The temperature-dependence of carrier mobility is not a reliable indicator of the dominant scattering mechanism Alex M. Ganose1 2Junsoo Park1and Anubhav Jain1

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The temperature-dependence of carrier mobility is not a reliable indicator of the

dominant scattering mechanism

Alex M. Ganose,1, 2, ∗Junsoo Park,1and Anubhav Jain1, †

1Energy Technologies Area, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

2Department of Chemistry, Molecular Sciences Research Hub,

White City Campus, Imperial College London, Wood Lane, London, UK

(Dated: October 5, 2022)

The temperature dependence of experimental charge carrier mobility is commonly used as a

predictor of the dominant carrier scattering mechanism in semiconductors, particularly in thermo-

electric applications. In this work, we critically evaluate whether this practice is well founded. A

review of 47 state-of-the-art mobility calculations reveals no correlation between the major scatter-

ing mechanism and the temperature trend of mobility. Instead, we demonstrate that the phonon

frequencies are the prevailing driving forces behind the temperature dependence and can cause it

to vary between T−1to T−3even for an idealised material. To demonstrate this, we calculate

the mobility of 23,000 materials and review their temperature dependence, including separating

the contributions from deformation, polar, and impurity scattering mechanisms. We conclusively

demonstrate that a temperature dependence of T−1.5is not a reliable indicator of deformation po-

tential scattering. Our work highlights the potential pitfalls of predicting the major scattering type

based on the experimental mobility temperature trend alone.

Ever since the ﬁrst theories of semiconductors were

developed, the temperature dependence of mobility has

been used to understand the quantum behaviour of ma-

terials. In 1931, Wilson derived expressions for charge

transport in semiconductors under the assumption that

lattice vibrations were the major cause of the electronic

resistivity [1,2]. His work proved highly successful at pre-

dicting the temperature dependence of mobility in n-type

germanium and lay the foundation for the modern theory

of band conduction in semiconductors [3,4]. The temper-

ature dependence of experimentally measured Hall mobil-

ity is now commonly used as a predictor of the dominant

scattering mechanism in thermoelectric materials [5–7].

Knowledge of the dominant scattering mechanism is of-

ten employed to ﬁt models of charge transport including

deformation potentials and eﬀective masses, and to ob-

tain estimates of the optimal doping concentration and

temperatures that maximise thermoelectric performance

[8–12].

Wilson’s 1931 paper demonstrated that the mobility of

a system dominated by acoustic lattice scattering should

exhibit a µ∝T−1.5dependence [2,3,13]. The same tem-

perature dependence was later demonstrated for optical

lattice scattering at high temperatures [14,15]. Since

then, a temperature dependence of µ∝T−1.5has widely

been considered an experimental signature of deforma-

tion potential scattering. Temperature trends ranging

from µ∝T−0.50–T−0.75 are thought to indicate scatter-

ing due to polar optical phonons [16–18], and even more

positive coeﬃcients are ascribed to piezoelectric (T−0.5),

alloy (T−0.5), and impurity (T1.5) scattering [19–21]. In

all of these cases, the expected temperature trends are de-

∗a.ganose@imperial.ac.uk

†ajain@lbl.gov

rived from highly-simpliﬁed models of electronic scatter-

ing in systems containing a single isotropic and parabolic

band and a single dispersion-less phonon frequency.

State-of-the-art approaches based on density func-

tional perturbation theory combined with Wannier inter-

polation (DFPT+Wannier) can now calculate the trans-

port properties of semiconductors with predictive accu-

racy [22–24]. As the number of materials studied using

DFPT+Wannier has grown — at the time of writing this

includes over 100 bulk and two-dimensional compounds

— an unexpected trend has emerged. Many materials

that were thought to be limited by acoustic deforma-

tion potential scattering based on the temperature de-

pendence of mobility of µ∝T−1.5have instead been

shown to have strong contributions from polar optical

and other scattering mechanisms (Fig. 1) [25,26]. Ac-

cordingly, these latest computational results are challeng-

ing the commonly held assumption that the temperature

dependence of mobility is a reliable indicator of the un-

derlying scattering processes.

In this work, we critically evaluate whether the tem-

perature dependence of carrier mobility is a reliable in-

dicator the dominant scattering type. We review 47

DFPT+Wannier calculations that reveals no correlation

between the major scattering mechanism and the temper-

ature trend of mobility. Instead, we demonstrate that the

phonon frequencies are largely responsible for the tem-

perature dependence of mobility and can cause it to vary

between T−1to T−3even in a simple parabolic band

structure. Finally, we extract the temperature depen-

dence of mobility for acoustic deformation potential, po-

lar optical, and impurity scattering in over 23,000 ma-

terials that we have calculated using the recently devel-

oped amset software [27]. Our results demonstrate that

the temperature dependence of mobility should not be

used to determine the dominant scattering mechanism.

We conclude with the potential pitfalls of assuming the

arXiv:2210.01746v1 [cond-mat.mtrl-sci] 4 Oct 2022

FIG. 1. The temperature (T) dependence of mobility does

not correlate with the dominant scattering mechanism. The

temperature-dependence of mobility calculated using density

functional perturbation theory with Wannier interpolation

(DFPT+Wannier) is plotted with the dominant scattering

type identiﬁed by the color of the points. There is clearly

no correlation between the temperature dependence and the

dominant scattering type. The temperature dependence is

plotted against the band edge eﬀective mass and separated

into abulk and btwo-dimensional semiconductors to help vi-

sually clarify the points. Theoretical results were extracted

from Refs. [22,24–26,28–43]. The scattering types consid-

ered are polar optical (PO, teal), deformation potential (DP,

pink), piezoelectric acoustic (PI, orange) and polaronic scat-

tering (PL, purple). A particular mechanism is considered

dominant if it reduces the mobility by an order of magnitude

or more. We have indicated the case where two scattering

mechanisms are competing as half ﬁlled circles. Only temper-

atures less than 550 K were considered. See Section S1 of the

Supplementary Material for the full temperature dependent

results and data extraction procedure.

dominant scattering mechanism based on the tempera-

ture dependence of mobility alone.

The temperature dependence of mobility calculated

by DFPT+Wannier (23 bulk and 24 monolayer mate-

rials) shows a wide range of values spanning −3.1(n-

type SrTiO3) to −0.42 (monolayer p-InSe) as presented

in Fig. 1. We have distinguished each compound by scat-

tering type, where a particular mechanism is considered

dominant if it reduces the mobility by an order of magni-

tude or more. We ﬁnd no observable correlation between

the dominant scattering mechanism and the temperature

trend. For example, most materials exhibit a temper-

ature dependence more negative than µ∝T−1.5. Al-

though this trend is commonly associated with deforma-

tion potential scattering, many of these materials are in

fact dominated by polar optical phonon scattering. Fur-

thermore, many materials limited by deformation scat-

100 200 300 400 500

Temperature (K)

101

102

103

104

Mobility (cm2/Vs)

n-GaAs (T−1.7)

p-Si (T−2.4)

n-SrTiO3 (T−3.1)

n-GaAs

Theory

Exp.

p-Si

Theory

Exp.

n-SrTiO3

Theory

Exp.

n-SrTiO3

Theory

Exp.

FIG. 2. Electron mobility as a function of temperature for

GaAs, Si, and SrTiO3. The theoretical mobility of GaAs was

calculated using density function perturbation theory com-

bined with Wannier interpolation (DFPT+Wannier) and in-

cludes two-phonon scattering (pink squares, [24]). The theo-

retical mobility of Si was obtained using DFPT+Wannier in

Ref. [22]. The theoretical mobility of SrTiO3was obtained

in Ref. [43] using DFPT+Wannier and a cumulant diagram-

resummation approach that includes eﬀects beyond the quasi-

particle regime. The experimental mobility of GaAs (grey up

triangles), Si (grey down triangles), SrTiO3(grey starts) were

obtained using Hall eﬀect measurements from references [44–

46], [47–50], and [51] respectively.

tering exhibit temperature trends more negative than

µ∝T−2. These results indicate that the mobility trends

derived for idealised scattering in simple parabolic bands

are not reliable in most materials.

We note that the majority of bulk materials are dom-

inated by polar optical phonon scattering, whereas the

monolayer materials are dominated by deformation po-

tential scattering. However, as most monolayers calcu-

lated using DFPT+Wannier to date have been elemen-

tal compounds that are inherently non-polar, this trend

may not reveal a fundamental diﬀerence in the scattering

physics between bulk and monolayer systems.

We stress that DFPT+Wannier is a state-of-the-art

approach that can yield excellent agreement with exper-

imental Hall eﬀect measurements. This is highlighted in

Fig. 2where we present the calculated and experimen-

tal carrier mobilities of n-GaAs, p-Si, and n-SrTiO3. For

each material, the theoretical mobility exhibits excellent

agreement with both the magnitude and temperature de-

pendence of the experimental measurements. However,

in each case the observed temperature trend diﬀers from

the trend predicted by the dominant scattering mecha-

nism. For example, p-Si exhibits a µ∝T−2.4, despite

being dominated by deformation potential scattering, a

very large deviation from the nominal T−1.5dependence.

In the following section we examine these materials in

more detail with the goal of uncovering why a given

scattering type can exhibit a wide range of temperature

po (THz)

T(K)

300 500

800 1000

T(K)

300 500

800 1000

T-dependence

GaAs

po SrTiO3

ne(cm 3)

1013

1018

120

Avg atomic mass (Da)

a b

40 80

r2= 0.509

m1/2

0 5 10 15 20

po (THz)

FIG. 3. Polar optical phonon frequency determines the temperature dependence of mobility for transport in a single parabolic

band. aPolar optical phonon frequency (ωpo) against the temperature (T) dependence of mobility, calculated at low (1013 cm−3,

open circles) and high (1018 cm−3, ﬁlled triangles) doping and between 300 K–500 K (blue) and 800 K–1000 K (orange). Angular

brackets (hωi) indicate averaged values. As Silicon is non-polar, we have used the average optical phonon frequency at the

Brillouin zone center (hωoi). Calculations were performed using AMSET on a single parabolic band (eﬀective mass of 0.2me)

with the scattering parameters detailed in Section S3 of the Supplementary Material. bThe atomic mass can be used as a

proxy to estimate the polar optical phonon frequency. Heatmap indicating the relationship between average atomic mass and

polar optical phonon frequency for all materials in the phonon frequency dataset (generated from density functional theory

calculations; machine learning predictions are excluded). The phonon frequency is roughly proportional to the inverse square

root of the averaged mass (ωpo ∝ hmi−1/2), as can be derived directly from the phonon dynamical matrix. Darker points

indicate more materials. The r2correlation coeﬃcient in the white box indicates reasonable correlation.

trends.

GaAs is a classic zinc-blende semiconductor with an

isotropic conduction band pocket centered at the Γ-point

and scattering dominated by polar optical phonons [24].

Despite its simple band structure and essentially single

source of scattering, GaAs exhibits a µ∝T−1.7depen-

dence that is very close to that associated with deforma-

tion potential scattering. As we shall demonstrate, the

value of −1.7is not intrinsic to the dominant scattering

mechanism itself but is instead a consequence of the phys-

ical properties of GaAs, in particular: (i) its large polar

optical phonon frequency and (ii) slight non-parabolicity

in the conduction band.

To investigate further, we performed mobility calcula-

tions for a single parabolic band with an eﬀective mass

m∗

e=0.2meusing the amset package [27] and only in-

cluded polar optical scattering (known to dominate in

GaAs). amset has been shown provide scattering rates

and mobility within ∼10 % of DFPT+Wannier when

benchmarked on 23 semiconductors [27]. Further details

on the amset methodology and the calculation proce-

dure are given in Section S3 of the Supplementary Ma-

terial.

Our transport calculations reveal that systems with

smaller phonon frequencies will show a weaker (less neg-

ative) temperature dependence of mobility for a ﬁxed

temperature range. Indeed, simply adjusting the polar

optical phonon frequency can cause the mobility to de-

cay as gradually as µ∝T−0.67 (ωpo =0.1 THz) or as

rapidly as µ∝T−3.34 (20 THz) for a single parabolic

band at low temperature and doping (T=300 K–500 K;

n=1013 cm−3, Fig. 3a, solid blue line). We note that

the temperature dependence for small ωpo falls within the

range of values broadly associated polar optical phonon

scattering (T−0.50–T−0.75). When calculations are per-

formed using the polar optical frequency of GaAs (ωpo

=8.16 THz), the mobility exhibits a dependence of µ∝

T−1.58 close to the experimental trend of µ∝T−1.7. Our

analysis indicates that simple compounds composed of

light atoms with high-frequency optical modes are likely

to exhibit a more negative T-dependence of mobility than

compounds composed of heavy atoms which generally ex-

hibit low phonon frequencies (as demonstrated in Fig. 3b

which reveals the inverse relationship between atomic

mass and ωpo).

We note that the temperature dependence of mobility

also depends on the temperature and the doping con-

centration. At higher temperatures, the temperature de-

pendence is weakened (made less negative). For exam-

ple, at a phonon frequency of 20 THz the mobility de-

pendence is reduced from T−3.34 between 300 K–500 K

to T−1.81 between 800 K–1000 K (Fig. 3a, solid orange

line). Thermoelectric materials generally exhibit opti-

mal performance at degenerate or near degenerate doping

(termed “high doping”). A higher doping concentration

results in a weakening of the temperature dependence of

mobility across all phonon frequencies (Fig. 3a, dashed

blue line, ne= 1018 cm−3). At high temperature and

FIG. 4. The temperature dependence of the electron lifetimes are determined by the Bose–Einstein phonon occupation number

(n) which is the main factor that controls the temperature dependence of mobility. aEnergy dependence of the electron

lifetimes (τ) for diﬀerent polar optical phonon frequencies, indicating pre-phonon emission and post-emission regimes. Larger

frequencies increase the pre-emission onset energy range. bTemperature against the pre-phonon emission electron lifetimes

(τpre) and the inverse phonon occupation (1/n). At higher temperatures, the phonon occupation increases and the lifetime is

shortened. The temperature dependence of ndirectly controls the temperature dependence of τpre . The phonon occupation and

scattering rate are normalised to their values at 300 K.cPre- and dpost-emission lifetimes, calculated at a carrier concentration

of 1013 cm−3. The temperature dependence of the lifetimes is stronger (more negative) for pre-emission states and at lower

temperatures. Percentage of states that contribute to mobility in the post-emission energy range against temperature for elow

doping (1013 cm−3) and fhigh doping (1018 cm−3) concentrations (see text for more details). The percentage of post-emission

states is lower at low temperatures and low doping. In c-f, grey numbers indicate the temperature dependence of the associated

curve. Calculations were performed using AMSET with scattering parameters detailed in Section S3 of the Supplementary

Material.

high doping, the mobility exhibits the weakest temper-

ature dependence and does not become more negative

than µ∝T−1.50 even at the largest phonon frequencies

(Fig. 3a, ωpo = 20 THz,ne=1018 cm−3,T=800 K–

1000 K). Note, however, that even at high-temperature,

high-doping conditions, mobility limited by polar-optical

scattering can exhibit a similar temperature dependence

as that broadly associated with lattice deformation po-

tential scattering (T−1.50). In Section S2 of the Supple-

mentary Material, we explicitly investigate the impact

of changing the temperature and carrier concentration

on the temperature dependence of mobility, and conﬁrm

the trends discussed above.

The relationship between phonon frequency and the

temperature dependence of mobility arises due to the

temperature dependence of the electron lifetimes (τ).

The temperature dependence of the electron lifetimes in

turn results from the Bose–Einstein occupation factor of

the phonons n= 1/[exp(~ω/kBT)−1], where ~is the

reduced Planck’s constant and kBis the Boltzmann con-

stant. Greater phonon occupation results in decreased

electron lifetimes. For example, Fig. 4b reveals the life-

times of the low energy electronic states (i.e., pre-phonon

emission scattering, at energies less than ωpo from the

conduction band minimum, Fig. 4a) are inversely pro-

portional to the Bose–Einstein phonon occupation. At

low temperatures, the phonons will not be suﬃciently

excited and their population will increase exponentially

with temperature, thereby rapidly decreasing the elec-

tron lifetimes (see blue region in Fig. 4c) and hence the

electron mobility. At higher temperatures, phonon oc-

cupation increases roughly linearly with temperature re-

sulting in more gradual decay of electron lifetimes (see

orange region in Fig. 4c). The temperature at which the

rate of occupation transitions from exponential to linear

increase is determined by the phonon frequency. A low

frequency (~ω < kBT) means the phonon occupation in-

creases more linearly and hence the mobility will show

a weaker temperature dependence (see teal line [4 THz]

in Fig. 4c). A higher frequency (~ω > kBT) means

the phonon occupation increases more exponentially with

temperature (see purple line [20 THz] in Fig. 4c). Ac-

cordingly, for a ﬁxed temperature range, the tempera-

ture dependence of mobility is less negative in systems

with smaller phonon frequencies, exactly as revealed in

Fig. 3a.

FIG. 5. The electronic band structure can control the temperature dependence of mobility. Eﬀect of Kane non-parabolicity

parameter (α) on athe electronic band structure and bthe temperature (T) dependence of mobility calculated using polar

optical phonon scattering (PO) and acoustic deformation potential scattering (AD). Eﬀect of a heavy anisotropic eﬀective mass

(m∗

h) on the celectronic band structure and dtemperature dependence of mobility. eFermi surface of the anisotropic eﬀective

mass where m∗

h=7meat 0.4 eV above the valence band edge. Fermi surface visualized using the ifermi package [52].

The weakening in the temperature dependence of mo-

bility at higher temperatures is also explained by the

more linear change of phonon occupation in the high tem-

perature regime. There is an additional eﬀect, arising

from the ratio of pre- and post-emission onset lifetimes,

that reinforces the impact of phonon occupations. This

is discussed further in Section S4 of the Supplementary

Material.

At higher doping concentrations, the temperature de-

pendence of mobility is also weakened (made less nega-

tive). This is because increased doping activates higher

energy electronic states (after the emission scattering on-

set, see pink lines [12 THz] in Figs. 4e for low doping and

4f for high doping) whose lifetimes have much weaker

temperature dependence. This can be seen in the blue

region of Fig. 4d, where the lifetimes of the post-emission

electronic states show dramatically reduced temperature

dependence (between τ∝T−0.98–T−0.81) compared to

the lower energy pre-emission states (Fig. 4c, τ∝T−1.27–

T−2.71).

Although thus far we have restricted our analysis to

polar optical phonon scattering, the same factors will

also determine the temperature dependence of systems

limited by optical deformation potential scattering, al-

beit with some caveats. The major complicating fac-

tor is that in polar materials, scattering occurs only by

polar longitudinal-optical modes near the zone center.

Accordingly, often only a few phonon frequencies con-

trol the entire scattering, and even just one frequency

in the simplest of polar systems. However, in optical

deformation potential scattering, both longitudinal and

transverse modes across the full Brillouin zone can scat-

ter carriers, leading to a wide spectrum of phonon fre-

quencies that contribute to scattering. Regardless, the

overall trends discussed above are expected to hold for

any systems dominated by electron-phonon interactions.

With this in mind, we return to the remaining sys-

tems previously highlighted, n-SrTiO3and p-Si. In each

case, the average phonon frequency (polar frequency for

SrTiO3) determines the temperature dependence of mo-

bility. In SrTiO3, the mobility is limited by a com-

bination of polar optical and soft-ferroelectric phonons

and polaronic eﬀects [43]. The mobility dependence of

µ∝T−3.1is one of the most negative in our dataset,

and is dominated by two optical phonons, with frequen-

cies 13.3 THz and 23.0 THz [43]. Using the arithmetic

mean of these phonon frequencies (18.2 THz) leads to a

mobility dependence around µ∝T−3as demonstrated

in Fig. 3a, very close to the experimental value. The mo-

bility of p-type Si exhibits a temperature dependence of

µ∝T−2.4and is dominated by optical deformation scat-

tering from phonons with frequencies between 12 THz–

15 THz [53]. Using the average of the optical mode fre-

quencies (13.5 THz) gives rise to an expected mobility

dependence of µ∝T−2.5, as illustrated in Fig. 3a. This

is in excellent agreement with the experimental trend

even though Si is not limited by polar optical phonon

scattering, but rather deformation potential scattering.

Accordingly, the temperature dependence of mobility in

GaAs, Si, and SrTiO3is controlled almost entirely by the

phonon frequencies irrespective of the dominant scatter-

ing mechanisms.

It is important to stress that the temperature depen-

dence of mobility also depends on features in the elec-

tronic band structure. For example, although GaAs pos-

sesses a single isotropic band, it is not perfectly parabolic

away from the band edge. By increasing the non-linearity

in a single isotropic band quantiﬁed by the Kane parame-

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Thetemperature-dependenceofcarriermobilityisnotareliableindicatorofthedominantscatteringmechanismAlexM.Ganose,1,2,JunsooPark,1andAnubhavJain1,y1EnergyTechnologiesArea,LawrenceBerkeleyNationalLaboratory,Berkeley,California94720,USA2DepartmentofChemistry,MolecularSciencesResearchHub,WhiteCityCampus,I...

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The temperature-dependence of carrier mobility is not a reliable indicator of the dominant scattering mechanism Alex M. Ganose1 2Junsoo Park1and Anubhav Jain1

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