The electrothermal conductance and heat capacity of black phosphorusParijat Sengupta,1 Saptarshi Das,2 and Junxia Shi3, a)

1)Department of Electrical and Computer EngineeringUniversity of Illinois at Chicago, Chicago, IL 60607.2)Department of Engineering Science and MechanicsPennsylvania State University, State College, PA 16802.3)Department of Electrical EngineeringUniversity of Illinois at Chicago, Chicago, IL 60607.

We study a thermal gradient induced current (Ith) flow in potassium-doped two-dimensional anisotropic blackphosphorus (BP) with semi-Dirac dispersion. The prototype device is a BP channel clamped between twocontacts maintained at unequal temperatures. The choice of BP lies in the predicted efficient thermolectricbehaviour. A temperature-induced difference in the Fermi levels of the two contacts drives the current (typifiedby the electro-thermal conductance) which we calculate using the Landauer transport equation. The currentshows an initial rise when the device is operated at lower temperatures. The rise stalls at progressivelyhigher temperatures and Ith acquires a plateau-like flat profile indicating a competing effect between a largernumber of transmission modes and a corresponding drop in the Fermi level difference between the contacts.The current is computed for both n- and p-type BP and the difference thereof is attributed to the particle-hole asymmetry. The utility of such calculations lie in conversion of the heat production and an attendanttemperature gradient in miniaturized devices to useful electric power and a possible realization of solid-statePeltier cooling. Unlike the flow of Ith that removes heat, the ability of a material to maintain a steadytemperature is reflected in its heat capacity which is formulated in this work for BP via a Sommerfeldexpansion. In the concluding part, we draw a microscopic connection between the two seemingly disparateprocesses of heat removal and absorption by pinning down their origin to the underlying density of states.Finally, a qualitative analysis of a Carnot-like efficiency of the considered thermoelectric engine is performeddrawing upon the previous results on thermal current and heat capacity.

I. Introduction

The miniaturization of circuit components introducesthe problem of localized heating1,2 that can give riseto temperature overshoots and degrade their overall lifespan. Such problems have necessitated the need for im-proved cooling schemes3 to hold the operating temper-ature to reasonable limits. The overarching goal is todrive the heat from localized high-temperature regions- more commonly known as hot-spots - to draw levelwith ambient conditions. While the generated heat canbe simply removed, a profitable spin-off is to transformthe heat current4 into electric power taking advantageof the Seebeck effect5 which describes current flow ina looped conductor under a finite temperature gradi-ent. This constitutes the basis for electric-thermal en-ergy conversion.6–8 The converse of this, the Peltier ef-fect, manifests as a reduction (or gain) in temperature viapumping of heat when charge current passes through ajunction of two dissimilar materials with distinct ther-mal behaviour. The optimization of the Seebeck andPeltier power conversion techniques have understand-ably received much attention, they have, however, ac-quired greater significance as newer materials - notablygraphene9 - hold promise of better thermoelectric op-eration, demonstrated by a higher figure of merit, ZT.In addition to the two-dimensional (2D) graphene whichhas a large Seebeck coefficient10,11, in part, attributable

a)Electronic mail: lucyshi@uic.edu

to its linearly dispersing bands, more unconventional 2Dcases not fully-explored yet also exist with ‘hybrid’ bandprofiles. For instance, Pardo et al. unveiled a semi-Dirac dispersion12,13 marked by a co-existence of the lin-ear Dirac-type linear and parabolic branches in VO2 lay-ers embedded in TiO2. In recent times, more evidenceof such dispersion was also established in an experimen-tal (and later through first-principles calculations) studyof potassium-doped multi-layered black phosphorus (BP)by Kim et al. ; they showed14,15 that the armchair (y-axis) direction carried a linear dispersion while acquiringa parabolic character along the zigzagged x -axis.

In this work, using BP as the representative materialwith a semi-Dirac dispersion, we quantitatively assess theelectrothermal conductance and the attendant currentthat flows solely on account of a temperature gradientinduced difference in Fermi levels. The thermally-drivencurrent calculations are done using the Landauer equa-tion.16 In principle, this also serves as an illustration ofthe microscale energy ‘harvesting’ paradigm (thermal-to-electric) via a prototypical Seebeck-like power generator(for a schematic representation, see Fig. 1) with applica-tions to solid state cooling through heat removal via thePeltier process. In the second half, a related calculationestimates the heat capacity to gauge its effectiveness as acoolant; an action that is principally converse to that ofheat removal via a temperature-driven current and essen-tially measures the material’s inherent ability to absorbheat. We compute the heat capacity for BP using thewell-known formula derivable from a Sommerfeld expan-sion.17 As for the choice of BP18–20 for these calculations,

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FIG. 1. The schematic illustrates the suggested arrangementof a BP channel flanked between left and right contacts main-tained at temperatures TL > TR and electrochemical poten-tials µL and µR, respectively. The curved lines denote thesmeared Fermi function at a finite temperature. A thermally-excited electron (hole) current (Ith) flows from the contactat a higher (lower) temperature. The inset shows the puck-ered unit cell of a single layer BP. The channel dimensions (inappropriate units) are W and L along the x - and y-axes.

we explain by briefly expounding on its unique thermaland electrical transport characteristics that promise im-proved thermoelectric behaviour. The efficiency of ther-moelectric processes, in general, are marked by their ZT,which quantifies the performance metric manifesting asan interplay between the electric and thermal conductiv-ities. A significant value for ZT is achievable for a highelectric conductivity (σ) and a concurrent low value of itsthermal counterpart (κ). In most materials, however, asimultaneous fulfillment of this condition is difficult toaccomplish, usually prompting a trade-off to optimizethe ratio, σ/κ. BP, in contrast, because of the intrin-sic anisotropy21 has a pronounced electric conductivityin the armchair direction while significant thermal trans-port occurs along the zigzag-axis.22 A combination of ex-ternally impressed thermal and potential gradients alongthe zigzag and the armchair, respectively, can thereforeaid in sharply enhancing σ/κ and consequently ZT. Theuptick in ZT - predicted to touch 2.5 - offers a viablealternative to existing thermoelectric materials.23–25

The calculations show that the thermal current ismarked by a steady increase when the prototype device(see Fig. 1) is operated at lower temperatures. However,the rise tails off for elevated temperatures and a plateau-like curve is obtained. This is explained by noting thatthe difference in Fermi levels between the two contactsshows a slow increase at higher temperature leading tosaturating current profile. We point out that the Fermilevel difference serves as the greater driving force vis-a-vis the increased number of transmission modes at highertemperatures. Additionally, we find, as anticipated, ahigher thermal current for a larger temperature differ-ence between the contacts. The heat capacity, derivedfrom the density of states (DOS), similarly shows an in-crease with energy around the Dirac crossing since atlow values of momentum the linear branch dominates

the quadratic contribution. The DOS increases linearlywith energy for linear bands. A comparative study ofthe heat capacity of BP with a two-dimensional quan-tum well of GaAs, a prototypical III-V compound withvery parabolic conduction bands around the Brillouinzone, reveals a three-order higher value for the formersuggesting superior performance as a coolant. In gen-eral, a semi-Dirac system like K -doped BP, in principle,has a higher heat capacity compared to materials iden-tified by a purely parabolic dispersion. We also explainhow the thermal current and the heat capacity, indepen-dent of the chosen material, and outwardly dissimilar intheir functionality, owe their modulations to the under-lying arrangement of electron states. A short qualitativetheoretical discussion visualizes the arrangement as animplementation of a Carnot-like heat engine with tun-able efficiency. A caveat about what follows must bemade explicit here: The current flow, including heat andcharge, are essentially non-equilibrium phenomena; how-ever, in this work, we tacitly assume that all thermoelec-tric phenomenon is modeled for situations not far awayfrom equilibrium such that the dispersion relation andFermi distribution function (which is strictly defined forequilibrium situations) are considered valid for all casesdescribed herein.

II. Analytic formulation

A semi-Dirac material26 is characterized by a set ofparabolic and linear bands along two mutually perpen-dicular directions. The minimal Hamiltonian is

H =(∆ + αk2

x

)τx + βkyτy. (1)

In Eq. 1, the coefficient α = ~2/2me and β = ~vf . TheFermi velocity is vf , the effective mass is me, and τx,yrepresent the iso-spin matrices. For a generalized case,we also include a band gap (∆) in conjunction with theparabolic set of bands. The dispersion is straightforwardto obtain, we have

E (k) = ±√

(∆ + αk2x)

2+ (βky)

2. (2)

The + (-) sign in the energy expressions denote the con-duction (valence) state. The dispersion by letting ∆→ 0in Eq. 2 clearly points to mass-less Dirac Fermions alongthe armchair direction (y-axis) while the zigzag axis (x -axis) hosts the conventional ‘massive’ parabolic electron.The band gap in Eq. 2 is tunable in case of BP throughK -doping which vanishes (see Supporting Info, Ref. 14)at a threshold value. The above dispersion relation whichholds true for a four-layered BP (adequately K -dopedsuch that the band gap vanishes) is plotted in Fig. 2with appropriate parameter values in the accompanyingcaption.

For a theoretical determination of the temperature-driven current, we must start from the dispersion re-lationship describing the electrons and holes. The dis-persion, however, must be asymmetric (does not possess

3

FIG. 2. An illustrative representation of a semi-Dirac disper-sion is shown using BP as a candidate example. Four BPlayers stacked together along an out-of-plane axis (inter-layerspacing is 5.3 A) give rise to a semi-Dirac dispersion with afinite gap adjustable through K -doping. The single-layer BPis actually a double layered structure and has two PP bonds.The shorter bond (bond length is 2.22 A) connects the near-est P atoms in the same plane while the longer bond (2.24A)connects atoms located in the top and bottom layers of theunit cell. The lower panel is the dispersion of a four-layer BP;for greater visual clarity that enhances the zigzag parabolicbranch vis-a-vis the linear term along armchair, we artificiallyset the effective mass to me = 0.06m0. Here, m0 is the freeelectron mass and vf is 5.6× 105ms−1.

particle-hole symmetry) for a non-vanishing current.27

This constraint arises, as is well-known, because electronsand holes guided by their respective dispersion flow inopposite directions under an established electrical bias;evidently a symmetric relationship exactly cancels eachcontribution leading to a vanishing thermal current. Forthe case of gapped BP, the contribution of the electronand hole parts would be unequal and dependent on theposition of the Fermi level. An n(p)-type BP would re-ceive significant contribution to the thermal current fromthe flow of electron (hole) in conduction (valence) states.However, in the eventuality that the gap diminishes tozero (by an electric field, which in our case comes fromK -doping) and the Fermi level is aligned exactly withthe semi-Dirac crossing (see Fig. 2), the symmetry of theelectron and hole parts would ensure a cessation of thethermal current. If the assertion that a finite current al-ways flows regardless of the band gap and Fermi levelis true everywhere, it is necessary to break the particle-hole symmetry via an external disturbance. For the mostgeneralized case, we consider the situation where the BPlayers are epitaxially grown on a ferromagnetic substrateand note that an asymmetry always exists. To see this,observe that the ferromagnetic substrate modifies theHamiltonian through an inclusion of the additional ex-change energy term, (∆exτ0 ⊗ σz). Here, τ0 is the 2 × 2identity matrix, ∆ex is the strength of the exchange field,and σz is the z -component of the Pauli spin-matrix. Therevised dispersion relation is then, Es (k) = η∆ex ± E,which is manifestly asymmetric for electron and holestates. Here, η = ±1 identifies the spin-up/down (+1/-

1) polarized dispersion curves. To distinguish from thepristine case described by Eq. 2, the superscript ‘s’ inthe energy expression indicates a modification by the fer-romagnetic substrate. In writing the exchange term, itis tacitly assumed that both sub-lattices (A and B) ofBP are subjected to the same exchange field. As a use-ful digression, we may note that asymmetry can also beintroduced in BP flakes through transfer of strain via astretched substrate, for example, polyethylene terephtha-late substrates.28 With this asymmetric dispersion as abasis, we attempt to compute the thermal current (seeFig. 1) for a film of K -doped semi-Dirac BP juxtaposedbetween two contacts maintained at different tempera-tures. This gives us the electrothermal conductance andcurrent behaviour.

III. Thermal current

The thermal current that flows between the two con-tacts is obtained using the Landauer-Buttiker formalism(LBF).29 For each spin component (assuming the bandsare spin-split when BP is grown on a ferromagnetic sub-strate), it is:

Ith =e

h

∫RdEM (E) T (E) (fL (E, TL)− fR (E, TR)) .

(3)The transmission probability in Eq. 3, for an electronto traverse the channel length (the longitudinal dimen-sion) is T (E), the function M (E) is the number ofmodes, and fL,R (E, TL,R) represents the Fermi distri-bution at the two contacts. As a first approximation, weset the transmission to unity assuming the target struc-ture (see Fig. 1) to be homogeneous throughout. To es-timate M (E), as is standard practice, we assume peri-odic boundary conditions along the y-axis such that thek-channels are equi-spaced by 2π/W .16 Here, W is thewidth (the transverse span along the y-axis) of the sam-ple. Each unique k-vector is a distinct mode and thenumber of such momentum vectors is determined fromthe inequality, −kf < ky < kf . The upper and lowerbounds of the inequality are the momentum vectors thatcorrespond to the Fermi energy, Ef . The approximatenumber of modes is therefore kfW/π. Also, note thatthe integral in Eq. 3 must be evaluated over two energy-manifolds, each in the vicinity of the conduction (Ec)and valence band (Ev) extremum. Explicitly, the en-ergy manifold of integration (Eq. 3) is a simple union oftwo disjoint intervals given by R1 : E ∈ −∞, Ev andR2 : E ∈ Ec,∞. The domain is then, R = R1 ∪ R2.Bearing these in mind, Eq. 3 for thermal current can berecast as

Ith =eW

πh

∫Rg (E) dE

∫ π/2

−π/2dθ cos θ (f1 (T1)− f2 (T2)) .

(4)The function g (E) in Eq. 4 is the analytic representationof the k-vector in energy space. Using Eq. 2 and the

4

exchange field, the function g (E) is

g (E) =

[1

2α2 cos4 θ

(Ω−

(2α∆ cos2 θ + β2 sin2 θ

))]0.5

.

(5)For brevity, we have used the short-hand notation

Ω =

√(2α∆ cos2 θ + β2 sin2 θ

)2+ 4 cos4 θ (E2 −∆2)α2

and E = (E − η∆ex). As a clarifying note, the sum ofmodes along the width (W ) is

∑kfW/π

∆ky which in the

continuum limit changes to∫ kf−kf dky. As usual, the az-

imuthal angle is θ while kx = k cos θ and ky = k sin θ.Note that the limits of angular integration satisfies thespan of a given k-vector (−kf < ky < kf ) or mode. Toobtain an estimate of the thermal current, a numericalintegration of Eq. 4 can be carried out. It is worthwhileto emphasize again on what we briefly alluded to above:The current in Eq. 4 has two components - from the con-duction band electrons and the valence band holes - thatflow in opposite directions for a certain potential drop.To further elucidate, when the left contact (in Fig. 1)is at a higher temperature than its right counterpart onthe right, we define the electron current to flow from theleft and empty in the right contact while the hole currentflows in the exact opposite sense.

A pair of noteworthy remarks is in order here: Firstly,we reiterate that the thermal current is purely an out-come of the temperature gradient induced difference inthe Fermi levels of the two contacts; therefore, before anynumerical value to a temperature gradient aided ther-mal current is sought, it must be borne in mind thatin an open-circuit steady state case, a build-up of elec-trons on one side (due to Ith) creates a potential dropwhich is countered by a oppositely-directed drift current.The net current flow is therefore zero in this arrange-ment. A voltmeter simply measures the voltage devel-oped across the sample attributed to the charge separa-tion brought about by the flow of the diffusive Ith. Fora finite current to flow under a temperature gradient, anelectrical load must be hooked between the two contactsmaintained at different temperatures. A qualitative de-scription centered around this point and thermoelectricefficiency appears later in Section V. The second obser-vation concerns the analysis presented until now whichassumes that the carriers seamlessly flow from the con-tacts to the channel indicating a perfect transmission(T = 1). In practice, scattering at the boundary of theBP channel and thermal reservoirs (contacts) causes areduction such that T < 1. The scattering effects can beestimated from several approaches including the classicalBoltzmann transport equation or the fully quantum me-chanical non-equilibrium Green’s function method. Typ-ically, the use of metal contacts affixed to semiconduct-ing channels creates a work function mismatch leadingto an ohmic contact for p-type sample and Schottky be-haviour for its n-doped counterpart. The Schottky bar-rier for the n-channel device shows a non-linear conduc-tion. BP transistors have been found to conform to this

observation.30,31 However, for now, we ignore such com-plications and adopt a simple ballistic regime allowingthe assignment of unity to T throughout. An obviousconsequence of such an assumption is an overestimationof the thermal-driven charge current.

Within the framework of the discussed formalism, wecan now numerically evaluate the temperature gradientdriven current; to do so, we begin by fixing the parame-ters starting with the temperatures of the two contacts.The temperature is swept between T ∈ [150, 350] Kmaintaining a constant difference of ∆T = 25K betweenthe left (hotter) and right contact. The exchange fieldof the ferromagnet that breaks the particle-hole symme-try is set to ∆ex = 30.0meV . As for BP, from previ-ously determined material constants, the effective massfor the parabolic branch is meff = 1.42m0 while theband gap with K -doping is approximately ∆ = 0.36 eV .The BP band gap is tunable via the concentration of theK -dopant.14 The Fermi velocity for the linear branch isvf = 5.6 × 105ms−1. Note that the out-of-plane mag-netization splits the conduction and valence bands intospin-up and spin-down ensembles; for the selected mate-rial constants, the bottom the conduction spin-up (down)band is 0.21 (0.15) eV. The corresponding top of thespin-up (down) valence band is −0.15 (−0.21) eV . TheFermi distribution functions (see Eq. 4) are computedby assigning an identical electrochemical potential (µ) toboth contacts; to simulate n- and p-type character, wetoggle µ between ±0.17 eV . In each case, µ is located be-tween the bottom (top) of the spin-split conduction (va-lence) spin-up and spin-down bands. The current (seeFig. 3) is obtained by a numerical integration (Eq. 4)considering all possible modes within 65.0meV from theconduction and valence band extremum and a range oftemperatures while holding a constant difference betweenthe contacts.

The analysis presented heretofore used semi-Dirac BPto estimate thermal current under a temperature gradi-ent. The same set of steps can be applied to undertakea comparison between the thermal currents carried bysemi-Dirac and Dirac materials. A gapped Dirac mate-rial, for instance, graphene, is described in the continuumlimit as

HDirac = ~vf (kxτx + kyτy) + ∆τz. (6)

. The symbols in Eq. 6 have their usual meaning. More-over, to uncover any contrasting behaviour between thesemi-Dirac and Dirac cases, we assign the Fermi veloc-ity of BP and its band gap to the graphene-like materialdescribed by Eq. 6. The exchange coupling as before isset to ∆ex = 30.0meV and the contact temperatures arevaried exactly as done for BP. The thermal current for theDirac case can be again obtained from Eq. 4 by simplychanging the function g (E) to reflect the transmissionmodes of the graphene-like material. It is

g (E) =1

~vf

√[(E − η∆ex)

2 −∆2]. (7)

5

FIG. 3. The numerically obtained current (I) that flows ina multi-layer BP sheet clamped between two contacts at dis-similar temperatures (see Fig. 1) is shown. Mirroring themagnetization-induced asymmetry of the conduction and holestates, I is lower for a p-doped material vis-a-vis an n-typestructure. The inset shows the current for an n-type graphenesheet with band gap and Fermi velocity identical to the lay-ered BP. The thermal current for gapped Dirac graphene islower than its semi-Dirac BP counterpart. Note that the tem-perature (TR) shown along the x -axis is for the left contact;the left contact is always set to TR + ∆T . Here, ∆T = 25K.

The two spin-states are distinguished by η = ±1. Insert-ing g (E) from Eq. 7 in Eq. 4 and positioning the Fermilevel identically as in the case of n-doped BP followed bya numerical integration, the Ith for a representative Diracmaterial is shown in the inset of Fig. 3. The thermally-driven current is feebler vis-a-vis a semi-Dirac material.However, the thermal current profiles for both Dirac andsemi-Dirac display a similar behaviour with a prominentplateau region in each case - a distinctive feature of cur-rents with a purely thermal origin.

We comment on a few noteworthy features of Fig. 3;firstly, it is immediately recognizable that as the tem-perature rises, the thermal current begins to saturatefor both n- and p-type, a behaviour attributable to thediminishing Fermi level difference (∆µ) with a rise intemperature for a given energy. At hand though, therealso exists a larger smearing of the Fermi function at ahigher temperature opening up additional modes ((M) inEq. 4) accessible for transport, however, in this case, thefall in ∆µ executes the more definitive role. In the samespirit, for a fixed temperature, the current falls (see inset,Fig. 3) at higher energies as ∆µ is again lowered for a pre-determined temperature difference between the contacts.To offer additional substantiation to this line of reason-ing, a plot (Fig. 4) of the thermal current for severaltemperatures differences is distinguished by an increas-ing behaviour, a fact that constitutes a simple demon-stration of an enlarged ∆µ. This increment to ∆µ trans-lates into a higher current. Notice that the current (forboth doping cases) shown in Fig. 3 receives contributionfrom four components: They are a pair consisting of spin-

FIG. 4. The thermal current behaviour as a function of tem-perature difference that exists between the two contacts. Thematerial parameters are identical to those used in Fig. 3. Thetwo temperatures (TR = 100, 300K) indicated on the plot arethe starting values for the right contact, the temperature ofthe left contact (TL = TR + ∆T ) is then swept up to enlargethe difference (shown on the x -axis) with its right counter-part that progressively enhances the Fermi drop (∆µ) andconsequently manifests as a larger thermal current.

up and spin-down electron components from the conduc-tion bands and a similar but counteracting hole-basedset originating in the valence bands. When BP is dopedn-type, the position of the Fermi level in the conductionband guarantees completely (almost, if smearing is ac-counted) filled valence bands, a consequence of which isa negligible hole current. By the same token, for a p-typematerial, the conduction states are nearly empty ensur-ing that bulk of the current is carried by holes. From anexperimental standpoint, both n- and p-type conductionhas been observed30 in BP consistent with the narrowband gap that permits tuning of the Fermi level close toeither the valence or conduction band.

We pointed to the electric field aided band gap mod-ulation of BP, either via K -doping or an applied electricfield; when such a mechanism is available, it is beneficialto check the associated variations in the thermal cur-rent. We use two temperature points (TR = 100, 150K)and for each such value, the difference, TL − TR, is as-signed a pair of numbers, ∆T = 20, 25K. The thermalcurrent is plotted (Fig. 5) for several band gaps (∆) in n-type BP. For the purpose of illustration, the Fermi levelis aligned with bottom of the conduction band (n-typematerial), which is ∆/2. The magnetization is not in-cluded and as explained above, the hole currents con-tribute negligibly for an n-type BP. The thermal cur-rent increases for a larger gap, an outcome simply ex-plained by noting that in the vicinity of a higher Fermienergy (remembering that it is aligned to the conduc-tion band minimum) a greater number of current carry-ing modes are present or equivalently a higher value forM (E) in Eq. 4. The thermal current expectedly rises.

6

For further elucidation, a more specific case can be ex-amined that offers a more concrete feel of the magnitudeof the thermal current. We consider a typical arrange-ment wherein the temperature is set to T = 300K whilethe difference between the contacts is maintained at asteady ∆T = 25K. The injected carriers are located inthe vicinity of the conduction (valence) band minimum(maximum) while the magnetization has been turned offin this case. The Fermi level, as in the preceding case isadjusted to µ = 0.17 eV , located between the mimima ofthe spin-split semi-Dirac BP’s conduction bands. Insert-ing these numbers in Eq. 4, the current approximatelyevaluates to Ith ≈ 31µA/mm. A caveat about this re-sult and in general pertaining to this work must howeverbe noted: First and foremost, the current solely drivenby a temperature difference is the maximum achievablefor the selected thermal settings; the quantum of cur-rent flow though in a realistic setup will degrade primar-ily through electron-phonon scattering which is active inmulti-layer BP for 100K < T < 300K. The impurityscattering dominates below the T = 100K mark. For acomparative account of current flow, it is useful to quotemeasurement data collected by Liu et al. in field-effecttransistor (FET) structures with stacked BP (stacked BPcan be thought of as a bulk form) wherein they report32

a maximum drain current of IDS = 194mA/mm in anFET structure at drain bias (VDS) of 2.0 V and 1.0µmin channel length. This current under an electric biasis roughly three orders larger over Ith. In passing, it ispertinent to mention that a converse of this thermionicprocess - the electrocaloric response - exists with similarapplications in solid state refrigeration and Peltier cool-ing. The electrocaloric effect centers around a reversiblechange in temperature by an electric signal and has beendemonstrated33,34 in ferroelectrics such as PbTiO3. Wedo not discuss it here.

IV. Electronic part of the heat capacity

The preceding analysis focused on an elementaryPeltier cooling35 model wherein a current flows under atemperature gradient removing the surplus heat. How-ever, certain applications require the ability to maintaintemperatures, an attribute tied to a large heat capacity.We examine the heat capacity of semi-Dirac BP employ-ing the Sommerfeld expansion (SE). The SE retaining upto only the second order term is17

∫ ∞−∞

H (ε) f (ε) dε =

∫ µ

−∞H (ε) dε+

π2

6(kBT )

2H

′(µ) .

(8)Briefly, ε denotes energy, kB is the Boltzmann constant,and the function H (ε) is expanded in the vicinity ofµ, the Fermi level. For a heat capacity derivation, westart by setting the function H (ε) = εD (ε) and insert-ing in Eq. 8 gives the internal energy U =

∫εD (ε) dε +

(πkBT )2D (ε) /6. The density of states (DOS) is D (ε).

Differentiating the internal energy expression w.r.t T, we

FIG. 5. The thermal current is shown for several band gaps inan n-type BP. The current is determined for two temperaturepoints (TR = 100, 300K) as noted in the figure. The temper-ature of the left and right contacts are therefore TR + ∆Tand TR, respectively. The current curves have been shownfor a pair of ∆T = 20, 25K. A higher band gap (that iscapped around the ∆ = 0.35 eV in BP) gives rise to a largerthermal current. The thermal current was calculated takinginto account conduction energy states lie within a 65.0 meVrange from bottom of the conduction band. Note that in ab-sence of any magnetization effects, the two spin componentscontribute equally to the thermal current. The correspondingcase of a p-type BP, again assuming zero magnetization anda Fermi level adjusted to top of the valence band, the thermalcurrent is identical in magnitude to the n-type case but withreversed polarity.

get the heat capacity at constant volume. It is given as

Cv =∂U

∂T=π2k2

BT

3D (ε) . (9)

A numerical value for the heat capacity therefore en-tails an evaluation of D (ε). For DOS calculation ofanisotropic BP, we use the dispersion in Eq. 2 to write

D(ε) =

∫d2k

(2π)2 δ (ε− E(k)) ,

=1

(2π)2

∑j

∫ 2π

0

dθkj

|g′(kj)|.

(10)

In Eq. 10, the azimuthal angle is θ. We have used the

identity δ (g(x)) =∑j

δ(x− xj)|g′(xj)|

such that g(xj) = 0

and xj is a simple zero of g(x). The function g(k)

in this ε −√

(∆ + αk2 cos2 θ)2

+ (βk sin θ)2. Setting

g(k) = 0 and solving, the positive root κ can be expressed

as κ2 =(√

p2 + 4α2 cos4 θ (ε2 −∆2)− p)/(2α2 cos4 θ

),

where p =(2∆α cos2 θ + β2 sin2 θ

). Inserting the posi-

tive root and the derivative (w.r.t k at κ) of g (k), the

7

expression for DOS is

D (ε) =1

4π2

∫ 2π

0

dθE (κ)

2α cos2 θ (∆ + ακ2 cos2 θ) + (β sin θ)2 .

(11)The DOS for semi-Dirac BP when substituted in Eq. 9gives the heat capacity at constant volume. This calcula-tion is not generalized and is true only when a Sommer-feld expansion (T Tf ) is possible. The Fermi temper-ature, Tf = εf/kB , εf denotes the Fermi energy, and kBis the Boltzmann constant. The crux of a CV calculationis then a determination of DOS; to that end, we performa numerical integration of the DOS integral (Eq. 11) fora range of energies in the vicinity of the conduction bandminimum. The numerically calculated DOS multipliedwith the appropriate pre-factor in Eq. 9 furnishes theCV for a two-dimensional semi-Dirac electron BP. In thiscalculation (see Fig. 9), the BP sample area was set toA = 1 cm2. The heat capacity within the framework ofthe Sommerfeld expansion is simply a portrayal of theDOS making it amenable to changes via processes andexcitations that can renormalize the armchair mass oralter the zigzag Fermi velocity.

At this stage, it is somewhat instructive to compare theheat capacity of semi-Dirac BP with an identically-sizedsample of two-dimensional electron gas (2DEG) with con-ventional parabolic dispersion. For ease of comparison,we select conduction electrons of zinc blende GaAs atT = 100K with an effective mass me = 0.067m0 anda direct band gap of Eg = 1.41 eV at the Γ point ofthe Brillouin zone. The heat capacity in this case (using

FIG. 6. The electronic contribution to the heat capacity (CV )of the two-dimensional semi-Dirac electron BP gas is plottedfor several Fermi energies and two band gaps. The tempera-ture for this calculation was set to T = 100K and the areaof the sample is assumed to be A = 1 cm2. For the range ofFermi energies assumed, the Fermi temperature (Tf ) is muchlarger than T = 100K. Note that in this regime, the heatcapacity is of the form CV = γ T , where γ is the so-calledSommerfeld coefficient.

Eq. 9) is CV = Aπmek2BT/(3~2). In writing this 2DEG

CV formula, we used the 2D DOS expression of me/π~2.Plugging in the appropriate material parameters and set-ting A = 1 cm2, we have CV = 1.1 × 10−11 J/K. As asimple check to determine the applicability of the Som-merfeld expansion for calculating the heat capacity ofGaAs, we set the electron density to n = 1011 cm−2

which corresponds to a Fermi wave vector of kF ≈0.008A−1. The Fermi energy is therefore (from the bot-tom of the conduction band which is at Eg) εf = 3.6meVand the equivalent Fermi temperature is approximately(Tf = 1.6× 104K

). Since Tf T , the use of an SE-

derived CV formula is correct. To compare, the heatcapacity for the model GaAs 2DEG is roughly three-orders reduced than a representative value (see Fig. 6)of 3 × 10−9J/K for semi-Dirac hybrid BP. It is of someconsequence to observe that the low CV values are inpart attributable to the shrunken-dimensionality (2D) ofthe system (compared to the three-dimensional bulk) andfurther accentuated in the case of GaAs due to the intrin-sic smallness of effective mass (0.067m0) of conductionelectrons around the Brillouin zone centre. In sharp con-trast, the corresponding value for K -doped BP at 1.42m0

along the parabolic branch is roughly twenty-fold higher.Additionally, while the CV for GaAs conduction electronsconfined in a quantum well is constant around the Bril-louin zone centre (assuming the effective mass does notvary too much), the hybrid BP shows a rise attributableto the linear branch of the dispersion along the zigzagaxis. The contribution of this branch to the DOS riseslinearly with energy; however, its complete manifestationis suppressed by the parabolic branch forbidding a trulygraphene-like behaviour. For small momentum valuesclose to the Dirac crossing, the linear branch is dominantand the DOS rises with energy marked by an accompa-nying increase in the heat capacity.

In light of the demonstrated higher CV of semi-DiracBP, it is reasonable to anticipate that a system with suchhybrid dispersion can serve as a better coolant to holdthe temperature steady for a longer period compared toa 2D nanostructure with parabolic materials. It there-fore also follows that techniques to augment the linearpart of the dispersion over the parabolic branch can offergreater advantage, especially when the DOS is inextrica-bly linked in the unfolding of a physical phenomenon. Afew notable examples in this regard could be quantumcapacitance and paramagnetic susceptibility, both quan-tities are linearly connected to the DOS. We briefly dis-cuss novel techniques to modulate the DOS in the closingsection.

V. Thermoelectric Efficiency

The thermal current and heat capacity calculations,besides their apparently ‘independent’ roles in power con-version and temperature control, can be brought togetherto compute a Carnot-like thermoelectric efficiency (TEE)under a temperature gradient. While we have consid-ered the case where the temperature gradient is a natural

8

FIG. 7. The schematic of an energy generator that ‘outputs’a thermoelectric current (Ith) as a response to an ‘input’ heatcurrent, Jq. The temperature gradient between the contactsis maintained by the heat current. The ratio Ith/Jq is theCarnot-like thermoelectric efficiency of this setup.

outcome of the microscopic processes in a miniaturizeddevice, it is possible to recreate a similar arrangementby pumping an external heat current (Jq) between thecontacts. The resulting temperature difference drives anelectric current (Je), a process to convert heat into use-ful work. A TEE for the operation of this prototypicalthermoelectric engine (for a schematic, see Fig. 7) canbe defined as η = Ith/Jq. In the following, we derivean expression for TEE that relevant to the energy cycleillustrated in Fig. 7. To begin, first note that for settingup a certain temperature gradient, ∆T , the desired heatcurrent from Fourier’s law is Jq = −κq∆T . Here, κq isthe complete thermal conductivity and is the sum of theelectronic (κel) and lattice (κel) contributions. We donot analyze the lattice contribution, simply noting thatit can be established from the kinetic theory using therelation36

κph =1

3Cv,phvphΛph. (12a)

The phonon velocity is vph while Λph indicates the corre-sponding mean free path. The specific heat for the latticepart (Cv,ph) can be expressed via the Debye model as37

Cv,ph =9NkBT

3

Θ3D

∫ ΘD/T

0

x4ex

(ex − 1)2 dx. (12b)

In Eq. 12b, N is the number of vibrational modes withfrequency ω, the Boltzmann constant is kB , ΘD is theDebye temperature, and x = ~ω/kBT . For the electronicpart, a similar kinetic theory description can be written,replacing Cv,ph with Cv,el in Eq. 12b, vph and Λph withthe Fermi velocity and the electronic mean free path, re-spectively. Notice that the electronic contribution to CVcan be obtained making use of Eq. 9 for semi-Dirac BP,the material of interest in this work. The input power,which is the pumped heat current, by clubbing the two(electronic and lattice) thermal conductivity contribu-tions (Eqs. 9 and 12b) and inserting in Fourier’s heat

law gives

Jq =1

3(Cv,phvphΛph + Cv,elvfΛel) ∆T. (13)

The output power is I2thR, where Ith can be determined

from Eq. 4 and R is the resistive load. The Carnot-likeTEE is therefore

η =3J2eR

(Cv,phvphΛph + Cv,elvfΛel) ∆T. (14)

A more careful analysis of thermoelectric efficiency musthowever start from the coupled energy (heat) and chargeequations38 and account for the overall Seebeck-inducedcurrent that may flow in a complete circuit, which couldbe quantitatively different from the thermionic contribu-tion predicted by Ith of Eq. 4. The coupled heat andelectric current equations are

Je = L11E + L12 (−∆T ) ,

Jq = L21E + L22 (−∆T ) .(15)

From Eq. 15, the Seebeck coefficient is S given byL12/L11 while the electric (thermal) conductivity is sim-ply L11

(L22). The thermoelectric current (Je) in absence

of any external bias (E = 0) has an additional contribu-tion σS∆T which must be added to the total outputthermal current thus modifying the TEE expression inEq. 14. The complete set of derivations accounting forthese processes that amends TEE is postponed to a sub-sequent work.

VI. Closing Remarks

We have presented theoretical calculations on thermalcurrent that flows under an impressed temperature gra-dient in a two-dimensional semi-Dirac system. The lay-ered black phosphorus, for reasons rooted in its innateanisotropy (briefly described in Section. I) was the cho-sen representative semi-Dirac material. The heat capac-ity of BP was also computed in the regime where theSommerfled expansion remains valid. The flow of a ther-mal current and the attendant heat removal represents apossible realization of the Peltier cooling while the heatcapacity, on the contrary, signifies the intrinsic effective-ness of absorbing heat and maintaining a steady temper-ature. These seemingly distinct phenomena, however, ina low temperature regime (that ignores phonon contri-butions) show a shared association at the microscopiclevel. We elaborate on this by recalling that while theLandauer formalism39 works with the density of modes,essentially an enumeration of the probable momentumpathways for carriers to flow, the heat capacity explicitlyuses the density of states. A dominant role for eitherheat removal or absorption is therefore a function of thedensity of states (D (ε)). A slightly different perspectivecan also be offered in way of connecting the D (ε) to thetemperature-driven current through the intrinsic Seebeckcoefficient (Qij) of the material. Observe that the ther-movoltage (VL,hot − VR,cold) developed between the two

9

ends of the material can be indirectly gauged via the See-beck coefficient, usually expressed in units of µV/K. Wedefine the functional form of Qij using the well-known

Mott relationship.40 In tensorial form, where i and j areunit vectors in the 2D plane, it is given by

Qij = −π2

3e

k2BT

σ

∂σij∂ε

. (16)

The Mott relation is electric conductivity (σ) dependent,which (for an isotropic case) in turn can be written as41

σ =e2v2

f

2

∫D (ε) τ (ε)

(−∂f∂ε

)dε. (17)

The D (ε) reappears in Eq. 17, from which we concludethat the thermal current and heat capacity are bound atthe microscopic level through the arrangement of elec-tronic states. In general, the thermopower is a tensorquantity with off-diagonal elements when set out in ma-trix form. It can assume a diagonal-only representationfor a truly isotropic (and zero magnetic field) system suchas pristine graphene, unlike semi-Dirac BP.

A suitable modification of the D (ε) may therefore, set-ting aside considerations of thermal conductivity, serve asan effective tool to tune the thermal current and heat ca-pacity. In context of semi-Dirac BP, it was shown ina recently published work42 the possibility of enhanc-ing the D (ε) using intense illumination. Succinctly, thedispersion of irradiated BP is revamped through the in-troduction of a frequency (ω) dependent linear term ofthe form F (ω) kx. The photo-induced F (ω) kx in con-junction with the intrinsic linear term in the Hamilto-nian (Eq. 1) for low values of momentum around thesemi-Dirac crossing dominates the quadratic componentimparting a quasi-linear character. Since for materi-als with a linear dispersion the D (ε) monotonically in-creases with energy, a similar behaviour may be expectedfor irradiated-BP. The modulation of DOS in the quasi-linear case evidently centers around the strength of thelinear terms β = ~vf and F (ω) and their combined out-weighing of the quadratic k2

x armchair dispersion.Briefly, it is relevant to state here that BP which

turns semi-Dirac with K -doping can undergo a topologi-cal phase transition with increasing dopant concentrationthrough a vanishing of the band gap. Beyond the bandgap ceasing, for a higher K -dopant concentration, pavesthe way for band inversion, a precursor to topologicallyprotected states; indeed, such states with massless andanisotropic Dirac fermions have been observed in BP.14

A simplified form of the Hamiltonian for the topologicalinsulator (TI) can be written as

HTI = ~ vx (kx − kD)σx + ~ vykyσy, (18)

where vx and vy are the velocities along the x - and y-axes, respectively at the Dirac point. A Dirac graphene-like dispersion (Eq. 18)is characterized by a DOS that lin-early scales with |ε| and may therefore exhibit (for a finite

energy) a larger heat capacity in comparison to a semi-Dirac or parabolic 2D nanostructure. It is decidedly anattractive proposition to observe topological phase tran-sitions via doping and the concurrent advatanges tht ac-crue; however, the benefits of the strong anisotropy ofBP, particularly suited for a large ZT may disappear. Acareful set of experimental measurements may uncoverthe precise connection between topological phase transi-tions and the overall thermoelectric behaviour in semi-Dirac BP.

VII. Summary

In closing, we have through analytic calculations pre-dicted the thermal current and heat capacity of BP withhybrid semi-Dirac structure. Further refinements to thecalculations can be carried out through inclusion of scat-tering in the channels suggesting that the assumed trans-mission is then no longer unity; besides, the role oftwo contacts maintained at different temperatures in amore accurate formulation must be accounted, for exam-ple, using the non-equilibrium Keldysh formalism.43 TheKeldysh formalism, apart from a more accurate modelingof contacts, is also necessary as demonstrated in a recentwork44 that analyzed flow of charge and heat currentin a nanoscale conductor taking into account transientparticle behaviour that develops from an abrupt appli-cation of a temperature gradient. This work ignores anynon-equilibrium heat and current density. Further, thesecalculations can also be applied to any two-dimensionalsystem, a prominent example of it being the honeycomblattice of a single layer transition metal dichalcogenide(TMDC). A TMDC single layer is naturally in accordwith the requirement of particle-hole asymmetry for a fi-nite thermal current, fulfilled via the strong spin-orbitcoupling operational in the valence bands45,46 and itsmuch weaker and inequivalent counterpart in the con-duction states.47 The inequality of the two spin-orbitcouplings asymmetrically splits the gapped single layerTMDC energy levels around the high-symmetry K andK

′valley edges. A magnetic field which may be required

in case of K -doped BP with zero band gap to createasymmetry is therefore not a necessary condition.

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