Phonon-limited carrier mobilities and Hall factors in 4H-SiC from rst principles Tianqi Deng1 2Deren Yang1 2and Xiaodong Pi1 2y

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Phonon-limited carrier mobilities and Hall factors in 4H-SiC from

ﬁrst principles

Tianqi Deng,1, 2, ∗Deren Yang,1, 2 and Xiaodong Pi1, 2, †

1State Key Laboratory of Silicon Materials and

School of Materials Science and Engineering,

Zhejiang University, Hangzhou, Zhejiang 310027, China

2Institute of Advanced Semiconductors & Zhejiang Provincial

Key Laboratory of Power Semiconductor Materials and Devices,

ZJU-Hangzhou Global Scientiﬁc and Technological Innovation Center,

Hangzhou, Zhejiang 311200, China

(Dated: October 7, 2022)

arXiv:2210.02813v1 [cond-mat.mtrl-sci] 6 Oct 2022

Abstract

Charge carrier mobility is at the core of semiconductor materials and devices optimization,

and Hall measurement is one of the most important techniques for its characterization. The

Hall factor, deﬁned as the ratio between Hall and drift mobilities, is of particular importance.

Here we study the eﬀect of anisotropy by computing the drift and Hall mobility tensors of a

technologically important wide-band-gap semiconductor, 4H-silicon carbide (4H-SiC) from ﬁrst

principles. With GW electronic structure and ab initio electron-phonon interactions, we solve

the Boltzmann transport equation without ﬁtting parameters. The calculated electron and hole

mobilities agree with experimental data. The electron Hall factor strongly depends on the direction

of external magnetic ﬁeld B, and the hole Hall factor exhibits diﬀerent temperature dependency

for Bkcand B⊥c. We explain this by the diﬀerent equienergy surface shape arising from the

anisotropic and non-parabolic band structure, together with the energy-dependent electron-phonon

scattering.

I. INTRODUCTION

Increasing demands in high-power and high-temperature electronic devices call for wide-

band-gap semiconductors as alternative functional materials to silicon. Silicon carbide (SiC)

has become one of the most promising materials in power electronic devices owning to its

unique combination of high carrier mobility, high critical ﬁeld strength, high saturation

velocity, and high thermal conductivity [1–6]. Among the more than two hundred polytypes,

4H-SiC is preferred for its wider band gap and higher critical electric ﬁeld than the cubic

3C-SiC and higher carrier mobilities and lower anisotropy as compared to 6H-SiC. Therefore,

it is more technologically relevant and has become the major functional SiC polytype for

applications in electronic devices [1].

Despite the recent surge of academic and industrial interests in 4H-SiC, many important

aspects of its physical properties and the underlying physics are not clariﬁed yet. For exam-

ple, as a hexagonal crystal, anisotropy is expected for its physical properties like mechanical

[7] and transport properties [1] such as carrier mobilities and Hall eﬀect. The carrier mobil-

∗dengtq@zju.edu.cn

†xdpi@zju.edu.cn

ity is a key functional property that determines device performance such as on-resistance [8].

However, it is diﬃcult to experimentally distinguish between the anisotropy contributions

from drift mobility and Hall factor in a Hall measurement, and the common practice is either

assuming a unity Hall factor rH= 1, estimating rHusing empirically parametrized models,

or estimating the true carrier concentration from dopant concentration and activation en-

ergies [8–13]. Additionally, the analysis of one of most important mechanisms underlying

its charge transport phenomena, i.e. the electron-phonon interactions and scatterings, still

relies on empirically determined, adjustable parameters with signiﬁcant uncertainty. These

adjustable parameters were also employed to explain exotic phenomena such as non-unity

hole Hall factors [9, 11, 12]. Therefore, it becomes increasingly important to investigate

such microscopic physics and conﬁrm their respective contributions in the charge transport

process without resorting to uncertain ﬁtting parameters.

Electron-phonon interactions from the density functional perturbation theory (DFPT)

calculations [14] emerged as a powerful tool for studying importance phenomena in solid

state and their underlying microscopic mechanisms, including phonon-limited charge trans-

port [15–17], superconductivity [18, 19], polaron [20, 21], phonon-assisted optical absorption

[22], band structure renormalization [23, 24], etc. In conjunction with Boltzmann transport

equation (BTE), the charge transport in the presence of electrical ﬁeld and magnetic ﬁeld

can be simulated self-consistently to obtain key quantities like drift mobility [15], break-

down ﬁeld [25], and thermoelectricity [26]. Recently, the Hall eﬀect in several typical cubic

semiconductors has been studied by solving the BTE in the presence of both electric and

magnetic ﬁelds, where quantitative agreement has been achieved in comparison with exper-

imental measurements [17]. It is thus intriguing to explore the possible anisotropy in the

hexagonal phase, to compute the Hall factors in the intrinsic limit, and to clarify the role of

electron-phonon interactions in 4H-SiC.

In this work, we performed in-depth analysis of electron-phonon interactions, phonon-

limited charge transport, and their anisotropy in 4H-SiC by ﬁrst-principles calculations.

Both short-ranged and long-ranged dipolar/quadrupolar electron-phonon interactions are

included from ﬁrst principles in combination with Wannier-interpolation technique [17, 27–

30]. We ﬁnd that the spin-orbit coupling (SOC) signiﬁcantly aﬀect the hole eﬀective masses,

even though the SOC splitting is small. The phonon-limited mobilities agree well with

experimental measurements of lightly-doped samples, and hole mobility exhibits a stronger

anisotropy than that of electron. The electrons are mainly scattered by optical phonons,

while the band-edge holes are mostly scattered by acoustic phonons. The Hall factors

depend on the directions of both the applied magnetic ﬁeld and the electric current. Hall

factors deviate from 1 for both electrons and holes, and distinct temperature-dependence

were predicted. The non-unity is explained by non-parabolic band structure, non-spherical

equienergy surface, and energy-dependent electron-phonon scattering strength. This work

thus clariﬁes the anisotropic charge transport phenomena in 4H-SiC and the impact of

electron-phonon interactions in the intrinsic limit from a microscopic, ab initio perspective.

The predicted Hall factors without empirical, adjustable parameters also allows possible

comparison between drift mobility and Hall mobility from experimental measurements.

II. METHODS

A. Carrier mobility and Hall eﬀect calculations

In a typical Hall measurement for Hall mobility along αdirection, an electric current

density jalong αand magnetic ﬁeld Balong γare applied, and the induced Hall ﬁeld Eor

Hall voltage along βis measured, where α,βand γare orthogonal. In the linear regime of

small Bγ, the Hall coeﬃcient is

αβγ =Eβ

jαBγ

=[σ−1(Bγ)−σ−1(0)]βα

Bγ

≈σ−1(0)σ(Bγ)−σ(0)

Bγ

σ−1(0)βα

.(1)

Therefore, calculation of the carrier mobility and Hall coeﬃcient involves computing the

magnetic ﬁeld B-dependent conductivity tensor

[σ(B)]αβ =−e

Vuc X

nZBZ

d3k

ΩBZ

vnkα∂EβfB

nk.(2)

Here Vuc is the unit-cell volume, vnkα=∂εnk

∂¯hkαis the band velocity deﬁned as the k-

derivative of eigen-energy εnkalong αdirection, and ∂EβfB

nkis the solution of the linearized

0.2M-ΓM L

0.0

0.2

=0.39

Energy (eV)

=0.33

0.2Γ-M ΓA

-0.2

0.0

=0.21

=1.38

=1.50

=1.28

=0.35

=1.07

Energy (eV)

(a)

(b)

(c)

FIG. 1. (a) The ﬁrst Brillouin zone of 4H-SiC and major high-symmetry points. The band

structure of (b) conduction and (c) valence bands near the band edges. The eﬀective masses along

(k) and perpendicular to the c-axis (⊥) are also given.

(BTE) with magnetic ﬁeld B

−evnkβ

∂f0

∂εnk

−e

¯h(vnk×B)· ∇k∂EβfB

mZBZ

d3q

ΩBZ τ−1

mk+q→nk∂EβfB

mk+q

−τ−1

nk→mk+q∂EβfB

nk,(3)

with f0

nkand ΩBZ being the equilibrium Fermi-Dirac distribution and ﬁrst Brillouin zone

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Phonon-limitedcarriermobilitiesandHallfactorsin4H-SiCfromrstprinciplesTianqiDeng,1,2,DerenYang,1,2andXiaodongPi1,2,y1StateKeyLaboratoryofSiliconMaterialsandSchoolofMaterialsScienceandEngineering,ZhejiangUniversity,Hangzhou,Zhejiang310027,China2InstituteofAdvancedSemiconductors&ZhejiangProvincialKe...

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Phonon-limited carrier mobilities and Hall factors in 4H-SiC from rst principles Tianqi Deng1 2Deren Yang1 2and Xiaodong Pi1 2y

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