Introduction

Research on atomically thin two-dimensional (2D) materials such as graphene, transition metal dichalcogenides (TMDCs), hexagonal boron nitride (h-BN), and black phosphorene is actively pursued due to their remarkable intrinsic properties with great interest in fundamental science and engineering applications.1 They offer great opportunities for a variety of applications, including nanoelectronics, optoelectronics, renewable energy and heat dissipation, which are highly expected for realization of atomically thin devices in the near future.2,3,4,5,6 With the quickly increased operation speed, and higher integration density, the power consumption in integrated nanoscale devices rapidly climb, which will cause degradation in device performance. In particular, for 2D materials based devices, owing to the atomic thickness, the localized heating in the confined volume is becoming more crucial issue.7,8 Moreover, understanding of thermal properties also directly relate to important application of 2D materials, including the developments of thermal interface materials,9,10 thermoelectric energy conversion,11 and thermal coating materials.12 Obviously, it is indispensable to understand the fundamental principles, and then improve the ability for manipulating thermal properties of 2D materials, which topic has attracted tremendous attention in recent years.13,14,15,16

In solid, the heat energy is not only carried by phonon, but also electron, because electrons likewise flow in the presence of a temperature gradient. However, in 2D materials, the contribution of these two heat carriers to total thermal conduction is not analogous.17,18,19,20,21 In 2D semiconductor such as MoS2 and black phosphorene, phonon dominates heat conduction.17,18,19 Even though group-IV 2D materials represented by graphene are gapless semi-metal hosting massless Dirac fermions makes them attractive, the low carrier concentration at Fermi level limits the electrical thermal conductance. Stanene, a monolayer of tin atoms, is reported to exhibit comparable electrical thermal conductance at room temperature with respect to the phonon counterpart.22 However, its lattice (electron) thermal conductance is only around 0.4 nWK−1 nm−2, which is only 10% of that of graphene (4.1 nWK−1 nm−2).23,24,25,26 Therefore, some important and fundamental questions remain unsolved, for instance, is there a 2D material which exhibits both high electrical and phonon thermal conductance? Is it possible to achieve higher thermal conductance over graphene?

In this work, we demonstrated the coexistence of high phonon and electron thermal conductance in the recently experimentally fabricated 2D hydrogen boride sheet,27 via a systematic study of electron and phonon quantum transport using first-principles calculation combined with non-equilibrium Green’s function theory. The computational details are shown in Methods section and supporting information, Fig. S1. Unlike the usually observed reduction in phonon thermal conductance induced by tensile strain, the orbitally driven electrical thermal conductance exhibits abnormal strain dependence. As carbon’s neighbor in the periodic table, boron has been expected to form graphene-like 2D structure borophene for many years.28,29,30,31,32 2D boron sheets, also called borophene, have been successfully synthesized on a Ag (111) surface.33,34 Following the realization of the buckled phase and the planar phase of borophene, extensive theoretical studies were conducted to explore the various applications of borophene such as hydrogen storage, rechargeable metal-ion batteries, superconductors, heat channel and mechanically robust components.35,36,37,38,39,40,41,42,43,44,45,46,47 In particular, the strain effects on the electronic, optical, and transport properties of borophene are explored.39,44,48,49,50 The fascinating behaviors revealed here in 2D hydrogen boride sheet not only widen our knowledge in thermal transport in 2D materials, but also provide a new route to manipulate the thermal conductance since it is much easier for controlling electrons than controlling phonons.

Results and discussion

Ideal strength and phonon instability

The hydrogenated graphene-like borophene, in another word, 2D hydrogen boride, is illustrated in Fig. 1a. Its unit cell consists of 4 hydrogen atoms and 4 boron atoms. It shows a graphene-like honeycomb lattice with a flat plane made from boron atoms, with two layers of hydrogen atoms are decorated on both sides of the B–B bonds. The optimized lattice constants are a = 5.298 Å and b = 2.987 Å. For convenience, we define the B–B bonds bridged by the hydrogen atoms as Bond-1, and the B–B bonds without hydrogen bridging as Bond-2. The lengths of Bond-1(r1) and Bond-2 (r2) are 1.82 Å and 1.72 Å, respectively. The B–H bond length was found to be 1.32 Å. The structural parameters are in excellent agreement with previous report.51

Fig. 1
figure 1

a The crystal structure of 2D hydrogen boride. The solid lines show the unit cell. b Phonon dispersion and density of states of 2D hydrogen boride

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The calculated phonon dispersion is shown in Fig. 1b. Correspondingly, we present atomic motions of vibrational modes in Fig. S2. There is no sign of imaginary frequencies in the phonon dispersion which accordingly confirms the thermodynamic stability of this attractive 2D material. Around the Γ point, there are three acoustic modes, two of them (longitudinal acoustic and transverse acoustic modes, LA and TA) have linear dispersion, and the third one (out-of-plane acoustic mode, ZA) has parabolic dispersion. The high-frequency optical modes (>1500 cm−1) are contributed by B-H bonds, and the rest low-frequency phonon modes (ω < 1000 cm−1) are contributed by B atoms, which are lower than those in graphene (~1600 cm−1). As illustrated in Fig. S2, in high-frequency region, atomic motions associated with \({\mathrm{B}}_{3{\mathrm{g}}}^4\), \({\mathrm{B}}_{3{\mathrm{g}}}^3\), \({\mathrm{B}}_{2{\mathrm{u}}}^3\), \({\mathrm{B}}_{2{\mathrm{u}}}^2\), occur mostly along the armchair-direction. On the other hand, atomic motions associated with \({\mathrm{B}}_{3{\mathrm{u}}}^3\), \({\mathrm{B}}_{1{\mathrm{g}}}^2\) occur mostly along the zigzag-direction.

Strain effect on electronic, optoelectronic, thermal, and thermoelectric properties of 2D materials has gained tremendous attention.52,53,54,55 Furthermore, strain dependent phonon dispersion provides direct criteria for the thermodynamic stability of layered materials. Next we explore the phonon dispersion of 2D hydrogen boride under uniaxial tensile and compressive strain. The strain is applied on the rectangular unit cell along the armchair- (or zigzag-) direction, and the lattice is optimized along the lateral direction. The strain is defined as \(\varepsilon _i = (L_i - L_{i0})/L_{i0}\), i = armchair/zigzag, where Li is the strained lattice constant along armchair- or zigzag-direction, and Li0 is the corresponding original lattice constant. When 2% uniaxial compressive strain is applied in armchair or zigzag direction, the structure becomes unstable (shown in Fig. S3). Therefore, in the following, we focus on the effect of tensile strain on the thermal transport of 2D hydrogen boride. Figure 2 shows the phonon dispersions of 2D hydrogen boride under the tensile strain along armchair- and zigzag-direction, respectively. For convenience, the negative y–axis is used to show the imaginary frequencies. The phonon mode exhibiting imaginary frequency will grow in amplitude until the structure shifts to a new stable state, demonstrating the instability of crystallographic nature. Under the uniaxial strain in the armchair-direction, softening of phonon modes occurring gradually, and ZA mode along the Γ−X direction softened and the frequencies become imaginary when the strain reaches a critical value (between \(\varepsilon _{\mathrm{armchair}} = 17\%\) and \(\varepsilon _{\mathrm{armchair}} = 18\%\)). For zigzag-direction strain, one phonon mode becomes unstable firstly with around 23% tensile strain. The unstable mode is also the out-of-plane ZA mode, indicating 2D hydrogen boride is unstable with respect to long wavelength distortion and is thus elastically unstable under large tensile deformation.

Fig. 2
figure 2

The phonon dispersion of 2D hydrogen boride under uniaxial strain along armchair- and zigzag-direction. a\({\mathrm{\varepsilon }}_{{\mathrm{armchair}}}\) = 17%, b\({\mathrm{\varepsilon }}_{{\mathrm{armchair}}}\) = 18%, c \({\mathrm{\varepsilon }}_{{\mathrm{zigzag}}}\) = 22%, d \({\mathrm{\varepsilon }}_{{\mathrm{zigzag}}}\) = 23%

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We also studied the ideal strength of 2D hydrogen boride using strain-stress curves under various strains, as shown in Fig. S4. When the strain is applied in the armchair-direction, the stress increases with increasing strain until \(\varepsilon _{\mathrm{armchair}} = 22\%\), beyond which the stress decreases sharply. In the zigzag-direction, the ultimate tensile strain is \(\varepsilon _{\mathrm{zigzag}} = 34\%\). The corresponding ideal strengths are 19.66 and 17.77 N/m, respectively. In each case the phonon instability occurs before the peak of the stress-strain curve, indicating the ideal strength of 2D hydrogen boride is dictated by instabilities in the acoustic phonon spectra, which is similar to the case for pristine borophene and hydrogenated borophene.56 From the optimized lattice constants along transverse and longitudinal directions, we obtained the Poisson’s ratio of 2D hydrogen boride as 0.26 (armchair direction) and 0.21 (zigzag direction), which are in good agreement with previous report (0.25 and 0.19, respectively).44

Phonon thermal conductance of 2D hydrogen boride

Next, we employ the phonon NEGF approach to investigate the lattice thermal conductance of 2D hydrogen boride. The scaled thermal conductance is introduced to characterize the thermal transport properties. Herein, the cross-sectional area S is defined as S = W × d, where W is the width and d is the thickness. For 2D hydrogen boride, its thickness is 3.49 Å, by adding the diameter of hydrogen atom (1.60 Å) with the distance of two hydrogen layers, which is 1.89 Å.27 Figure 3a shows the scaled thermal conductance of 2D hydrogen boride as a function of temperature along both armchair- and zigzag-direction. The increasing trend of thermal conductance with temperature is a common phenomenon for ballistic phonon transport.24 It can be noticed that the phonon thermal conductance at room temperature along armchair- and zigzag-direction are 3.93 nW K−1 nm−2 and 4.07 nW K−1 nm−2, which is slightly smaller than that of graphene (~4.1 nW K−1 nm−2),24 but much higher than most 2D materials. In 2D hydrogen boride, the effective atomic mass (B–H group) can be considered as same as that of graphene, and the strength of B–B bond and C–C bond are comparable. Furthermore, the honeycomb lattice of 2D hydrogen boride is much similar to that of graphene. As consequence, 2D hydrogen boride and graphene have comparable lattice thermal conductance.

Fig. 3
figure 3

a Phonon thermal conductance of 2D hydrogen boride versus temperature. b Phonon transmission coefficient T[ω] per unit area against phonon frequency for 2D hydrogen boride

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Figure 4 shows the room temperature phonon thermal conductance (σp-armchair and σp-zigzag) as a function of tensile strain along armchair- and zigzag-direction, respectively. It is clear that armchair-strain leads to quick decrease in σp-armchair and σp-zigzag within the whole range of considered strain. On the other side, zigzag-strain reduces σp-armchair, while σp-zigzag increases slightly when strain is less than 14%, and then decreases when εzigzag > 16%. Overall, tensile strain results in reduction in lattice thermal conductance, excluding σp-zigzag which exhibits weak dependence to strain along zigzag-direction.

Fig. 4
figure 4

The room temperature phonon thermal conductance as a function of tensile strain. a Armchair-direction strain and b zigzag-direction strain. c The differential thermal conductance Δσ under 8% armchair-strain. d The differential thermal conductance Δσ under 8% zigzag-strain

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To understand the nontrivial strain effect on thermal conductance, we present the phonon transmission along the armchair- or zigzag-direction, as shown in Fig. 3b and Fig. S5. When the strain is applied in the armchair-direction, tensile strain results in dispersive phonon modes converged to the low-frequency region. As shown in Fig. S5(a). and (b), the transmission coefficient in the frequency range of 100–1000 cm−1 decreases, and there has a significant red-shift in the maximum frequency (ωmax) in transmission coefficient along armchair- and zigzag-direction. This is consistent with the shifting down in acoustic modes as shown in Fig. 2. All these features result in reduction of thermal conductance upon armchair-strain. On the other side, for zigzag-direction strain, there is also an obvious decrease in the transmission coefficient of armchair-direction, combined with red-shift in the high-frequency phonon (750 ~ 1000 cm−1) transmission. For phonon transmission along zigzag-direction, aside to these negative influences, there is slight increase in phonon transmission at the zone center (ω < 100 cm−1), leading to the weak zigzag-strain dependence observed in thermal conductance along zigzag-direction (Fig. 4b).

To further understand how strain affects the relative contribution from phonons at different frequencies, next we investigate the differential thermal conductance, which is defined as

$$\Delta {\boldsymbol{\sigma }}(\omega ) = \frac{{\hbar \omega }}{{2\pi S}}\frac{{\partial f\left( \omega \right)}}{{\partial T}}\Delta {\mathrm{\Xi }}(\omega ),$$
(1)

where \(\Delta {\mathrm{\Xi }}(\omega )\) is the difference in phonon transmission at 8 and 0% strain. Here we focus on the case at room temperature 300 K. Δσ(ω) identifies the contribution to the change in thermal conductance by phonon mode with frequency ω, with respect to the 8% tensile strain (armchair- or zigzag-strain).

With strain in the armchair-direction (Fig. 4c), the reduced thermal conductance (Δσp-armchair and Δσp-zigzag) is due to the drop in the phonon transmission in low-frequency regime. However, when tensile strain is applied in the zigzag-direction (Fig. 4d), for σp-zigzag, there is an increase in phonon transmission in the frequency range of 0–900 cm−1. On the other side, for σp-armchair, there is obvious drop over the same frequency range. Therefore, σp-zigzag increases with zigzag-direction strain, while σp-armchair decreases significantly.

Giant electrical thermal conductance

Figure S6 (a) illustrates the band structure of strain-free 2D hydrogen boride. It shows that this 2D material yields metallic character and our predicted band structure matches excellently with the previous theoretical reports.44,51

Based on the electronic band structure, we calculated the intrinsic electrical thermal conductance σe with zero gate voltage (μ = Ef). Figure 5a, b shows the armchair-direction electrical thermal conductance σe-armchair and zigzag-direction electrical thermal conductance σe-zigzag as a function of uniaxial strain at room temperature (T = 300 K). σe-armchair and σe-zigzag for strain-free 2D hydrogen boride are 3.56 and 2.88 nW K−1 nm−2, respectively, which are much higher than that of graphene (~0.3 nW K−1 nm−2).22 When the strain is applied in the armchair-direction, σe-armchair and σe-zigzag increase with tensile strain. In contrast, under strain along zigzag-direction, σe-armchair and σe-zigzag decreases rapidly. The electronic conductance Ge and electrical thermal conductance σe can be described by the Wiedemann-Franz law, \(\sigma _e = L_eG_eT\), where Le is the Lorenz constant, T is temperature. The electronic conductance Ge is \(G_e = ne^2\tau /m\), here n is the carrier concentration, τ is the carrier relaxation time, and m is the effective mass of charge carrier. To understand the strain effects on electrical thermal conductance, we show the electron density of states (DOS) under armchair-direction strain (Fig. 5c) and zigzag-direction strain (Fig. 5d), respectively. For strain along the armchair-direction, with tensile strain increases from 0 to 16%, the metallic electronic character is completely preserved, and the DOS at the Fermi level increases from 0.78 to 1.12, revealing higher carrier concentration induced by tensile strain, consequently leads to increase in electronic conductance and electrical thermal conductance. With armchair-strain increases from 0 to 16%, σe-armchair (σe-zigzag) increases from 3.56 (2.88) nW K−1 nm−2 to 5.02 (5.18) nW K−1 nm−2, with about 41% (79%) increment. On the other side, the effect of zigzag-strain on the modification of the electrical thermal conductance is more pronounced, which is related to the bonding features of 2D hydrogen boride. With zigzag-strain, σe-armchair (σe-zigzag) decreases quickly because of the reduction in density of states at Fermi level. More interesting, when strain exceeds 16%, σe-armchair (σe-zigzag) decreases to 0 nW K−1 nm−2, representing fully off-state of the heat channel by electron. As shown in Fig. S6, when tensile loading along zigzag-direction is applied, there is metal-to-semiconductor transition, results in significantly reduction in the DOS at the Fermi level. The band gap is 0.15 eV at 16% tensile strain in the zigzag direction. In this material, there are three different types of bond, B-B covalent bond along the zigzag-direction, B–B bond with hydrogen bridging along the armchair-direction, and B–H bonds. For tensile strain along the zigzag-direction, the covalent B–B bonds are oriented along the loading direction. As the electron localization occurs at the center of B–B covalent bond, it is sensitive to the loading along zigzag-direction, which results in the metal-to-semiconductor transition.44 The metal-to-semiconductor transition leads to significant reduction in the electrical thermal conductance, from about 3 nW K−1 nm−2 to 0 when tensile strain exceeds 16%.

Fig. 5
figure 5

The room temperature electrical thermal conductance as a function of strain for a armchair-direction strain and b zigzag-direction strain. The DOS of 2D hydrogen boride as a function of strain for c armchair-direction strain and d zigzag-direction strain

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High total thermal conductance over graphene

From the phonon and electron thermal conductance, finally we explore the total thermal conductance of 2D hydrogen boride \(\left( {\sigma = \sigma _{ph} + \sigma _e} \right)\) at room temperature, and the strain dependence. For strain-free 2D hydrogen boride, the total thermal conductance along armchair- and zigzag-direction are 7.48 and 6.96 nW K−1 nm−2, respectively. It is worth stressing that the thermal conductance along armchair-direction is almost two-fold higher than that of graphene,22,24 and it is the highest value among all known 2D materials. This is contributed by the similar lattice thermal conductance with respect to graphene, and the significantly higher electrical thermal conductance compared with graphene.

As shown in Fig. 6, total thermal conductance of 2D hydrogen boride along both armchair- and zigzag-direction increase with armchair-strain, totally unlike the reduced phonon thermal conductance in the same range of tensile strain. In general, tensile strain will lead to reduction in thermal conductance because the reduced phonon group velocity and phonon transmission. However, for 2D hydrogen boride, abnormal tensile strain dependence occurs, because of the increased electrical thermal conductance. With armchair-strain increases from 0 to 16%, phonon thermal conductance decreases from about 4 nW K−1 nm−2 to about 3.2 nW K−1 nm−2. However, in the same range of tensile strain, electrical thermal conductance increases from 3 ~ 3.5 nW K−1 nm−2 to ~ 5 nW K−1 nm−2. The large increase in electrical thermal conductance offsets the reduction in phonon thermal conductance, and results in this abnormal strain dependence. Moreover, the effect of zigzag-strain on total thermal conductance is more remarkable. Using armchair-direction thermal conductance as an example, we can find that zigzag-strain can reduce phonon thermal conductance along armchair-direction about 40%, while the reduction ratio is up to 70% for total thermal conductance. This happens because of the large contribution of electron to total thermal conductance, and the electronic bandgap opening due to zigzag-strain. Therefore, compared with other 2D materials in which phonons dominate thermal conductance, there is larger degree of freedom to modulate thermal conductance of 2D hydrogen boride.

Fig. 6
figure 6

The room temperature total thermal conductance as a function of strain for a armchair-direction strain and b zigzag-direction strain

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In conclusion, we investigated phonon, mechanical and electronic properties of 2D hydrogen boride. Based on the first-principles calculation provided phonon property and electronic band structure, combined with quantum transport theory, we predicted phonon and electrical thermal conductance. The phonon thermal conductance of 2D hydrogen boride is close to that of graphene, while its electrical thermal conductance is almost ten times over graphene. This happens due to the high electron density of state at Fermi level. Moreover, at the Fermi level, the tensile strain along armchair-direction gives a higher electronic density of states, which effectively brings more states toward electrical thermal conductance. In contrast, zigzag-direction tensile strain induces bandgap opening at the Γ-point, which can fully “close” the electron channel, leading to zero electrical thermal conductance. This is the opposite of the armchair-direction strain effect. Our work reveals giant thermal conductance in 2D hydrogen boride, a new boron based 2D material synthesized experimentally.

Methods

DFT calculations

In this paper, we studied electron and phonon thermal conductance of the experimentally fabricated 2D hydrogen boride sheets. The structural parameters, lattice vibrations and interatomic force constants (IFCs) are calculated from the density functional theory (DFT) by using the Quantum-Espresso code,57 within the generalized gradient approximation (GGA) of the Perdew-Burke-Ernzerhof (PBE). Cutoff energies of 50 Ry for plane wave and 500 Ry for electronic density are adopted. The first Brillouin zone was sampled with a 35 × 25 × 1 for the unit cell, and a vacuum layer with 15 Å thickness is adopted. Each atomic structure of the 2D hydrogen boride was fully relaxed until the forces are less than 0.01 eV/Å. The dynamic matrix based on a 7 × 5 × 1 grid of q points is calculated by density functional perturbation theory (DFPT), and giving the through inverse Fourier transform to real space.

Strain effect

For uniaxial tensile strain, we applied a series of tensile strains on the rectangular unit cell along the armchair- (or zigzag-) direction and relaxed the lattice along the zigzag- (or armchair-) direction until the corresponding conjugate stress component is less than 0.01 GPa. In the structure optimization, periodic boundary condition is applied to both armchair- and zigzag-direction, and all atoms are allowed to move. The tensile strain is defined as \(\varepsilon _i = (L_i - L_{i0})/L_{i0}\), i = armchair/ zigzag. Here Li is the strained lattice constant along armchair- or zigzag-direction, and Li0 is the corresponding original lattice constant, respectively.

Phonon thermal conductance

Phonon mean free path is the key quantity for classifying transport feature. It is defined as the average transport distance of phonon between two successive scatterings arises due to the coupling between phonons and other quasi-particles, as well as phonons themselves. In physics, the phonon-phonon scattering process requires the conservation of energy and quasi-momentum. In low-dimensional nano materials, confinement effects induce energy level splitting, make the satisfaction of the conservation of energy and quasi-momentum difficult. Moreover, because of the high Debye temperature in carbon-based and boron-based materials,58 room temperature can be treated as low temperature. At room temperature, the phonon excitation is low, thus the possibility of phonon-phonon anharmonic scattering process is greatly suppressed. Consequently, carbon-based and boron-based materials usually possess long phonon mean free path. When the size of material is smaller than phonon mean free path, phonons experience almost no scatterings during transport, leading to ballistic phonon transport. In such case, thermal conductivity is not a constant but significantly depends on the system size. This phenomenon has been experimentally observed in graphene59 and carbon nanotubes.60 However, thermal conductance is well defined and does not change as the system length increases. Therefore, in this work, we focus on thermal conductance of this novel material.

The phonon thermal conductance can be obtained from the phonon non-equilibrium Green’s function (NEGF) method based on the calculated force constant matrix. As shown in Fig. S1, the sample is connected to left and right heat reservoirs, which are assumed to be in thermal equilibrium with temperature TH and TC, respectively. The thermal conductance can be described by the Landauer formula:61

$$\sigma (T) = \frac{1}{{2\pi S}}\mathop {\int }\limits_0^\infty d\omega \hbar \omega \frac{{\partial f\left( \omega \right)}}{{\partial T}}{\mathrm{\Xi }}(\omega ),$$
(2)

where f(ω) is the Bose-Einstein distribution function for a phonon with frequency ω at the reservoirs, S is the transverse cross-sectional area, \({\mathrm{\Xi }}({\mathrm{\omega }})\) is phonon transmission as a function of phonon frequency ω, which can be evaluated according to the IFCs within the NEGF framework.61,62

The cross-section area scaled thermal conductance σ has relationship to thermal conductivity κ as \(\sigma = \frac{\kappa }{L}\), where L is the longitudinal length of the sample. Obviously, as thermal conductance σ is length independent, in the ballistic regime, thermal conductivity increases linearly with system length.

Electrical thermal conductance

The electrical thermal conductance is calculated using NEGF-DFT codes in the Atomistix ToolKit (ATK) under the equilibrium conditions.63,64 The first Brillouin zone is sampled with 1 × 100 × 100 Monkhorst-Pack k-mesh, and the cutoff energy is set to 150 Ry. All atomic positions are relaxed until the maximum inter-atomic force is smaller than 0.01 eV/Å. For electron transport calculation, two semi-infinite leads are adopted as left and right electrodes, and the studied system is the central part. The electron transmission is calculated as:

$${\mathrm{\Xi }}_{\mathrm{e}}\left[ \omega \right] = {\mathrm{\Xi }}_{\mathrm{r}}({\mathrm{G}}^{\mathrm{r}}{\mathrm{\Gamma }}_{\mathrm{L}}{\mathrm{G}}^{\mathrm{a}}{\mathrm{\Gamma }}_{\mathrm{R}}),$$
(3)

where Gr = (Ga)† is the retarded surface Green function of the central of the scattering region and \({\mathrm{\Gamma }}_{{\mathrm{L}}({\mathrm{R}})}\) represents the coupling interaction with the left seim-infinite lead. Thus electronic thermal conductance σe can be obtained by,

$$\sigma _e = \frac{1}{T}\left[ {L^{\left( 2 \right)} - \frac{{\left( {L^{(1)}} \right)^2}}{{L^{(0)}}}} \right],$$
(4)

with

$$L^{(n)} = \frac{2}{h}{\int} {\left[ { - \frac{{\partial f\left( {E,T} \right)}}{{\partial E}}} \right]T_e\left( E \right)\mathrm{d}E} ,$$
(5)

where f(E, T) is the Fermi-Dirac distribution function at temperature T.