*2.2. Landauer–Buttiker and Quantum Transport*

The Landauer–Buttiker's formalism (LB) is an elegant and economic way to study transport through molecular devices [26–28]. The idea at its core is deceivingly simple: The overall device is separated into three or more regions of space, one of which the central, usually characterizes the physical properties of the overall system. The other regions serve as sink or reservoir of electrons. The only property that is required from the reservoirs is that they connect smoothly with the central region, thus avoiding backscattering due to the contacts, and that they are either perfect emitters or absorbers, in the sense that one electron entering them cannot leave through the central region again. This requires also that their spectral density is essentially flat over a wide range of energies around the Fermi level of the central system. LB then describes the currents flowing between the leads, through the probability that an electron entering through the lead *i* is scattered by the central region, and thus leaves through the lead *j*, *Ti*,*j*. In this respect thus the LB formalism is a scattering theory and

the transmission probability is its central ingredient. Finally, one sums over all the allowed electron energies [26–28]. The electrical current is therefore given by

$$I\_{i,j} = \frac{2\varepsilon}{h} \int \mathrm{d}\mathfrak{e}\left(f\_i(\mathfrak{e}) - f\_j(\mathfrak{e})\right) T\_{i,j}(\mathfrak{e})\,\,\,\tag{17}$$

where *fi* is the Fermi function describing the electron occupations of the lead *i*, *e* is the electron charge, *h* is the Plank's constant, and the factor 2 takes into account spin degeneracy. Generally, the Fermi functions depend upon the local temperature *Ti* of the lead *i*, their chemical potential *μi*, and the bias applied to the lead, *Vi*. Without loss of generality, in the presence of the bias *Vi* we can assume that *μi* = 0. It is important to realize the important approximation of the model: we are assuming that the occupation of the electrons entering from the lead *i*, depends only on the equilibrium physical parameter of the *i* lead. Finally, we have also assumed that the central system is essentially one dimensional—in this case, the density of states and the electron velocity cancel out to achieve the universal result of Equation (17). A thorough discussion of the derivation and the physical implications of the Landauer transport theory goes way beyond the aim of this review and the topic has been discussed in a large number of publications, including reviews and monographs [1,2,4,26–29].

Given the current *I* from the different terminals, one can easily calculate the two-terminal conductance, in the linear-response regime *<sup>δ</sup>Vi*,*<sup>j</sup>* → 0,

$$\sigma\_{i,j} = \lim\_{\delta V\_{i,j} \to 0} \frac{I\_{i,j}}{\delta V\_{i,j}} = \frac{2e^2}{\hbar} \int \mathrm{d}\varepsilon f'(\epsilon) T\_{i,j}(\epsilon) \,. \tag{18}$$

where *f* () is the first derivative of the Fermi function with respect to the energy, and *<sup>δ</sup>Vi*,*<sup>j</sup>* = *Vi* − *Vj* the bias difference between the leads *i* and *j*. The simplest most striking prediction that the LB theory makes is the quantization of the conductance. Indeed, in the low temperature limit, *f* → *<sup>δ</sup>*( − *EF*) where *EF* is the Fermi energy of the leads, so that [26–28]

$$
\sigma\_{\mathbf{i},\mathbf{j}} = \frac{2e^2}{h} T\_{\mathbf{i},\mathbf{j}}(E\_F). \tag{19}
$$

If we assume that *Ni*,*<sup>j</sup>* states are fully open to transport current, i.e., their transmission probability goes to 1, we have

$$
\sigma\_{i,j} = \frac{2c^2}{h} N\_{i,j} \tag{20}
$$

as it has been carefully verified in many experiments, for example in the quantum Hall effect [30,31] or in the ballistic regime of a quantum point contact [32–34]. Equation (19) is what is normally called the "Landauer's formula" and it is deceivingly simple. However, its physical interpretation has puzzled the community for many years. In particular, one question that arises naturally is from where this conductance originates. Clearly, we are describing an almost ideal system. The electrons flow seamlessly from the leads to the central region, and they are transmitted to the other lead with probability 1, so there is no scattering in these states. In addition, yet, we have a finite conductance thus associated with energy loss and potential drops. The solution of this apparent paradox lies in the different dimensionality of the leads with respect to the central system. The "adjustment" of the wave-function to adapt to the reduced dimensionality of the center causes a charge accumulation at the interfaces between the center and the leads, no matter how smooth these interfaces are. This extra charge that accumulates as soon as we contact the central region, creates a finite bias that opposes the one applied to the reservoir and finite conductance appears.

One of the main advantages of the Landauer's formalism lies in treating on the same footing different physical problems. For example, the study of thermal transport both by electrons and

phonons can be easily recast in the form of a scattering problem for the electrons and phonons trough the central region. Therefore, the energy current due to the electrons is evaluated as

$$J\_{i,j}^{\epsilon} = \frac{2}{\hbar} \int \mathrm{d}\mathfrak{e}(\mathfrak{e} - \mu) \left( f\_i(\mathfrak{e}) - f\_j(\mathfrak{e}) \right) T\_{i,j}(\mathfrak{e}) \,\,\,\,\,\tag{21}$$

while the phonon contribution is given by

$$J\_{i,j}^p = \frac{\hbar}{2\pi} \int \mathbf{d}\omega \,\omega \left( n\_i(\omega) - n\_j(\omega) \right) T\_{i,j}(\omega),\tag{22}$$

where *ni* is the Bose distribution of the phonons in the lead *i* kept at temperature *Ti*. In the following we assume that *Ti* = *T* + Δ*Ti*, where Δ*Ti* defines the difference between the temperature of the lead *i* and the reference temperature *T*. As we did for the electrical conductance, we can calculate the thermal conductance for the phonons and the electrons by simply assuming that a small thermal gradient is present between the leads. Moreover, if one assumes that both the potential and thermal gradient are small, the Landauer formalism reduces to the Onsager linear-response out-of-equilibrium transport theory, with the advantage of providing a reliable way of calculating the Onsager's coefficients [1,35]. Indeed, if one defines the functions (Lorentz's integrals)

$$L\_{\rm II} = \int \mathrm{d}\boldsymbol{\varepsilon} (\boldsymbol{\varepsilon} - \boldsymbol{\mu})^{\boldsymbol{n}} f'(\boldsymbol{\varepsilon}) T\_{\bar{\boldsymbol{l}}, \boldsymbol{j}}(\boldsymbol{\varepsilon}) \tag{23}$$

we have the Onsager's relations (restricting ourselves to only two leads)

$$I\_{\perp} = -\frac{2\varepsilon^2}{h\_{\perp}} L\_0 \Delta V - \frac{2\varepsilon}{hT\_{\perp}} L\_1 \Delta T\_{\prime} \tag{24}$$

$$J\_{\varepsilon} \quad = \quad -\frac{2\varepsilon}{h}L\_1\Delta V + \frac{2}{hT}L\_2\Delta T. \tag{25}$$

The electron transport coefficients, namely the electrical conductance, the thermal conductance and the Seebeck's coefficient then follow from their physical definitions and are expressed solely in terms of the functions *Ln*. Indeed, the electrical conductance is *σ* = *<sup>I</sup>*/Δ*V*|<sup>Δ</sup>*T*≡<sup>0</sup> = <sup>2</sup>*e*2*L*0/*h*, the Seebeck's coefficient is *S* = <sup>−</sup>Δ*V*/Δ*T*|*<sup>I</sup>*=<sup>0</sup> = −*L*1/*eTL*0, while the thermal conductance is *κ* = *J*/Δ*T*|*<sup>I</sup>*=<sup>0</sup> = <sup>2</sup>(*<sup>L</sup>*2 − *<sup>L</sup>*21/*<sup>L</sup>*0)/*hT*. Notice that the physical definition of the transport coefficients is in general not restricted to the linear-response regime, so one could define the Seebeck coefficient as *S* = −Δ*V*/Δ*T* also for large Δ*T*, but the expression in terms of the Onsager's coefficients and Lorentz integrals is valid only in linear response.

Moreover, the formula for the phonon energy current Equation (22) predicts the surprising result that the low energy phonons, usually responsible for large part of the energy transport since they have the longer wavelength, have a quantized thermal conductance,

$$
\kappa\_p = \frac{k\_B^2 \pi^2}{3h} T\_\prime \tag{26}
$$

per each open channel, as predicted by Rego and Kirczenow [36] and verified by Schwab et al. [37].

Although the Landauer's theory appears quite natural, its physical implications are far reaching as we have seen with the introduction of the quantum of conductance [30–34] and the quantum of thermal conductance that has been observed [37]. However, the maximum strength of the theory is reached when it is coupled with the standard method for electronic structure calculations, namely static Density Functional Theory (DFT), and the non-equilibrium Green's function formalism (NEGF). These two methods made the Landauer's theory and the Boltzmann's transport theory, the base for almost any recent transport calculations. DFT indeed produces a reliable description of the system energy and electron density and of the electronic band structure, both necessary for the evaluation of the transmission probability although its use on transport modeling should be taken with care

as we will discuss shortly. Starting from the DFT description the NEGF method can express the transmission coefficients *Ti*,*<sup>j</sup>* in terms of the Green's functions of the leads and those of the central system. This combination allows for an almost parameter-free description of quantum transport at the atomistic level and it is the actual reference method for this kind of physical problems.

So far we have not discussed how to calculate the transmission probabilities between the leads *i* and *j*, *Ti*,*j*. This is usually a difficult problem since to make an accurate description of the scattering process we need modeling the states of the electrons inside the central region. Fortunately, we can combine the predictive power of DFT, which precisely describes the states of the system with the NEGF formalism which (as we will see) allows calculation of the transmission probability. A detailed derivation of all the results would be outside the scope of this review. There are a significant number of detailed monographs dedicated to the subject and we refer the interested reader to them [29,38]. Given a Hamiltonian *H*, we formally define its advanced and retarded Green's functions through (we set from now on ¯*h* = 1)

$$\left[-i\frac{d}{dt} - H\right]G^{R(A)}(t, t') = \delta(t - t'),\tag{27}$$

with the conditions *<sup>G</sup><sup>R</sup>*(*<sup>t</sup>*, *t*) = 0 if *t* < *t* and *<sup>G</sup><sup>A</sup>*(*<sup>t</sup>*, *t*) = 0 is *t* > *t*. An equivalent definition of the retarded and advanced Green's functions is based on their representation in terms of energy, i.e., after a Fourier transform,

$$\left[ (E \pm i\eta) - H \right] G^{R(A)}(E) = 1\tag{28}$$

where *η* is an infinitesimal positive quantity that serves to establish the analytical properties of *G<sup>R</sup>*(*A*). When we select a basis set for the Hilbert space, the Green's functions (as well as the Hamiltonian) can be represented as infinite matrices. However, normally only a certain number of states will be relevant for the dynamics (in our case those close to the Fermi energy or electrochemical potential *μ*) and therefore only a submatrix of the total Green's function will be needed.

For the study of electron transport through a nanoscale device, we need to describe the system (a central region) coupled to at least two external reservoirs. A pictorial representation of a device is given in Figure 2.

**Figure 2.** A central region (C) is connected to two external energy and particle left and right reservoirs (L and R) by metallic contacts. Current flows between the reservoirs when a temperature or bias gradient is established. The reservoirs are semi-infinite, namely the left proceeds from −∞ to C, the right goes from C to +<sup>∞</sup>.

If separated, each of these objects can be described by their Hamiltonian and Green's functions. When we couple the central region with the reservoirs, we are effectively introducing an interaction potential that couples states in the reservoirs with states in the leads. We introduce the following notation: *HA* with *A* = *L*, *R*, or *C* is the Hamiltonian of the left or right reservoir, or the central region, respectively; *VAB* with *AB* = *RC*, *LC*, *CR*, or *RL* represent the coupling between the central region with the left and right reservoirs, respectively. *VAB* can be for example a tunneling Hamiltonian between the lead and the central system. We assume there is not direct coupling between the reservoirs, so electrons must travel through the central region. In a matrix representation, we can think of the total Hamiltonian as a matrix whose sub-matrices are *HA* and *VAB*,

$$H = \begin{pmatrix} H\_L & V\_{LC} & 0 \\ V\_{CL} & H\_C & V\_{CR} \\ 0 & V\_{RC} & H\_R \end{pmatrix} . \tag{29}$$

Clearly, one could define a Green's function associated with this Hamiltonian whose dynamical equation can be written as

$$
\begin{pmatrix} E - H\_L + i\eta & -V\_{LC} & 0 \\ -V\_{CL} & E - H\_C + i\eta & -V\_{CR} \\ 0 & -V\_{RC} & E - H\_R + i\eta \end{pmatrix} \begin{pmatrix} G\_L^R & G\_{LC}^R & 0 \\ G\_{CL}^R & G\_C^R & G\_{CR}^R \\ 0 & G\_{RC}^R & G\_R^R \end{pmatrix} = \mathbf{f},\tag{30}
$$

where we have used the previous notation for the elements of *<sup>G</sup>R*, and 1ˆ is the identity matrix. This system of equations can be solved exactly to first express *GRLC* and *GRRC* in terms of *GRC* and then, solve for the latter. The exact result is

$$\boldsymbol{G}\_{\mathbb{C}}^{\mathbb{R}} = \left[\boldsymbol{E} + i\boldsymbol{\eta} - \boldsymbol{H}\_{\mathbb{C}} - \boldsymbol{\Sigma}^{\mathbb{R}}\right]^{-1} = \left[\boldsymbol{E} - \boldsymbol{H}\_{\mathbb{C}} - \boldsymbol{\Sigma}^{\mathbb{R}}\right]^{-1} \tag{31}$$

where we have defined the self-energy

$$
\Sigma^{\mathbb{R}} = V\_{\rm LC}^{\dagger} \left( E + i\eta - H\_{\rm L} \right)^{-1} V\_{\rm LC} + V\_{\rm RC}^{\dagger} \left( E + i\eta - H\_{\rm R} \right)^{-1} V\_{\rm RC} \tag{32}
$$

where with a † we indicate the Hermitian conjugate of *V*. Notice that in *GRC* the analytic properties are finally determined by those of the self-energy Σ and one can therefore neglect the terms *iη*. A similar equation can be derived for *GAC* . It is important to point out that in this theory, the leads enter both in the interaction with the central region through *VLC* and *VRC* and their isolated Green's function *<sup>G</sup>RL*(*R*)= *E* + *iη* − *HL*(*R*)−1.

To solve this set of equations, one assumes that there is no particle–particle interaction in the leads. This approximation can be justified by observing that they are thought of as normal metals and thus the screening length is relatively small, thus particles can be treated as weakly interacting if not independent. Within this scheme, *GL*(*R*) are uniquely determined and can be used to arrive at the lead self-energies to solve for *GRC* and *GAC* . In these last quantities, however, we cannot neglect particle–particle interaction. Due to the complexity of the problem, the exact many-body Hamiltonian *HC* is replaced with the so-called Kohn–Sham (KS) Hamiltonian, where an external single-particle potential (generally unknown) serves to mimic the effect of the interaction (see also the following sections for somewhat deeper discussion) [39–42]. The KS Hamiltonian is tailored to reproduce the exact ground-state energy and density, but it often produces an accurate description of the total Green's function of the isolated central region.

Finally, we connect the Green's function formalism with the Landauer's approach to quantum transport, since the former gives direct access to the transmission probability *Ti*,*j*. This step can be done by, for example, introducing the states for the left and right leads and solve the scattering problem with the central region by using the Green's functions. After some manipulations, one writes the transmission function *T*(*E*) as [43]

$$T(E) = \text{Tr}\left(\Gamma\_L G\_C^A \Gamma\_R G\_C^R\right),\tag{33}$$

where <sup>Γ</sup>*L*(*R*) = *i* Σ*RL*(*R*) − <sup>Σ</sup>*AL*(*R*)is the so-called spectral function of the leads. Other approaches or models to the calculation of the transmission probability are clearly possible, and provide the tools

to investigate novel and interesting phenomena, see for example Ref. [44] and the references in that focus issue.

This formalism is suited to describe both thermal and electrical transport since we have never specified the kind of gradient we maintain between the reservoirs. We are therefore entitled to consider both a bias voltage, and a temperature gradient that modifies the particle distribution functions in the reservoirs.

Quite naturally the formalism can be extended to consider the transport of energy through phonons or more generally vibrations. Formally, the only difference lies in replacing the particle Green's function *G* with the vibration Green's function *D* which is a solution of

$$\left(\omega^2 + i\eta - H\_v\right) D^R(\omega) = 1\tag{34}$$

where *ω* is the frequency of the vibration, and *Hv* the Hamiltonian function describing the vibration dynamics. Following the same steps as before, one can introduce the transmission function of the vibration *<sup>T</sup>*(*ω*) and derive a formula similar to Equation (33).

It is fair to examine here some of the problems one might face when using DFT + NEGF for the electronic transport calculations:

