First-principles method for electron-phonon coupling and electron mobility:Applications to 2D materials

Tue Gunst,1, ∗ Troels Markussen,2 Kurt Stokbro,2 and Mads Brandbyge1

1Department of Micro- and Nanotechnology (DTU Nanotech), Center for Nanostructured Graphene (CNG),Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark

2QuantumWise A/S, Fruebjergvej 3, Postbox 4, DK-2100 Copenhagen, Denmark(Dated: November 9, 2015)

We present density functional theory calculations of the phonon-limited mobility in n-type mono-layer graphene, silicene and MoS2. The material properties, including the electron-phonon inter-action, are calculated from first-principles. We provide a detailed description of the normalizedfull-band relaxation time approximation for the linearized Boltzmann transport equation (BTE)that includes inelastic scattering processes. The bulk electron-phonon coupling is evaluated by asupercell method. The method employed is fully numerical and does therefore not require a semi-analytic treatment of part of the problem and, importantly, it keeps the anisotropy informationstored in the coupling as well as the band structure. In addition, we perform calculations of thelow-field mobility and its dependence on carrier density and temperature to obtain a better under-standing of transport in graphene, silicene and monolayer MoS2. Unlike graphene, the carriers insilicene show strong interaction with the out-of-plane modes. We find that graphene has more thanan order of magnitude higher mobility compared to silicene. For MoS2, we obtain several ordersof magnitude lower mobilities in agreement with other recent theoretical results. The simulationsillustrate the predictive capabilities of the newly implemented BTE solver applied in simulationtools based on first-principles and localized basis sets.

I. INTRODUCTION

Two-dimensional (2D) materials are promising candi-dates for future electronic devices.1–6 Examples of threesuch materials are illustrated in Fig.1, showing a mono-layer of graphene (A), its Silicon based counterpart,silicene7–10 (B), and monolayer MoS2 (C). In such sys-tems two-dimensionality allows very precise control ofthe carrier density by a gate which enables tuning ofthe electron-phonon interaction.3 Electron-phonon inter-action in graphene has been studied previously11–13, butwith the recent advances in fabrication of devices basedon other 2D materials, like the first demonstration ofa silicene transistor4, further studies of interaction phe-nomena in 2D materials are necessary.

A)

B)

C)

FIG. 1. The monolayered two-dimensional systems that areconsidered are A) graphene, B) silicene and C) MoS2.

When comparing the electrical performance of devicesone often considers the carrier mobility of the materials.

Mobility is a key parameter for the semiconductor indus-try describing the motion of electrons when an electricfield is applied. Experiments can approach the ’intrin-sic’ phonon-limited mobility by several means. One ex-periment combine defect-free edge contacting14 of gate-tunable graphene electrodes with MoS2 encapsulated inhexagonal boron nitride layers.15 Several other experi-ments screen the scattering from charged impurities bya high-κ gate dielectrics16 or through device suspensionin high-κ liquids.17–19 In general, van der Waals het-erostructures may pave the way for devices with reducedextrinsic scattering, like charged impurity scattering, ren-dering modeling of electron-phonon scattering in thesedevices even more important.

Conventional mobility modeling usually considers ef-fective mass approximations20 in the case of semiconduc-tors or linear bands for semimetals11,21,22 combined withempirical deformation potentials and semi-analytical so-lutions of the Boltzmann equation.11,23 The Boltzmanntheory is a useful approach to model the low-field/linearresponse mobility23–25, as well as the high-field transportthrough Monte Carlo simulations.26–28 Several studieshave examined effects related to screening29,30, scatter-ing from out-of-plane vibrations31 and performed atom-istic calculations of the mobility from tight-binding32–34

as well as electron-phonon interactions role in facilitatinginterlayer conduction35 and current-induced heating.36,37

Density functional theory (DFT) and atomistic meth-ods can be used to assess the electronic structureand electron-phonon coupling in novel 2D materialswhere fitted deformation potential parameters are notavailable.38–44

Recently, several groups have combined DFT with den-sity functional perturbation theory38,39,45–47 to evaluate

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the mobility from first-principles. Parameter-free meth-ods can be used to address how close experiments are toideal conductivities and if further optimization of fabri-cation techniques and device designs for novel 2D ma-terials could improve device performance. In addition,first-principles calculations of the bulk electron-phononinteraction may be used for comparing deformation po-tential values to those obtained from experiments48,49

and used to conclude which scattering effects are dom-inant. The dominant effect is not always directly clearfrom experiments and published deformation potentialscan vary significantly.11

Several techniques which differ from contacted elec-trical measurements are in development that can pro-vide detailed knowledge of the electron-phonon interac-tion; broadening of Raman peaks50, kinks in the angle-resolved photoemission spectra51 and non-destructive op-tical methods.52 Hereby a first-principles method to eval-uate bulk electron-phonon interactions may provide use-ful data for comparison with experiments.

Methods to obtain the bulk electron-phonon interac-tions include Wannier functions together with a general-ized Fourier interpolation scheme53, perturbation theoryand empirical pseudopotentials54, finite-differences in theprojector-augmented wave method55 and density func-tional perturbation theory.45,46,56,57

We will later give a simple derivation of an expressionfor the bulk electron-phonon coupling applicable in anysetup based on localized basis sets. From this we have im-plemented a supercell method used to calculate the bulkelectron-phonon interaction employing finite differences.This method is analogous to that used for calculatingthe phonon dispersion both in terms of methodology andcomputational cost.

In this paper, we present a detailed description of theimplementation of the bulk electron-phonon coupling,and a Boltzmann Transport Equation (BTE) solver inthe Atomistix ToolKit (ATK) simulation tool.58 We ap-ply atomistic simulations with ATK to study electron-phonon coupling in 2D materials from first-principles.We formulate a normalized full-band relaxation time ap-proximation (RTA) for the linearized BTE that includesinelastic scattering processes. The bulk electron-phononmethodology employed makes simulations possible thatdo not require a semi-analytic treatment of part of theproblem and, importantly, we keep the anisotropy infor-mation stored in the coupling as well as the band struc-ture. In addition, we perform calculations of the low-fieldmobility and its dependence on carrier density and tem-perature to obtain a better understanding of transportin graphene, silicene and MoS2.

Despite the fact that several papers have presentedschemes for obtaining electron-phonon coupling and mo-bilities from first-principles calculations, only few appli-cations exist and the methods are by no means standard.The focus of our work is to make an efficient and prac-tical scheme for such calculations in order to make themas accessible as performing a bandstructure calculation.

In the methods section we will discuss the differencesin our technical implementation compared to previouswork, and here we only briefly mention some general ad-vantages of our work which we believe will be importantfor the wide spread usage. The method is based on lo-calized basis sets which allows for exploiting locality inreal space and we have made a great effort of optimiz-ing the implementation and apply efficient parallelizationand interpolation schemes. The method is therefore fastand able to handle large systems. Secondly, it is highlyaccurate and as we will show in this paper, it gives resultsconsistent with previous DFT simulations. Finally, it isimplemented in a versatile framework with an easy to usepython interface which allows for performing all the dif-ferent parts of the calculations with a single script, thus,it is easy to setup the calculation and it requires minimalhuman interference for performing the calculation.

The paper is organized as follows. In Sec. II we presentthe theoretical and numerical methods used. We deriveexpressions for the linearized BTE and the bulk electron-phonon interaction implemented in the ATK simulationtool. In Sec. III we present our results for the bulkelectron-phonon coupling in graphene, silicene and MoS2.In addition, we discuss the dependence of the mobility onthe carrier density and temperature for all three materi-als. Finally, the results are summarized and discussed inSec. IV.

II. METHODS

In the diffusive transport limit, the mobility can be ob-tained by solving the semiclassical BTE for the electronicdistribution function, f(εkn) = fkn:

∂fkn∂t

+ vkn · ∇rfkn +F

~· ∇kfkn =

∂fkn∂t

∣∣∣∣coll

. (1)

Here k, n labels the k-point and band index, respectively.The velocity is defined as vkn = 1/~∇kεkn and F =q(E+v×B) gives the external force. The right hand sidein Eq. (1) describes different sources of scattering anddissipation that drives the system towards steady state.In the case of a homogeneous system, zero magnetic field,and a time-independent electric field in the steady statelimit, the BTE simplifies to:

qE

~· ∇kfkn =

∂fkn∂t

∣∣∣∣coll

. (2)

Assuming instantaneous, single collisions, which are in-dependent of the driving force, the collision integral canbe expressed using transition rates, Pnn

′

kk′ ,

∂fkn∂t

∣∣∣∣coll

=−∑k′n′

[fkn (1− fk′n′)Pnn

′

kk′

− fk′n′ (1− fkn)Pn′n

k′k

]. (3)

3

The transition rate due to phonon scattering from a state|kn〉 to |k′n′〉 is obtained from Fermi’s golden rule (FGR):

Pnn′

kk′ =2π

~∑q,λ

|gλnn′

kq |2[nλqδ (εk′n′ − εkn − ~ωqλ) δk′,k+q

+ (nλ−q + 1)δ (εk′n′ − εkn + ~ω−qλ) δk′,k−q], (4)

describing absorption (first term) and emission (lastterm) of a phonon. The last term includes spontaneousemission which remains at zero temperature. The sumruns over phonon momentum (q) and phonon branch in-dex (λ). In Eq. (4) we have explicitly stated the momen-tum conservation coming from the bulk electron-phononinteraction matrix element. We have implemented a su-percell method to calculate the bulk electron-phonon in-teraction, gλnn

′

kq , employing finite differences in a local-

ized basis set (derived in section. II C).We will assume unperturbed phonons and apply the

equilibrium Bose-Einstein distributions, nλq = n0,λq , in

which case n0,λ−q = n0,λq since ω−qλ = ωqλ. The transition

rates Pλnn′

kk′ and Pλn′n

k′k are linked through the ”detailedbalance equation”24,[

f0kn(1− f0k′n′

)Pnn

′

kk′ − f0k′n′

(1− f0kn

)Pn

′nk′k

]= 0 .(5)

Here f0 is the equilibrium Fermi distribution function.

This equation secures that ∂fkn

∂t

∣∣∣coll

= 0 in equilibrium.

We linearize the BTE in the electric field. The lefthand side of the BTE, Eq. (2), is approximated to linearorder in the electric field by changing to the equilibriumdistribution:

qE

~· ∇kfkn ≈

qE

~· ∇kf

0kn = qE · vkn

∂f0kn∂εkn

. (6)

The right hand side is linearized by assuming a formof the distribution function that is linear in the elec-tric field. Defining a generalized transport relaxationtime24,39, τkn, so that

fkn = f0kn + qE · vknτkn(−∂f0kn

∂εkn) , (7)

and combining and inserting Eqs. (3), (4), (5), (6) inEq. (2), we arrive at the linearized BTE,

1 =∑k′n′

Pnn′

kk′(1− f0k′n′)

(1− f0kn)

×[τkn − τk′n′

nk′n′

nkn

f0kn(1− f0kn)

f0k′n′(1− f0k′n′)

]. (8)

Where we defined the direction projections nkn =E · vkn. Equation (8) is still a full integral equa-tion. However, several approximations, termed re-laxation time approximations (RTA), exist throughoutliterature38,59,60 to reduce the problem to a k′-spaceintegration. For instance the term in the brackets inEq. (8) is replaced by τkn times the non-normalized fac-

tors[1− k·k′

k2

],[1− vk′n′ ·vkn

|vkn|2

]or the normalized factors

[1− k·k′

kk′

]and infrequently

[1− vk′n′ ·vkn

|vk′n′ ||vkn|

].60 The non-

normalized conditions are related to the assumption thatτk′n′ ≈ τkn while the normalized expressions are relatedto the assumption that τk′n′ |vk′n′ | ≈ τkn|vkn|, and inboth cases that the last Fermi-factor is equal to unity.The expressions based on group velocities have the ad-vantage that they do not depend on the chosen referencek-point. In addition, non-normalized expressions maylead to unphysical negative momentum relaxation times.Here we define a normalized full-band RTA of the lin-earized BTE, including inelastic scattering processes, as:

1

τkn=∑k′n′

(1− f0k′n′)

(1− f0kn)(1− cos(θkk′))Pnn

′

kk′ . (9)

Here the scattering angle is defined by61

cos(θkk′) =nk′n′

nkn=

vk′n′ · vkn

|vk′n′ ||vkn|. (10)

For an angle-independent transition rate then small-anglescattering, where cos(θkk′) ≈ 1, does not obstruct theflow of electrons whereas large-angle scattering, wherecos(θkk′) ≈ −1, significantly increases resistivity. How-ever, the selection rules stored in the bulk electron-phonon coupling matrix element complicates this gen-eral trend. The current density is related to the averagevelocity, J = qn0〈vkn〉, obtained from the nonequilib-rium distribution function in Eq. (7). From the transportrelaxation time, Eq. (9), we then evaluate the low-fieldelectron mobility59

µe = −2q

∑kn∈c |vkn|2 ∂f

0kn

∂εknτkn

n0, (11)

where a factor of two accounts for the spin degeneracy.The hole mobility is obtained by replacing the electroncarrier density, n0 =

∑kn∈c f

0kn, and the summation over

conductions bands (c) with the hole carrier density,p0 =

∑kn∈v(1 − f0kn), and a summation over valance

bands (v). The total conductivity is given by σ = qn0µe+qp0µh.

We also mention the difference of the transport scat-tering time discussed here versus the lifetime of electronicquasiparticles. The lifetime can be measured by angle-resolved photoemission spectroscopy and is evaluated as

1

τ lkn=

2π

~∑

k′qn′λ

|gλnn′

kq |2

×[(nλq + f0k′n′

)δ (εk′n′ − εkn − ~ωqλ) δk′,k+q

+(1 + nλq − f0k′n′

)δ (εk′n′ − εkn + ~ωqλ) δk′,k−q

].

Performing the k′ sum we obtain

1

τ lkn=

2π

~∑qn′λ

|gλnn′

kq |2[(nλq + f0k+qn′

)δ (εk+qn′ − εkn − ~ωqλ)

+(1 + nλq − f0k−qn′

)δ (εk−qn′ − εkn + ~ωqλ)

]. (12)

This equation can be derived by keeping only all termsproportional to fkn in Eq. (3) and corresponds approx-imately to neglecting the scattering angle transport fac-tor, (1− cos(θ)), in Eq. (9).13,59,62

4

A. Numerical details

The full k-dependent scattering rate, in contrast to asimplified energy-dependent expression, leads to severalnumerical complications which has not been addressedin the literature to the best of our knowledge. There-fore, we will outline the main technicalities needed toaccount for the anisotropy information stored in the fullband-structure and the electron-phonon coupling. Werepresent the delta-functions in Eq. (4) by Lorentzians,

δ(ε) ≈ 1π

γ/2(γ/2)2+ε2 , with a finite broadening γ. Con-

sequently, we find that it is important to evaluate theFermi-factors f0k′n′ at the exact final energy, εkn ± ~ωas opposed to εk±qn, to get correct results. This alsomeans that absorption and emission terms need to behandled independently when it comes to the peculiarFermi-prefactor in Eq. (9). In addition, we rewrite theprefactor in the case of absorption, to avoid numericalinstabilities:

(1− f0k′n′)

(1− f0kn)=

f0knf0k′n′

1 + nB(εk′n′ − εkn)

nB(εk′n′ − εkn). (13)

This secures a stable denominator, since absorption willdominate the low energy spectrum, ε < µF where f0 → 1and 1−f0 → 0. Considering the k- and q-grids a conver-sion factor of ∆q/∆k is needed if the k-grid and q-gridare not equivalent. One can show that if the two gridsare equivalent the full linearized BTE, Eq. (8), simpli-fies to a linear matrix equation. However, it is advan-tageous to allow for different grid resolutions in order toapply smart-chosing of the grids and resulting simulationspeedup. A fine resolution of the final state q-grid securesa correct result for each k-point even at a rough k-grid.Our approach has therefore been to use fine q-grids withthe possibility of interpolation to even higher resolutionfor all q-dependent variables.

As a final remark we mention that the velocities,vkn,vk′n′ , are obtained from perturbation theory. Froma change dkα, we obtain the derivative

dHk

dkα=∑I

iRI,αHνµeik·RI , (14)

of the Hamiltonian matrix in a basis of localized orbitals(ν, µ), and correspondingly for the overlap matrix S byreplacing H → S. Here I labels the unit cells, withlattice vectors RI , of a supercell discussed thoroughly inthe next sections. A perturbation calculation then gives:

vkn,α =1

~dεkndkα

=1

~〈nk|dHk

dkα− εkn

dSk

dkα|nk〉 , (15)

where εkn and |nk〉 are the energy and Bloch state ofband n at wave vector k. We hereby avoid finite differ-ence errors from crossing bands when the bands are notsorted correctly.

B. Phonons

The phonon polarization vectors (eλq) and energies~ωq,λ are obtained as solutions to the equation

D(q)eλq = ω2q,λe

λq , (16)

where q is the phonon momentum and λ labels thephonon branch (band) index and D(q) is the Fouriertransformed dynamical matrix. We initially compute thereal-space dynamical matrix from a standard finite dif-ference approach63, where the elements are given by

Diµjν =1√MiMj

∂2Etot(R)

∂xiν∂xjµ

≈ 1√MiMj

Fjµ(+∆xiν)− Fjµ(−∆xiν)

2∆xiν, (17)

where i, j are atom indices and µ, ν denote cartesian di-rections. Etot(R) is the total energy written as a functionof all atomic coordinates. The force Fjµ(+∆xiν) is act-ing on atom j in direction µ when atom i is displaced by∆xiν in direction ν. The approximate equality sign indi-cates the approximation inherent in the first order finitedifference method.

When computing the forces, we first construct a super-cell by repeating the unit cell (nA, nB , nC) times alongthe directions of the primitive lattice vectors. Secondlywe perform the finite difference derivative by calculatingthe forces in the entire supercell, while only displacingthe atom in the central unit cell, as schematically shownin Fig. 2.

FIG. 2. Schematic illustration of the supercell method. Aunit cell is repeated 5x5 times. Only the atoms in the centralunit cell are being displaced, while forces are evaluated on allthe atoms in the supercell. Outside a certain range (elipsoidalarea) the forces are zero.

The normalized phonon eigen modes in Eq. (16) are di-mensionless. The transformation to modes with physicaldimension is uλq = lqe

λq, where the characteristic length is

calculated from the polarization vectors and the diagonalmass matrix, m:

lλq =

√~

2ωλ eλ†q ·m · eλq

. (18)

5

C. Bulk electron-phonon coupling

We here provide a simple derivation of the bulkelectron-phonon coupling in the case of a localized ba-sis by applying the periodicity of the problem.

We want to calculate the coupling matrix element be-tween Bloch states |nk〉 and |n′k′〉 due to a phonon withmomentum q and branch index λ perturbing the Hamil-tonian:

gλnn′

kk′q = 〈n′k′|δHqλ|nk〉 . (19)

The perturbation to the Hamiltonian can be expressedas:

δHqλ = lλq∑α

∑I

∂H

∂xI,αeλq,I,α , (20)

where the I-sum runs over the periodic unit cells and theα-sum runs over the spatial degrees of freedom (atomindex and Cartesian direction) within each cell. Usingthe Bloch periodicity of the phonon polarization vectorwe can rewrite this as

δHqλ = lλq∑α

eλq,α∑I

∂H

∂xI,αeiq·RI , (21)

where now eλq,α is components of the polarization vectorin the unit cell with index ’0’ (the reference cell). Theunit cell with index I is displaced from the reference cellby lattice vector RI .

We evaluate the derivative of the Hamiltonian in a sim-ilar manner as the dynamical matrix described above. Aunit cell is repeated to form a supercell, but only theatoms in the central unit cell are displaced. The termsthat contribute to the derivative is the (local) effectivepotential Vlocal(r) and the non-local (NL) Kleinmann-Bylander term VNL. Details of the derivative of theHamiltonian are given in Appendix A.

The electronic Bloch-states are expressed as

|nk〉 =1√N

∑I

∑µ

cµn,keik·r|φµ;RI〉 , (22)

where the I-sum runs over N unit cells in the macro-scopic system, µ labels the basis orbitals within a singleunit cell, cµn,k are expansion coefficients, and |φµ;RI〉 isthe µ’th basis orbital in the unit cell displaced from thereference cell by the lattice vector RI . Inserting (21) and(22) in (19) we get:

gλnn′

kk′q =lλqN

∑IJK

∑µν

∑α

e−ik′·RJ eik·RK (cνn′,k′)∗c

µn,k

× eλq,α〈φν ;RJ |∂H

∂xI,αeiq·RI |φµ;RK〉 . (23)

Due to the periodicity of the system, the derivative ofthe Hamiltonian matrix with respect to atom positionsin RI , can be shifted as follows:

〈φν ;RI + Rm|∂H

∂xI,α|φµ;RI + Rl〉 = 〈φν ;Rm|

∂H

∂x0,α|φµ;Rl〉 .

where we defined the relative vectors Rm/l connectingcell K,J to the cell I. As for the force derivative cal-culation described in Section II B, the derivative of theHamiltonian will also be non-zero in a region around theatoms being displaced. The J,K-sums in Eq. (23) whichruns over all cells in the macroscopic sample can be lim-ited to the cells included in the supercell calculation ofthe ∂H/∂x0,α. We thus replace the J,K-sums with sumsover neighbouring cells m, l relative to I in Eq. (23):

gλnn′

kk′q =lλqN

∑Iml

∑µν

∑α

e−ik′·(RI+Rm)eik·(RI+Rl)eiq·RI

× (cνn′,k′)∗cµn,ke

λq,α〈φν ;Rm|

∂H

∂x0,α|φµ;Rl〉 , (24)

where the derivative of the Hamiltonian is only carriedout for spatial degrees of freedom (α) in the reference unitcell. The I-sum can now be carried out to simply givea factor of

∑I e

i(k−k′+q)·RI = N δk′,k+q which enforces

momentum conservation. Defining gλnn′

kk′q = gλnn′

kq δk′,k+q,

Eq. (24) is then simplified to the final expression for thebulk electron-phonon coupling in a supercell setup:

gλnn′

kq = lλq∑ml

∑µν

∑α

eik·Rl−i(k+q)·Rm(cνn′,k+q)∗cµn,k

× eλq,α〈φν ;Rm|∂H

∂x0,α|φµ;Rl〉 . (25)

In summary, Eq. (25) provides a procedure for calcu-lating the bulk electron-phonon coupling in any localizedbasis setup: One has to evaluate the finite differences of asupercell Hamiltonian where atoms in the center cell aredisplaced and a summation over unit cells is performedwith corresponding phase factors.64

III. SIMULATIONS AND RESULTS

The simulations were performed using the ATKDFT code with the PBE-GGA functional for exchange-correlation in the cases of graphene and silicene, and LDAin the case of MoS2. In all cases we use a Double-Zeta-Polarized (DZP) basis-set. The real-space grid cutoff was110 Ha. The geometries were relaxed until all forces weresmaller than 0.001 eV/A, and 51 × 51 in-plane k-pointswere used in the electronic structure calculations. A vac-uum gap of 30 A was used in the direction normal tothe material plane and Dirichlet boundary conditions wasused in the Poisson equation for this direction. The bulkelectron-phonon interaction and phonon dispersion wasobtained from a 11× 11 supercell calculation in the caseof graphene and a 9× 9 supercell for silicene and MoS2.The delta-functions in Eq. (4) were numerically repre-sented by Lorentzians with a broadening of γ = 3 meV.

6

A) B)

D) C)

1.89eV

ZA

TA LA

TO1

LO1

TO2 LO2

HP (ZO2)

ZO1

ZA

TA

LA

ZO

TA

LO TO

ZO

ZA

LA

TO LO

FIG. 3. (Color online) Electron bandstructure of A) grapheneand silicene and C) MoS2 and phonon dispersion of B)graphene and silicene and D) MoS2.

A. Bandstructures

One obtains linear valence and conduction bands nearthe Dirac point, K, in both graphene and silicene, asshown by the band structures in Fig. 3A. We obtainFermi velocities of 0.9 × 106 m/s and 0.57 × 106 m/sof graphene and silicene, respectively. Both materialshave six phonon branches. The three acoustic modeswill dominate the low temperature scattering where twomodes (LA, TA) have a linear q-dependence and the thirdout-of-plane acoustic (ZA) mode has a q2-dependencenear the Brillouin zone center65, see Fig. 3B. We ob-tain sound velocities of 20.4(12.6) × 103 m/s for theLA(TA) mode of graphene and 9.1(6.1) × 103 m/s forthe LA(TA) mode of silicene. MoS2 is found to be adirect-gap semiconductor66,67 with a band gap of 1.89 eV,see Fig. 3C. The electron- and phonon bandstructures,in Fig. 3C-3D, are consistent with previous theoreti-cal results.55 MoS2 has three acoustic and six opticalbranches. The three acoustic branches are the in-planelongitudinal acoustic (LA), the transverse acoustic (TA)as well as the out-of-plane acoustic (ZA) modes. We ob-tain sound velocities of 6.6(4.2)×103 m/s for the LA(TA)mode of MoS2. The six optical branches are two in-planelongitudinal optical (LO1, LO2), two in-plane transverseoptical (TO1, TO2) and two out-of-plane optical (ZO1,ZO2) modes. The two lowest optical branches (LO1,TO1) are nonpolar modes which do not couple to thecharge carriers. The next two branches (LO2, TO2) arepolar optical modes where the Mo and S atoms vibratein counterphase. The dispersionless out-of-plane mode,

ZO2, is also called the homopolar mode. It is character-istic of layered structures and is related to fluctuationsin the layer thickness68. For further discussion of phononmodes in MoS2 we refer the reader to Refs. 69 and 70.

B. Bulk electron-phonon coupling

We now turn to the calculated bulk electron-phononcoupling. It is common practice to plot the scatteringmatrix element for a fixed k-point as a function of q.In this way one can visualize the detailed suppressionof the scattering, as opposed to a constant or linear-in-q deformation potential, depending on the symmetry ofthe involved phonon and electron states. The units areconverted to eV/A, often used for extracted deformationpotentials from experiments, by dividing by the charac-teristic length prefactor, lλq, in Eq. (25). In Fig. 4 we

TA LA B) A)

LO TO D) C)

FIG. 4. (Color online) Bulk electron-phonon coupling ingraphene for the four modes with nonzero coupling. The in-teraction is illustrated as a function of phonon q-vector at ak-point shifted 300 meV from the Dirac K-point towards theΓ-point. We refer the reader to Ref. 13 for a detailed dis-cussion of the interpretation of the plots for the TA and LAmodes. The scattering rate is obtained as integrals aroundthe constant energy circles satisfying εk′ = εk ± ~ω. Inserts:phonon modes are vizualized by arrows indicating the atomicdisplacements.

illustrate the q-variation of the bulk electron-phonon in-teraction obtained for the four modes coupling with elec-trons in graphene. The atomic motion of the consideredmode is illustrated as an insert in the upper right cor-ner, by arrows indicating the atomic displacement. Theinteraction is obtained around 300 meV from the DiracK-point, and the interaction seems slightly higher for theacoustic modes (between 10−20%) and very close in mag-nitude for the optical modes (within 10%), but in bothcases with the same symmetry as two previously pub-lished results.13,38 The maximal values for the TO andLO modes are in energy units approximately 0.4−0.5 eV

7

for comparison. Importantly, we see that the couplingelements are highly anisotropic. In the case of acousticmodes we see that back-scattering (qx < 0, qy ≈ 0) is sup-pressed for the LA mode, while the situation is reversedfor the TA mode where forward scattering is suppressed.In addition, other directions with complete suppressionalso appear and the q-dependence is highly nontrivial. Ingeneral, the anisotropy and scattering suppression of thebulk electron-phonon coupling is determined by the com-bined symmetry of both phonon and electronic states.

TA LA B) A)

TO LO D) C)

ZA ZO F) E)

FIG. 5. (Color online) Bulk electron-phonon coupling in sil-icene. Unlike graphene also the out-of-plane modes (ZA, ZO)couple significantly with electrons. The interaction is illus-trated as a function of phonon q-vector at a k-point shifted110 meV from the Dirac K-point towards the Γ-point. Thescattering rate is obtained as an integral around a constantenergy circle satisfying εk′ = εk±~ω. Inserts: phonon modesare vizualized by arrows indicating the atomic displacements.

Unlike graphene, the carriers in silicene display astrong interaction with the ZA mode,42 see Fig. 5. Thisis related to the buckling of the silicene sheet, where onebasis atom is displaced approximately 0.44 A out-of-planehence breaking the planar symmetry. Otherwise we seethat the situation for the acoustic modes is similar tothat of graphene. Back-scattering is suppressed for theLA mode, while the situation is reversed for the TA modewhere forward scattering is suppressed. Again, other di-rections with complete suppression of the scattering alsoappear.

It is important to realize that the reduction of the sym-metry of the lattice leads to additional scattering mech-

anisms and modes interacting with the charge carriers.This was the case for silicene where there was no pla-nar symmetry, and it is even more so for the case ofMoS2 which lacks inversion symmetry. In the last men-tioned case, all kinds of electron-phonon coupling maytake place; like different orders of deformation poten-tial, piezoelectric and Frohlich couplings. This makesa strong case for a fully numerical solution of the BTEtogether with a first-principles method to evaluate thebulk electron-phonon interactions. We avoid having ahigh number of free parameters in an analytical modeland instead every type of interaction is directly takencare of. In Fig. 6 we show the bulk electron-phonon in-

TA LA B) A)

TO2 Homopolar (ZO2) D) C)

LO2 E)

FIG. 6. (Color online) Bulk electron-phonon coupling inMoS2. The interaction is illustrated as a function of phononq-vector at the conduction band minimum K-point. Inserts:phonon modes are vizualized by arrows indicating the atomicdisplacements.

teraction obtained for the five modes coupling with elec-trons in MoS2. Unlike graphene and silicene, which aresemimetals, MoS2 is a semiconductor and the interac-tion is evaluated at the conduction band minimum, sinceonly n-doping is relevant in MoS2. Again we find veryanisotropic couplings where the symmetry compares wellto previously published results.55

The coupling with the TA, LA and TO modes are ofthe same order of magnitude as in Ref. 55, cf. Fig. 6A,Band C, but we obtain a somewhat lower coupling forthe homopolar mode (approximately 65% lower), seeFig. 6D. Fig. 6E shows the Frohlich interaction for thepolar optical LO2 mode. The Frohlich interaction isdifficult to converge with respect to supercell size. In

8

the long-wavelength limit this element should increaselinearly.55,68 We find that the peaks increase in magni-tude by approximately 4% and move toward |q| → 0 asexpected if the supercell size is increased from 9 × 9 to15× 15.

C. Scattering rates

We obtain the scattering rate by integrating the lin-earized BTE within the RTA, cf. Eq. (9). The electron-phonon coupling is evaluated for every k-point up to anenergy-cutoff in a 100 × 100 q-mesh. The coupling, en-ergies and velocities were subsequently interpolated totwice this q-space resolution before the BTE was solved.In Fig. 7 we show the result obtained for graphene. Belowthe Bloch-Gruneisen temperature (approximate 57

√nK

for graphene with the carrier density measured in unitsof 1012cm−2), only those phonons with short q are effec-tively excited. This manifest itself in the dip in the scat-tering rate around the Fermi-level. We clearly see the ex-pected low-temperature Bloch-Gruneisen dips around theFermi-level (100meV in the present case) and the open-ing of optical phonon interaction (emission) at µF + ~ω,Fig. 7A. It is illustrative to plot the rate along a single k-line as in Fig. 7A. However, the full two-dimensional de-pendency is needed to capture the anisotropy of the scat-tering rate. Fig. 7B illustrate the scattering rate and in-verse lifetime of the LA and TA modes found in a full two-dimensional k-mesh. The spread of the points illustratea significant dependence on directions of the scatteringrate. This if further highlighted in Fig. 7C,D where thescattering rate was interpolated to clearly illustrate theanisotropy of especially the LA and TA modes. Part ofthe anisotropy originates from the bulk electron-phononcoupling. This is the main contribution to the anisotropyof the inverse lifetime. However, the anisotropy is furtheramplified by the transport scattering angle 1− cos(θ) inEq. (9), which is seen by comparing scattering rate andinverse lifetime in Fig. 7B.

We conclude that the scattering rate depends signifi-cantly on the k-space directions. In addition, our imple-mentation gives results for the graphene scattering ratethat are consistent with previous theoretical results in thelow-temperature Bloch-Gruneisen regime, as well as thehigh temperature equipartition regime where the scatter-ing rate should depend linearly on energy.11,55

Previous studies have calculated the lifetime of carriersin silicene.42 In Fig. 8 we show the scattering rate alonga single k-line for silicene and MoS2.

We find a significant scattering with the ZA mode forsilicene. However, this mode is difficult to describe ac-curately; The electron-phonon interaction is only partlyscreened in the present formalism and the linear dis-persion of the ZA modes results in a constant den-sity of states which is therefore not able to cutoff long-wavelength interactions. We expect the scattering ratewith the out-of-plane ZA mode to be significantly re-

duced due to interaction with a substrate and a moreprecise description of screening. Therefore, this interac-tion will not be included in the mobility calculations. Theobtained mobilities should be taken as upper limits andif the ZA scattering is included as is it would reduce themobility by two orders of magnitude. For both siliceneand MoS2, we find that the rates only changes slightlyif the Fermi-level is reduced further. For MoS2 the scat-tering rate is almost independent of µF for µF < Ec;only the scattering rates for the LO2 and LA modes re-duces slightly at high energy. Unlike the scattering re-lated to microscopic changes in potential (deformationpotential or nonpolar optical scattering), the scatteringrelated to the macroscopic electric-field (piezoelectric orFrohlich interaction) will be partially screened by a di-electric environment. One approach to handle this dif-ference in screening is to partition electron-phonon in-teraction in real-space into the short-ranged deforma-tion potential contributions and long-ranged piezoelec-tric and Frohlich interactions and apply the screeningindividually49. However, as shown in Fig. 8B the Frohlichinteraction (LO2 mode) is not dominating the transportand the piezoelectric coupling is most important at lowtemperatures24,55.

D. Mobility

In Fig. 9 we illustrate the obtained carrier concen-tration and mobility71 of graphene, silicene (A, B) andMoS2 (C, D). We only include intravalley scattering inthe present analysis. The same q-mesh is used as forthe scattering rate analysis. For the k-space we evaluatethe scattering rate in the k-points that contribute, froma very dense 1500 × 1500 Monkhorst-Pack sampling ofthe first Brillouin zone, up to a given Fermi-level/carrier-density. As an example, the resulting k-points are illus-tated by markers in Fig. 7 in the case of graphene. Theresulting number of k-points treated within the valley is316, 378 and 514 for graphene, silicene and MoS2, respec-tively.

The phonon scattering limited mobilities calculatedhere shows that graphene can have a mobility closeto 106 cm2/V s at 100 K and a carrier density of 3 ×1011cm−2, cf. Fig. 9. At 300 K, we obtain a mobil-ity decreasing from 145, 000 cm2/V s at a carrier den-sity of 1 × 1012cm−2 to 55, 000 cm2/V s at a carrierdensity of 3 × 1012cm−2. In comparison, experimentshave so far achieved room temperature values decreas-ing from roughly 90, 000 cm2/V s to 45, 000 cm2/V s atthe same carrier densities.14 The same experiment alsoobtain mobilities up to 106 cm2/V s at lower temper-atures. In addition, we find the mobility of siliceneto be more than an order of magnitude lower thangraphene but still very high. We obtain a mobility ofroughly 2100 cm2/V s at 300 K and a carrier density of3 × 1012cm−2. Previous calculations have only consid-ered ungated silicene and obtained values in the range of

9

TA C)

B) A)

LA D)

FIG. 7. (Color online) Scattering rate as a function of energy/k for graphene. A) At a temperature of 5.7 K and a Fermi-levelµF = 100 meV along the k-points from K to Γ up to 0.5 eV. B) Comparison of the anisotropy in the scattering rate and inverselifetime at a temperature of 300 K and a Fermi-level µF = 100 meV. C,D) Two-dimensional scattering rate for the highlyanisotropic TA and LA phonon modes. The k-points are relative to the Dirac point, K, and the temperature is 300 K andFermi-level µF = 100 meV. The points illustrate the original mesh up to 0.5 eV and the contours are obtained from cubic splineinterpolation.

10− 1000 cm2/V s at 300 K.42,72 The only experiment onsilicene presently published achieve a mobility of roughly100 cm2/V s at 300 K.4 For MoS2 our calculated mobil-ity decreases from a value of 1700 cm2/V s at 100 K toapproximately 400 cm2/V s at 300 K. These results arein good agreement with published first-principles sim-ulations and experimental values.46,55,57 In comparison,experiments have achieved up to 200 cm2/V s at 300 K.16

We note that the electron-phonon coupling as a func-tion of q, as was shown for all materials for a singlek-point, was evaluated at all k-points in the mobilitycalculations in Fig. 9. Another approach that is oftenapplied is to evaluate the scattering rate along a singlek-line obtained from a |gλnn′

kq | at a fixed k-point. Thiscan be transformed to a generic energy-dependence whichis used to evaluate the mobility. This approach how-ever, neglects part of the anisotropy in the bulk electron-phonon coupling, which was included in the present re-sults.

E. Intervalley scattering

We only included intravalley scattering in the previousfigures. However, it is straightforward to evaluate the in-tervalley scattering rate separately by doing a calculationwith the same k-mesh and shifting the q-mesh to q+K.We have done so for all three materials along a singlek-line to evaluate at what temperatures and carrier den-sities the intervalley scattering might start contributing.

In Fig. 10 we compare the obtained intra- and interval-ley scattering rates summed over all modes, except the in-travalley ZA mode as discussed previously. For graphene,we find that intervalley scattering starts contributingaround 200− 300 K and at a Fermi-level above 140 meV,corresponding to a carrier density of n0 ≈ 1012cm−2,cf. Fig. 9A. The reason is that intervalley scattering re-quires a phonon momentum connecting the two valleys|q| ≈ |K − K′| ≈ |K| where the lowest energy phononmodes (ZA and ZO around q = K) only exist above65 meV in the case of graphene, cf. Fig. 3B. A simi-lar onset exist for silicene and MoS2 where we at 300 K

10

B)

A)

MoS2

Silicene

FIG. 8. (Color online) Scattering rate as a function ofenergy/k for silicene (A) and MoS2 (B) relative to the con-duction band minimum Ec. In both cases the temperature is300 K. The Fermi-level is tuned to µF = 100 meV for siliceneand 500 meV below the conduction band minimum for MoS2.We used k-points along a line from K to Γ up to 0.5eV forsilicene and K to M for MoS2.

find from Fig. 10 that intervalley scattering starts con-tributing at a Fermi-level of 30 meV and 100 meV abovethe conduction band minimum, respectively. For MoS2

this corresponds to n0 ≈ 1014cm−2 so that intervalleyscattering does not contribute significantly, whereas forsilicene intervalley scattering seems to make a significantcontribution above n0 ≈ 0.5× 1012cm−2.

Our definition of the onset for intervalley contributionis that we obtain an intervalley scattering rate of thesame order of magnitude as for the intravalley scatter-ing. Using Matthiessen’s rule 1/µ = 1/µintra + 1/µinter

we can estimate that the total mobility will decrease byapproximately a factor of two at carrier densities aboven0 ≈ 1012cm−2 if intervalley scattering was includedfor graphene. For MoS2 the contribution will be muchsmaller at assessable carrier densities, whereas silicenecan have as much as an order of magnitude lower mo-bility due to intervalley scattering for carrier densitiesabove n0 ≈ 0.5× 1012cm−2.

IV. CONCLUSION

In summary, we have presented a Boltzmann TransportEquation (BTE) solver implemented in the AtomistixToolKit (ATK) simulation tool. The method allows forcalculation of material properties, including the electron-phonon interaction, from first-principles. We have ap-plied the tool to calculate the phonon-limited mobilityin n-type monolayer graphene, silicene and MoS2. Ourresults compares well to published theoretical results andexperiments for MoS2 and graphene and extends previoustheoretical calculations for silicene. The bulk electron-phonon coupling is highly anisotropic due to scatteringsuppression in different q-directions, as well as a nontriv-ial q-dependence, related to the combined symmetry ofthe electron and phonon states. The simulations providean upper bound for the electron mobilities of the selected2D materials. The ab initio approach demonstrated inthis paper can be directly applied to other materials inboth 1D, 2D and 3D, larger nanostructures and may bestraightforwardly extended to study cases with electron-impurity scattering. In addition, we have illustrated howthe reduction of the lattice symmetry, when going fromgraphene to silicene with no planar symmetry and furtheron to MoS2 with no inversion symmetry, leads to addi-tional scattering mechanisms and modes interacting withthe charge carriers. This makes a strong case for the needof a fully numerical solution of the BTE together witha first-principles method to evaluate the bulk electron-phonon interaction - especially, when going to systemswith many modes, such as systems with low lattice sym-metry, slab structures and nanostructured materials.

ACKNOWLEDGMENT

We thank K. Kaasbjerg (DTU) and T. Frederiksen(San Sebastian) for useful discussions on the methodsemployed. The authors acknowledges support from In-novation Fund Denmark, grant Nano-Scale Design Toolsfor the Semiconductor Industry (j.nr. 79-2013-1). TheCenter for Nanostructured Graphene (CNG) is sponsoredby the Danish Research Foundation, Project DNRF58.

Appendix A: Finite difference derivative ofHamiltonian

The derivative of the Hamiltonian operator is calcu-lated in a way similar to the dynamical matrix followingthe approach of Kaasbjerg et al.55. We construct a su-percell by repeating the primitive configuration a num-ber of times in each periodic direction. Subsequently wedisplace the atoms in the central cell by ±δ in all Carte-sian directions (forward and backward) and calculate thederivative with respect to e.g. atom i in the x-direction

11

300K

200K

100K

100K

200K

300K

MoS2 MoS2

B) A)

D) C)

FIG. 9. (Color online) Carrier concentration as a function of Fermi-level for graphene and silicene (A) and MoS2 (C). Theconduction band edge of MoS2 is at 0.95 eV. The mobility as a function of carrier density is also shown for graphene (B) andsilicene (B, inset) and MoS2 (D).

E-Ec [eV]

FIG. 10. (Color online) Scattering rate as a function ofenergy/k for graphene, silicene and MoS2 relative to the con-duction band minimum Ec. In all cases the temperature is300 K. The Fermi-level and k-route is the same as for theintravalley scattering rates presented in Fig. 7B and Fig. 8.

as

∂H

∂xi≈ H(xi + δ)− H(xi − δ)

2δ, (A1)

where H(xi + δ) indicates the Hamiltonian operator ob-tained for the configuration where atom i is displaced by

δ in the positive x−direction. Note that Eq. (A1) appliesfor the Hamiltonian operator, and not for the Hamilto-nian matrix expressed in a basis of LCAO basis functions.In the latter case it is necessary to correct also for thedisplacement of the basis orbitals63.

The Hamiltonian operator has several terms:

H = T + Vlocal + VNL , (A2)

where T is the kinetic energy operator, Vlocal is the lo-cal potential including the exchange-correlation poten-tial, the Hartree potential as well as the local pseudo-potential, while VNL is the non-local Kleinman-Bylanderpseudo-potential. Since the kinetic energy does not de-pend on the atomic coordinates the derivative of thatis zero. The derivative of the local potential can be di-rectly evaluated using Eq. (A1). The non-local potentialis written as

VNL =∑i

∑αβ

|χiα〉viαβ〈χiβ | , (A3)

where |χiα〉 is a projector function centered on atom iand viαβ are (fixed) projector coupling elements. The

12

derivative is written as

∂VNL∂xj

=∂

∂xj

∑i

∑αβ

|χiα〉vαβ〈χiβ |

=∑αβ

(∂|χjα〉∂xj

vαβ〈χjβ |+ |χjα〉vαβ

∂〈χjβ |∂xj

).(A4)

The derivatives of the projector functions are evaluatednumerically as in Eq. (A1).

Having calculated the derivative of the Hamiltonianoperator we evaluate the derivative in the LCAO basis(

∂H

∂xi

)µν

= 〈φµ|∂H

∂xi|φν〉 . (A5)

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ment gλnn′

kk′q . Correspondingly, the opposite selection rule

δk′,k−q is obtained from the matrix element gλnn′

k′kq . Since

|gλnn′

kq | = |gλnn′

qk | only one element needs to be evaluatedin the calculation of the mobility.

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