*3.2. Strongly Correlated System*

The Landauer approach coupled with Density Functional Theory through the Non-Equilibrium Green's function formalism provides a reliable and presently standard method to calculate the transport properties of many devices. Countless are the successes of the theory, far beyond its actual range of applicability. Nonetheless there are many reasons to go beyond this state of the art. A simple reason is that we need alternative methods to provide benchmarks for the DFT + LB theory and learn from them. Indeed, the strength of DFT lies also in adapting the KS potential to the cases under investigation. On the other hand, we might need to go beyond some of the fundamental approximations and thus build a different theory.

Meanwhile, a striking example of the limits of the DFT + LB theory is related to one of the most fascinating outcomes of the so-called mesoscopic physics. Imagine that the device we place between two leads is a small molecule or a quantum dot. Both these systems are thought of having just a few states close to the Fermi energy (or the electrochemical potential). Imagine that one electron enters the device and occupies the lowest energy state. The next electron then faces an increased energy barrier, since beside the energy to occupy the lowest available energy states it also musts overcome the Coulomb interaction with the other electron. Normally, for large devices this energy is small since the electron density is "diluted", but when we consider small dots or molecules, the Coulomb interaction might be the dominating energy scale and transport can be "blocked" until either the first electron leaves the device or the second electron has enough energy to overcome the Coulomb interaction. A standard DFT + LB approach to this problem is most likely going to fail. Indeed, DFT describes the electrons through their density and therefore it does not produce the sharp energy transition due to the addition of a single electron. This effect goes normally under the name of "Coulomb blockade", and it is the epitome of a strongly correlated system where essentially the dynamics is dictated by electron interaction and correlation.

However, it is important to point out that these limits are related to our ability of inventing KS potentials able to describe certain physical regime. Per se, DFT can describe the ground-state properties of the system and thus give the exact energy for the single and double occupied electron states. It is our inability of encoding these effects into the KS potential that makes the theory fails. Indeed, progress has been made to include strong correlation into the KS potential into a pure DFT scheme. Here, we will consider the extension of DFT to deal with the Coulomb blockade regime. To do that, we first need to extend the theory to include the transport properties in a more accurate way. As we will see this step corrects the electronic conductance that is calculated from the standard DFT approach.

Our starting point is the observation that generally speaking, the quantities we want to investigate in studying the device of Figure 2 are the local electron density *n*(**r**) and the total current *I* flowing from one reservoir to the other due to a thermal gradient or a bias voltage. For the moment, we will focus on the steady state, i.e., we assume the system has evolved from an initial state and approached, as time goes by, a constant density and current. The external fields that we are applying are the bias voltage *V* and the gate voltage *v* which controls the electron density and the total number of particles. We assume that the nuclear potential is not affected by the electron distribution and it is, therefore, constant (we assume uniform temperature). We see *V* and *v* as a perturbation and *n*(**r**) and *I* as response. To make a DFT theory for this set of variables, we need to prove that they are uniquely connected. Specifically, that the pair *n*(**r**) and *I* uniquely determine both *V* and *v*. Moreover, we are interested in *n*(**r**) and *I* only inside a finite region, R, surrounding the device. The proof of the theorem entails the evaluation of the Jacobian of the mapping {*n*(**r**), *I*} → {*v*(**r**), *V*} for any **r** ∈ R. One can prove that around *V* = 0 this mapping is invertible since the Jacobian does not vanish [53]. We can therefore follow the KS construction and find a system of non-interacting particles which can reproduce *n*(**r**) and *I* of the original system by replacing the interaction with the external potentials

$$V\_{\mathbb{S}}[\mathfrak{n}\_{\prime}I] \quad = \quad V[\mathfrak{n}\_{\prime}I] + V\_{\text{xc}}[\mathfrak{n}\_{\prime}I] \tag{38}$$

$$
v\_{\mathcal{S}}[\mathfrak{n}, I] \quad = \quad v[\mathfrak{n}, I] + v\_{H\text{xc}}[\mathfrak{n}, I] \tag{39}$$

where *Vxc* and *vHxc* are the xc potentials (in *vHxc* we include also the Hartree mean-field potential). *n*(**r**) and *I* in the KS system are determined by

$$n(\mathbf{r}) \quad = \ 2 \int \frac{d\mathbf{e}}{2\pi} \left[ f\left(\mathbf{e} - \frac{V + V\_{\mathbf{x}\mathbf{c}}}{2}\right) A\_L(\mathbf{r}, \mathbf{e}) + f\left(\mathbf{e} + \frac{V + V\_{\mathbf{x}\mathbf{c}}}{2}\right) A\_R(\mathbf{r}, \mathbf{e}) \right] \tag{40}$$

$$I\_{\varepsilon} = -2 \int \frac{\mathbf{d}\varepsilon}{2\pi} \left[ f\left(\varepsilon + \frac{V + V\_{\text{xc}}}{2}\right) - f\left(\varepsilon - \frac{V + V\_{\text{xc}}}{2}\right) \right] T(\varepsilon),\tag{41}$$

where *AR*(*L*) = **r**|*G*()<sup>Γ</sup>*R*(*L*)*G*†()|**r** is the right (left) KS partial spectral function, *<sup>G</sup>*() the Green's function of the KS system in the energy representation, and *<sup>T</sup>*() the transmission function (see Equation (33)). Notice that the right-hand sides are expressed solely in terms of KS quantities, while in the left-hand side we have the many-body quantities. In these expressions, the gate voltage *v* enters in the KS partial spectral function and the transmission coefficient only. We want now to derive an expression for the electrical conductance *σ* and the Seebeck's coefficient *S* solely determined from the KS quantities. We are focusing here to the linear-response regime, i.e., we will take at the end the limit *V* → 0. In this limit *Vxc*(*n*) vanishes for any density *n*, since otherwise we would have a finite current when no external perturbation is present in contradiction with the theorem of uniqueness. We have

$$
\sigma = \left. \frac{\mathrm{d}I}{\mathrm{d}V} \right|\_{V=0} = \left( 1 + \left. \frac{\mathrm{d}V\_{\mathrm{xc}}}{\mathrm{d}V} \right|\_{V=0} \right) \int \frac{\mathrm{d}\epsilon}{2\pi} f'(\epsilon) \, T(\epsilon) = \left( 1 + \left. \frac{\mathrm{d}V\_{\mathrm{xc}}}{\mathrm{d}V} \right|\_{V=0} \right) \sigma\_{\mathrm{S}} \tag{42}
$$

where we have introduced the KS conductance *σS* = d/2*π f* () *<sup>T</sup>*(). To evaluate the term in the round bracket we use the standard chain rules, remembering that our "variables" are *n*(**r**) and *I*,

$$\left. \frac{\delta V\_{\text{xc}}}{\delta V} \right|\_{V=0} = \frac{\partial V\_{\text{xc}}}{\partial I} \frac{\partial I}{\partial V} + \int d\mathbf{r} \frac{\delta V\_{\text{xc}}}{\delta n(\mathbf{r})} \frac{\partial n(\mathbf{r})}{\partial V} = \frac{\partial V\_{\text{xc}}}{\partial I} \sigma \tag{43}$$

since the last term vanishes in the linear-response regime, *Vxc*(*n*)*<sup>I</sup>*=<sup>0</sup> = 0. Inserting this expansion into the previous result, we find finally

$$
\sigma = \frac{\sigma\_{\rm S}}{1 - \frac{\partial V\_{\rm rc}}{\partial I} \sigma\_{\rm S}}.\tag{44}
$$

Notice that in general we should expect that *∂Vxc ∂I* = 0, and therefore *σ* = *σS*. Although for the conductance we can find a general correction, for the Seebeck's coefficient we can derive an analytical formula only in the Coulomb blockade regime. In this particular physical condition, we are interested only in the total number of particles, *N* = Rd**r***n*(**r**) and we can prove that

$$S = -\frac{\frac{\text{dN}}{\text{d}N}}{\frac{\text{dN}}{\text{d}\mu}}\tag{45}$$

if *μ* is the electrochemical potential. In a similar way as for the conductance, we find for the Seebeck coefficient in the Coulomb blockade regime, the expression

$$S = S\_S + \frac{\partial v\_{Hxc}}{\partial T} \tag{46}$$

where *SS* is the Seebeck's coefficient calculated from the Landauer's approach [73]. We can compare our result with the standard DFT approach and with the exact solution provided by the rate equations [7,74,75]. Other approaches are possible to directly calculate the transmission probability, with and without interaction in the central region [44,76,77]. A direct comparison of the different methods would be desirable, but difficult, since one needs to map each set of parameters appropriately.

Figure 3 report the Seebeck's coefficient as calculated from the exact many-body theory, the rate equations, the correction Equation (46) and the standard DFT approach *SS*. We notice that the dynamical correction brings *SS* to coincide with the exact results. Notice that *N* in this case is well reproduced both by the dynamical approach as well as by the standard DFT, therefore in this case is a variable less sensitive to the approximations used. When considering more than one level, we use a constant interaction model. For the case of two levels, we find some discrepancies from the exact theories and the present approach (see Figure 4). This discrepancy originates from using the total number of particles *N* rather than the single occupation of each states *n*1 and *n*2 as our basic variables.

**Figure 3.** The Seebeck's coefficient for a single quantum dot in the Coulomb blockade regime as a function of the gate voltage *v*. The exact many-body (MB), the rate equation (RE), and the present theory (DFT) agree quite well in the whole range of the gate voltage *v* considered. The standard KS gives the correct asymptotic but fails in the central region. In cyan, we plot the effect of the dynamical correction *∂vHxc ∂T* . In the inset, we report the total number of particles *N*. All the theories agree quite well for this quantity. Reprinted from [73]. Copyright (2015) American Physical Society.

**Figure 4.** Left pane, the total number of particles *N* and the occupation of the second (highest) energy level as a function of the gate voltage *v*. While the first is exactly reproduced by both DFT and rate equations, the second differs. This originates differences also in the Seebeck's coefficient which is sensitive to the single particle occupations (right pane). These differences are however small compared with the correction brought about by the dynamical approach. Reprinted from [73]. Copyright (2015) American Physical Society.
