A unified ballistic transport relation for anisotropic dispersions and generalized dimensions

An analytical formula is derived for particle and energy densities of fermions and bosons, and their ballistic momentum and energy currents for anisotropic energy dispersions in generalized dimensions. The formulation considerably simplifies the comparison of the statistical properties and ballistic particle and energy transport currents of electrons, acoustic phonons, and photons in various dimensions in a unified manner. Assorted examples of its utility are discussed, ranging from blackbody radiation to Schottky diodes and ballistic transistors, quantized electrical and thermal conductance, generalized ballistic Seebeck and Peltier coefficients, their Onsager relations, the generalized Wiedemann-Franz law and the robustness of the Lorenz number, and ballistic thermoelectric power factors, all of which are obtained from the single formula.


Introduction:
The need for analytical expressions for particle, energy, and current densities arises frequently in various branches of science and engineering. They are typically handled separately for each case of interest. This is because the densities depend on the quantum statistics of the type of particle or field of interest (i.e., whether they are fermions or bosons), on their specific energy dispersions (e.g. E =h 2 |k| 2 /2m or E =hv F |k|), or the specific dimensionality under consideration (e.g. d = 1, 2, 3). A single unified analytical expression is found in this work for all the above densities and their ballistic momentum and energy currents for anisotropic dispersions. This enables particle and energy densities, and ballistic particle and energy transport currents of electrons, phonons and photons to be treated in a unified manner amplifying their similarities and differences, the need for which has been advocated [1].
Setup: For particles in a box of dimension d = 1, 2, 3 and volume L d , wave-particle duality allows discrete wavevectors k i = p i (2π/L) where p i = 0, ±1, ... are integers. The resulting energy dispersion is written as Here the type t = 1 represents linear (or conical) dispersion with α i =hv i and t = 2 represents parabolic dispersion with α i =h/ √ 2m i , whereh = h/2π is the reduced Planck's constant. Table I shows that this formulation captures anisotropic dispersions via direction-dependent wave velocities v i (e.g. anisotropic, non-dispersive and transparent optical or acoustic media) or effective masses m i (e.g. the electron energy bandstructure of the semiconductor Silicon). Though the table and the following discussion is restricted to massless Dirac-like and massive parabolic dispersions, the formulation holds for other t. Extensions to other dispersions ought to be feasible along similar lines.
'Source' (1) and 'drain' (2) reservoirs, characterized by dimensionless parameters η 1 = β 1 µ 1 and η 2 = β 2 µ 2 are connected to the box of particles of dimension d on opposite * js3452@cornell.edu faces of dimension d − 1. Here β = (k b T ) −1 where k b is the Boltzmann constant. The chemical potentials µ 1 and µ 2 and temperatures T 1 and T 2 of the source and drain may in general be different. The particles in the source and drain reservoirs follow the equilibrium distribution functions f ± (E) = 1/(exp[β (E − µ)] ± 1) with + for fermions and -for bosons with the corresponding chemical potential and temperature.
The particles in the box are in quasi-equilibrium with two reservoirs via ballistic transport: for example, particles injected from the source share the same distribution as the source. Let x 1 denote the coordinate along which the potential difference is applied across the source and drain reservoirs. This generates a current J 1 = gL −d ∑(vg1(k)) a E b f ± (E) in each valley of the dispersion, where v g1 (k) = (h −1 ∇ k E) ·x 1 is the group velocity projected along the x 1 coordinate, a and b may be fractions or integers, and g combines degeneracies (e.g. valley, spin, polarization) and physical constants (e.g. electron charge, mass). The sum runs over all k states of all valleys in the dispersion. The parameters β and µ in f ± are dictated by the respective reservoirs, and the group velocity neglects Berry-phase contributions.
Main Result: The generalized current is recast as linear combinations of sums of type I u,s d,t = ∑ Ω k k u 1 E s f ± (E) that run over grid points in the d−dimensional hemisphere Ω k 1 for k 1 ≥ 0. This converts to the integral Splitting off k 1 using k 2 0 = k 2 1 + k 2 , passing into spherical coordinates d d−1 k = S d−1 k d−1 dk where S d = 2π d/2 /Γ(d/2) and Γ(...) is the Gamma function, this becomes which upon switching the order of integration evaluates to , and λ dB i = √ 4πα i β 1/t is the generalized anisotropic thermal de-Broglie wavelength in the direction i that characterizes the spatial spread of the wavepacket carrying the current. For example, is the Fermi-Dirac or Bose-Einstein integral. The generalized current expressed in terms of Equa- , which takes the compact form which is the main result of this work. Here j = a + b + r − 1, are constants that depend on (d,t, a, b). Physically, this is the current flowing in the x 1 direction. Though uninspiring at first look, its value lies in the fact that it is rather versatile in unifying the treatment of several disparate physical phenomena across dispersions and dimensions, as illustrated with assorted examples.
I: Particle Densities: From Equation 4 the generalized particle density for various statistics, dispersions, and dimensions is obtained with a = 0, b = 0: The number density of photons of g = 2 polarizations in thermal equilibrium with a radiation source at temperature T is obtained using F − . The chemical potential µ = 0 for photons which are bosons whose particle number is not conserved in thermodynamic equilibrium with matter at temperature T . In d = 3 it is 2J 0,0 3,1 = 16πζ (3)( k b T hc ) 3 where c is the speed of light and ζ (...) is the zeta-function, and in d = 2 is 2J 0,0 2,1 = 2π 2 ( k b T hc ) 2 . Because the photon has a positive branch dispersion, no energy gap, and Bose-Einstein statistics, no mass action law exists unlike for electrons and holes in semiconductors.
For a t = 2 parabolic conduction band energy dispersion valley degeneracy g c and the +ve sign for fermions gives the generalized volume density of electrons in d- where the band-edge density of states N d c = 2g c /λ d dBc is twice the inverse of the conduction band edge thermal de Broglie volume [2,3]. The equivalent d−dimensional distribution for the valence band If the Fermi level is at the Dirac point µ = 0 for metallic carbon nanotubes (d = 1), monolayer graphene (d = 2), and HgCdTe (d = 3), the intrinsic thermally generated electron density in each valley is n di = 4 This density sets the lowest carrier density (and hence highest electrical resistivity) that may be reached in such materials at any temperature. For E = ±hv F |k| where two cones touch, the sum of electron and hole densities is n d (+µ) + n d (−µ), resulting in a corresponding mass-action law for Dirac dispersions. The temperature dependence of the intrinsic electron/hole densities for conical bandstructure is therefore identical to the density of photons.

II: Energy Densities:
The volume density of energy stored in a photon field in equilibrium with a radiation source of tem- 3 , with corresponding results for other dimensions. For long-wavelength acoustic phonons, the thermal energy stored in a solid is similarly obtained by choosing g = 1 for each branch of sound velocity v s via α i =hv s , with t = 1 and a = 0, b = 1. This gives the thermal energy density 2J 0,1 3,1 = ( , and a heat capacity per atomic density n of C v n = 2∂ J 0,1 3,1 /∂ T = ( is the generalized law of the equipartition of energy, in which the approximation is valid for the Boltzmann approximation of η << 0 for both Fermions and Bosons. For particles in d−dimensions with mass and t = 2, there is k b T /2 energy per each dimension. For linear dispersion t = 1 on the other hand, there is k b T energy per each dimension as identified by Tolman [5] in the relativistic limit and investigated further for other dispersions [6,7]. For degenerate fermions characterized by η ≫ +1, the equilibrium average energy is u d ≈ µd/(d + t) and the resulting electronic specific heat [5,8]. For example, for electrons in metals with d = 3, g = 2, and t = 2, III: Ballistic Charge Currents: Suppose a solid with electronic bandstructure valleys of the types of Table I is connected to two reservoirs held at the dimensionless potentials η 1 and η 2 . By setting g = 2q where q is the electron charge of spin degeneracy=2, a = 1, b = 0, and f + for Fermions gives the charge current density for each valley in quasi-equilibrium with the source reservoir: where is the net macroscopic current, where the characteristic J 0 depends on t, d and λ dB , and is independent of the potential difference across the terminals.
The generalized form enables direct computation of ballistic currents in diodes and transistors of various dimensions and bandstructures. Applying Equation 7 to a Schottky diode of electron barrier height qφ b between a metal and a semiconductor with anisotropic bandstructure of dispersion type t = 2 yields a generalized current density for β (η 1 − η 2 ) = qV in the limit of η 1 , η 2 << −1 as is typically the case in experiments. The case for d = 3 was first derived by Bethe [9]; A d,2 is the d−dimensional Richardson coefficient, and the dimensionless form factor f = Π √ m l m t m e [10,11]. Similarly a lateral monolayer NbSe 2 /WSe 2 junction forms a 2D-2D ballistic Schottky diode for which the current is J schottky ≈ Equation 7 also applies for ballistic electron transport in 2-terminal resistors, or 3-terminal field-effect transistors (FETs). For example, for a 2D electron gas channel with d = 2 and bandstructure type t = 2, the current per unit width per each valley is , in Natori's form [12].
For bandstructure type t = 1 and d = 2 encountered in monolayer graphene or surface-bands of topological insulators, the current is , which in the limit η 1 , η 2 >> +1 typically encountered in experiments reduces to the Landauer limit [13] given by J = 2q 2 h V , indicating the conductance J/V is quantized to 2q 2 /h regardless of the type of bandstructure. For ballistic currents for t = 2, simultaneously fixing the total charge n d = J 0,0 d,t (η 1 ) + J 0,0 d,t (η 2 ) (say via capacitive gate control) requires a self-consistent solution for η 1 and η 2 for charge and current, resulting in the saturation of the ballistic current beyond a certain voltage difference between the source and drain, the hallmark of ballistic transistors that provide electronic gain for signal amplification, and switching for digital logic.
IV: Ballistic Heat Currents: The energy current density is obtained directly from the entropy in the ballistic case using a Landauer approach (see for example [14]) or in the scattering-limited diffusive case using the Boltzmann approach in the relaxation-time approximation (see for example [15]). The ballistic heat energy current from an electrode is Q = gL −d ∑ v g1 (k)(E − µ) f ± (E), where µ is the chemical potential and T the temperature of that electrode. The generalized ballistic energy current density in quasi-equilibrium with the source reservoir is then Q 1 = J 1,1 d,t − µ 1 J 1,0 d,t : and the net heat current density is Q = Q 1 −Q 2 . Since µ = 0 for bosons whose particle number is not conserved, for t = 1 and v i = c the net energy current with f − becomes  [16,17], a spectral integral over the Planck blackbody radiation density in the photon field. The corresponding currents for blackbody radiators in d = 2 is 3h )T 2 . The case of d = 1 is special since it does not depend on the speed of light; indeed it is independent of the energy dispersion altogether because the velocity cancels the density of states. Identical behavior exists for phonons and electrons, as discussed next.
For each branch of acoustic phonons, Equation 10 also gives the ballistic heat current between electrodes, with the speed of light replaced by the corresponding sound velocity. When the drain electrode is at T 2 = 0 K, the d = 1 heat current by an acoustic phonon branch of polarization g = 1 is J 1,1 1,1 = (π 2 k 2 b /6h)T 2 , identical to the photon current per polarization. Though the ballistic phonon heat currents depend on temperature non-linearly, for T 2 = T 0 and a slightly hotter source at is linear in temperature difference Q = G d ∆T . For d = 1 the thermal conductance quantum G 0 = π 2 k 2 b T /(3h) is obtained. This was theoretically anticipated [18,19] and subsequently experimentally observed [20].
Because for electrons µ = 0, Equation 9 gives an energy current dependent non-linearly on both µ and T of the source and drain reservoirs. For small differences, for t = 2 dispersion and d = 1 to leading order in η = β µ ≫ 1 it is which linearizing around a temperature T and µ 1 = µ 2 gives the same heat conductance quantum π 2 k 2 b T /(3h) per spin channel as for photons and phonons. In spite of the cancellation of the group velocity and the density of states in d = 1, the heat conductance quantum due to electrons derives from its Fermionic statistics, yet is identical to the heat conductance quantum of phonons and photons that follow Bosonic statistics. This strange similarity was recognized in [21][22][23], and Haldane's fractional exclusion statistics [24,25] was invoked to explain its possible origin [26]. The similarity of the 1d energy conductance quantum as a physical quantity independent of bosonic or fermionic statistics arising in the formulation here is traced to the following identities connecting the Fermi-Dirac and Bose-Einstein integrals: Unlike photons and phonons though, the electron chemical potential difference also drives an energy current, which is captured well in the generalized linear transport coefficients.
V: Linear Response Coefficients: Linearizing the above exact generalized formulations for ballistic transport for small differences in the reservoir chemical potentials µ 1 − µ 2 = ∆µ and temperatures β 1 − β 2 = ∆β brings correlations between particle and energy currents into sharper focus. Instead of linearizing the distribution function (e.g. see [27] for ballistic and diffusive thermoelectric coefficients), here the unified generalized currents embodied by various choices of (a, b) in Equation 4 are expanded to linear order J a,b d,t ≈ g a,b µ ∆µ + g a,b β ∆β around the average chemical potential µ 0 = (µ 1 + µ 2 )/2 and the average temperature T 0 given by β 0 = 1/k b T 0 = (β 1 + β 2 )/2. The linear coefficients are directly obtained as g a,b µ = (∂ J a,b d,t /∂ µ)| µ=µ 0 and g a,b Π =S · T 0 , and ]. (14) where g 0 is the product of spin and valley degeneracies, η = µ 0 β 0 , and r = (d − 1)/t generalizes the expressions for the several bandstructure types and dimensions. A conceptual difference of the ballistic coefficients is that the diffusive coefficients represent local properties, whereas the ballistic ones represent terminal (or system) properties as discussed lucidly for d = 1 by Butcher in [28]. The quantization of both σ and κ in d = 1 for η ≫ +1 is explicit for all t in Equation 14. The Onsager symmetry relation Π = ST 0 is seen to remain valid for the ballistic situation for all d,t. The generalized Lorenz number L d,t = κ/(σ T 0 ) obtained from Equations 14 goes to L d,t → π 2 3 ( k b q ) 2 in the degenerate fermion limit of β µ ≫ 1 for all d and t, highlighting the robustness of the Wiedemann-Franz law in the ballistic limit [28,29]. In the non-degenerate limit of β µ << −1 relevant for semiconductors, L d,t → ( d−1 t + 1)( k b q ) 2 . For all d,t the ballistic themoelectric power factor S 2 σ shows a maximum near µ = 0, except for the d = 3,t = 1 conical electron energy dispersion, in which case it increases monotonically with µ and saturates to 0 , an observation that warrants further study.

Particle Density
Energy Density Particle Current Energy Current Conclusions and Future Directions: The generalized ballistic current expression obtained in Equation 4 is found to be a versatile tool to compute and compare in a unified manner the particle and energy densities, charge and energy currents, thermoelectric coefficients and more for fermions and bosons of various energy dispersions. Such a compact formulation is well suited for optimization problems, in which the extrema of one or more densities, currents, transport coefficents, or their combinations need to be determined as a function of the dimensionality, type of dispersion, effective masses, wave velocities etc. To facilitate such studies, the generalized ballistic currents J a,b d,t for various a, b are summarized in Table II, and  Table III shows the linear response coefficients.
The energy dispersion types are not restricted to the specific cases of t = 1, 2 discussed, or to integers. The ballistic current expression may be extended for mixed dispersions of the tight-binding type E = E 0 + 2t cos ka ≈ E 0 + 2t[1 − (ka) 2 /2 + (ka) 4 /24...] near band edges, and to those that involve k i k j and k a i + k b j present in realistic systems, and topologically nontrivial terms may be introduced. Extending the formulation to multi-terminal cases in the spirit of the Landauer-Büttiker formalism [30,31], and especially for generalized nonlinear response in a magnetic field for various dimensions and dispersions is of high interest. So is exploring the various non-linear response predictions for ballistic electronic and thermoelectric transport phenomena. Extension of this approach to ballistic particle and energy transport in hetero-dimensional situations (mixed d), and for mixed dispersions and statistics (e.g. plasmons or phonon-polaritons) is also suggested as future work. The formulation is not limited to electrons, photons and phonons discussed here, and is applicable to molecular systems that undergo ballistic motion. Ballistic electron transport in condensed matter systems is seen primarily in nanoscale structures, which also have small numbers of particles, sometimes on the verge of failing the large number requirements on which traditional thermodynamic relations rest. The implications of recently revealed non-equilibrium thermodynamics equalities in nanoscale systems and on fluctuations of the densities, energies, and currents discussed here are therefore of significant theoretical and practical interest [32,33].