Quantum Transport of Particles and Entropy

Abstract

A unified view on macroscopic thermodynamics and quantum transport is presented. Thermodynamic processes with an exchange of energy between two systems necessarily involve the flow of other balancable quantities. These flows are first analyzed using a simple drift-diffusion model, which includes the thermoelectric effects, and connects the various transport coefficients to certain thermodynamic susceptibilities and a diffusion coefficient. In the second part of the paper, the connection between macroscopic thermodynamics and quantum statistics is discussed. It is proposed to employ not particles, but elementary Fermi- or Bose-systems as the elementary building blocks of ideal quantum gases. In this way, the transport not only of particles but also of entropy can be derived in a concise way, and is illustrated both for ballistic quantum wires, and for diffusive conductors. In particular, the quantum interference of entropy flow is in close correspondence to that of electric current.

Keywords: quantum transport; thermodynamics; thermoelectricity; transport equations.

Figure 1

(a) An energy current $$I_E$$ carried by and entropy current $$I_S$$ and a particle current $$I_N$$ current flows through two isothermal surface elements with constant temperatures $$T_1$$ and $$T_2\lesssim T_1$$ and electrochemical potentials $${\overline{\mu }}_1$$ and $${\overline{\mu }}_2\lesssim {\overline{\mu }}_1$$. Although $$I_E$$ and $$I_N$$ are constant, $$I_S$$ is not. The rate of entropy production between the two surfaces becomes negligible compared to the entropy currents through the surface elements, as the distance d and the differences $$T_1-T_2$$ and $${\overline{\mu }}_1-{\overline{\mu }}_2$$ go to zero. (b) A container with a gas of (quasi)-particles can be decomposed into small subvolumina with almost arbitrary size. Each subvolume represents another realization of the system ‘gas’, which continuously exchanges energy, entropy and particles with its neighbors. In the presence of a gradient of T or $$\mu$$ a subvolume can be considered to be in local equilibrium, provided that its size is about $${\Lambda }^3$$.

Figure 2

Elementary derivation of the diffusion constant in three dimensions: summing up the four contributions $$j_{Xz}=\pm \frac{1}{6}x\left(z\right)·\langle |\overrightarrow{v}|\rangle$$ to the z-component $$j_{Xz}$$ of the X-current density through the top and bottom surface of a cube of dimension $$\Lambda$$, one arrives in linear approximation at Equation (20), if the z-direction is chosen parallel to the gradient $$\nabla x(t,\overrightarrow{r})$$ of the X-density.

Figure 3

Thermopower over temperature $$\mathcal{S}/T$$ versus molar entropy over temperature $$\widehat{s}\left(T\right)/T$$ in the limit $$T\to 0$$ for many different metallic compounds. For electron-like $$(\widehat{q}<0)$$ conduction $$\mathcal{S}$$ is negative (lower panel), while for hole-like $$(\widehat{q}>0)$$ conduction $$\mathcal{S}$$ is positive (upper panel); Solid circles (squares) represent Ce (Yb) heavy-fermion systems. Uranium-based compounds are represented by open circles, metallic oxides by solid triangles, organic conductors by open diamonds, and common metals by open squares. For some data points, due to the lack of space, the name of the compound is not explicitly mentioned. The two solid lines represent $$\widehat{s}/\left(\widehat{q}T\right)$$ and are motivated by the discussion leading to Equation (87) below (adapted from [22]).

Figure 4

Entropy $$S_k$$ (solid lines) and entropy per particle $${\widehat{s}}_k$$ (dashed lines) of elementary Fermi- and Bose-systems with characteristic energy $$\epsilon \left(k\right)$$.

Figure 5

Schematic of single channel quantum wire connected to two reservoirs for energy, entropy, and particles. The quasi-one-dimensional character of the transport is ensured, once the transverse width of the wire is comparable to the Fermi wavelength. It can also be realized in wider strips via the formation of edge states in a quantizing magnetic field, i.e., in the quantum Hall regime. In this case, the magnetic field also provides a spatial separation between left- and right-movers. The elementary Fermi-or Bose-systems are charged with energy, entropy, and particles via the left (red) and the right (green) reservoir, respectively. They are in thermodynamic equilibrium with their source reservoirs, but not with each other.

Figure 6

(a) Schematic of irreversible relaxation processes in the reservoirs following the transmission of a particle. Each transmission event causes inelastic scattering processes (wiggly lines) in both reservoirs that change the particle numbers of the elementary Fermi-systems until the reservoirs are in equilibrium again. (b) Colored dashed and dash-dotted lines: rates $$S_{L,R}$$ of entropy change in the left and right reservoir together with the entropy current $$I_S$$ (black solid lines) leaving the left and entering the right reservoir. The T-difference $$\Delta T$$ is varied, while the electrochemical potentials are kept equal. At very low $$\Delta T$$ the transport of entropy dominates over its production, while at larger $$\Delta T$$, entropy production by the irreversible relaxation processes in the reservoirs governs their entropy change.

Figure 7

The left- (right-) propagating elementary Fermi- or Bose-systems labeled ‘$$\overrightarrow{k}$$’ emanating from the right (left) volume element of size $${\Lambda }_k$$ contribute an amount $$v_k{x}_k(T_{L,R},{\overline{\mu }}_{L,R})$$ to the total X-current density $${\overrightarrow{j}}_X$$.

Figure 8

The same quantum wire as in Figure 5, but interrupted by a quantum point contact (QPC) with transmission coefficient $$\mathcal{T}\left(\epsilon \right)$$. The scattering states of the particles emanating from the left (red) and the right (green) reservoir, respectively, must be considered to be one elementary Fermi- or Bose-systems, which cannot be further decomposed into subsystems.