior of T s when the spectrum of the nanowire is either scanned by varying V g at = E 0 ). This means that the divergence of the dimensionless thermopower S observed in figure 4(a) is only valid in an infinitely small range of temperature above 0 K. It would be worth figuring out wether or not a singular behavior of the thermopower at the band edges of the conduction band persists at larger temperature.  Finally, let us give an order of magnitude in Kelvin of the temperature scale T s . In figure 8(b), the lowest T s reached around Asking for a precision of ϵ = 1% on , we get ∼ Nk T t 0.016 Those temperatures being commonly accessible in the laboratories, the results discussed in this paper should be amenable to experimental checks.

Conclusion
We have systematically investigated the low-temperature behavior of the thermopower of a single nanowire, gradually depleted with a gate voltage in the field effect transistor device configuration. Disorder-induced quantum effects, unavoidable in the low-temperature coherent regime, were properly taken into account. We have provided a full analytical description of the behavior of the typical thermopower as a function of the gate voltage and have confirmed our predictions by numerical simulations. Our results show that the typical thermopower is maximized when the Fermi energy lies in a small region inside the impurity band of the nanowire, close to its edges. Moreover, since thermoelectric conversion strongly varies from one sample to another in the coherent regime, we have carefully investigated the mesoscopic fluctuations of the thermopower around its typical value. We have shown that the thermopower is Lorentzian-distributed inside the impurity band of the nanowire and that its fluctuations follow the behavior of the density of states at the Fermi energy when the gate voltage is varied. In the vicinity of the edges of the impurity band and outside the band, the thermopower was found Gaussian-distributed with tiny fluctuations.
The thermopower enhancement which we predict around the edges looks to be in qualitative agreement with the recent experimental observation reported in [17], using silicon and germanium/silicon nanowires in the field effect transistor device configuration. We stress however that those measurements were carried out at room temperatures, and not in the low temperature coherent regime which we consider. To describe them, inelastic effects must be included. It will be the purpose of our next paper [7]. The low temperature coherent regime considered in this paper has been studied in [18], where the conductances G of half a micron long Si-doped GaAs nanowires have been measured at = T mK 100 in the field effect transistor device configuration. Assuming equation (10) for evaluating the thermopower  from G V ln ( ) g , the typical behavior and the fluctuations of G V ln ( ) g given in [18] are consistent with the large enhancement of  near the band edges which we predict.
Electron-electron interactions were not included in our study. A comprehensive description of the thermopower of a 1D disordered nanowire should definitely consider them. Nevertheless, we expect that the drastic effects of electronic correlations in 1D leading to the formation of a Luttinger liquid are somehow lightened by the presence of disorder. Second, the gate modulation of the thermopower we predict here is mainly due to a peculiar behavior of the localization length close to the edges of the impurity band. And experimentally, coherent electronic transport in gated quasi-1D nanowires turned out to be well captured with one-electron interference models [18]. Of course, one could think of including electronic interactions numerically with appropriate numerical 1D methods but regarding the issue of thermoelectric conversion in nanowires, we believe the priority lies rather in a proper treatment of the phonon activated inelastic regime.
Finally, let us discuss the potential of our results for future nanowire-based thermoelectric applications. To evaluate the efficiency of the thermoelectric conversion [42] in a nanowire, one needs to know also its thermal conductance K. Below the temperature T s , the electron contribution K e to K is related to the electrical conductance G by the WF law. This gives for the typical value of K e . The evaluation of the phonon contribution K ph to the thermal conductance of a nanowire is beyond the scope of the used Anderson model, since static random site potentials are assumed. In one dimension, one can expect that K ph should be also much smaller than the thermal conductance quantum which characterizes the ballistic phonon regime [43,44]. However, it remains unlikely that K could be as small as G for giving a large figure of merit ZT in a single insulating nanowire at low temperature.
Similarly, if we were to look to the delivered (electric) output power, we would find that a large length N would make it vanish, as the electrical conductance in this regime would be exponentially small. Indeed, looking at the power factor   = G 2 , which is a measure of the maximum output power [45], we realize that the enhancement of  at the edge of the impurity band would not be enough to face the exponentially small values of G. Obviously, the optimization of the power factor  for a single nanowire requires one to take shorter lengths ( ξ ≈ N ), while the optimization of the thermopower  requires one to take long sizes ( ξ ≫ N ). Moreover, because of the strong variation of the localization length as the energy varies inside the impurity band, the optimization of the power factor for a given size N also requires one to not be too close from the edges of the impurity band. This illustrates the fact that a compromise has always to be found when thinking of practical thermoelectric applications. A way to optimize the efficiency and the output power could consist of taking a large array of nanowires in parallel instead of a single one. Since the conductances G in parallel add while the thermopower  does not scale with the number of wires (at least if we take for  its typical value, neglecting the sample to sample fluctuations), the compromise could favor the limit of long nanowires with applied gate voltages such that electron transport occurs near the edges of impurity bands. Nowadays, it is possible to grow more than 10 8 InAs nanowires [46] per cm 2 , a large number which could balance the smallness of the conductance of an insulating nanowire.
Actually, when thinking of practical applications, the results of this paper are rather promising regarding Peltier refrigeration. Indeed, our conclusions drawn here for the thermopower at low temperature also hold for the Peltier coefficient, the two being related by the Kelvin-Onsager relation  Π = T . One could imagine to build up Peltier modules with doped nanowires for cooling down a device at sub-Kelvin temperature in a coherent way. Besides, whether it be for energy harvesting or Peltier cooling, it would be worth considering more complicated setups using the nanowire as a building block (e.g. arrays of parallel nanowires in the field effect transistor device configuration) in order to reach larger values of output electric/cooling power. 2) ( 2) symmetric tridiagonal matrix with all diagonal elements equal to − E V g and all elements on the first diagonals below and above the main one equal to t. Using the Fisher-Lee formula (14), we write the transmission function  E ( ) as where G A is the 2 × 2 submatrix in the top left-hand corner of G, G A N (1 ) its top right element and γ σ = −2 Im ( ). To calculate G A , we first notice that where □ − C 1 is a 2 × 2 submatrix of − C 1 made up of the four elements located at its four corners. Secondly, we make use of [49] for computing the inverse of the symmetric tridiagonal matrix C. . Plugging equations (B4-B7) into equation (B3), we deduce the exact transmission function  E ( ), and hence the thermopower S defined by equation (15). An expansion at large N yields  ζ ∝ − N exp ( 2 ) (as expected for a tunnel barrier) and the expression (28) for the thermopower. The same demonstration can be made for the energy range ⩽ − E V t 2 g .