Surface polar optical phonon scattering of carriers in graphene on
various substrates
I-Tan Lin,1 Jia-Ming Liu,1,*
1
Electrical Engineering Department, University of California, Los Angeles, Los Angeles, CA
90095, U.S.A.
Online Supporting Information
1. Scattering rate equations
2. Surface optical phonon scattering and approximations
3. Elastic scattering models
1. Scattering rate equation
To derive the scattering rate arising from the interaction between electrons and surface optical
phonons, we start from the Boltzmann equation in the steady state for a homogeneous system:1
1
eE 
f 0
   S (k , k ) f (k ) 1  f (k )   S (k , k ) f (k ) 1  f (k )   ,
k k 
(S1)
where ħ is the reduced Planck’s constant ( h / 2 ), e is the electronic charge, E is an applied
homogeneous electric field, f is the carrier distribution function, k and k  are respectively the 2D
wave vectors of carriers before and after the collision, and S (k , k ) is the transition rate from
state k to state k  . Equation (S1) describes the steady state achieved by the balance of the inscattering rate and the out-scattering rate of states k due to the externally applied electric field
and scattering collisions. If we further assume that the electric field is sufficiently weak, we can
write
f  k   f0 
f
eEvF
cos  0   E  ,
kBT
E
(S2)
where k B is the Boltzmann constant, T is the temperature, vF (  106 m/s ) is the Fermi velocity
of graphene, φ is the angle between k and E , f0 is the equilibrium carrier distribution function
(Fermi-Dirac function), and  ( E ) is the energy-dependent scattering time constant with  1 ( E )
being the energy-dependent transport scattering rate we intend to obtain. The carrier energy E is
given by E   kvF , where the positive and negative signs are for carrier energy on the
conduction band and on the valence band, respectively. Using Eq. (S2) and the principle of
detailed balance, Eq. (S1) can be rewritten as1
A
4 2
2
 d k
1  f0 ( E)
 (E)  s  cos  E S k, k  1,
1  f0  E 
(S3)
where A is the area of graphene under consideration, and s  1 for intraband scattering and
s  1 for interband scattering.2,3 In Eq. (S3), the identity f0 / E  f0 (k )  f0 (k ) 1 / kBT has
been used, and cosθ comes from the term cos   / cos   cos   tan  sin  , where   is the
angle between k  and E , and θ is the angle between k  and k . The integration of Eq. (S3) for
the second term  tan  sin   is zero because S  k, k is an even function of θ. As can be seen
from Eq. (S3), to obtain the scattering rate  ( E )1 of the carrier that has energy E, one has to
know a priori the scattering rate  ( E )1 of the final state of the scattering event. Different
techniques, such as the Monte Carlo4 and the iteration methods,1,5 can be used to solve Eq. (3). In
the letter, we adopt the iteration method for its simplicity and efficiency in terms of the
numerical running time. Another popular way to approximate the solution of Eq. (S3) in the
literature is to assume the elastic limit E  E .6,7 In this way, Eq. (S3) can be much simplified:
 1 ( E ) 
A
4
 d k  1  s  cos   S  k , k  .
2
2
(S4)
This equation is the quasiparticle scattering rate with an extra angular term (1  s  cos  ) in the
integrand.3
2. Surface optical phonon scattering and approximations
To obtain the iterative formula of Eq. (3) in the letter, we plug Eq. (1) into Eq. (S3):
e2
4
d
v,
2
k
1  f 0 ( E )
 ( E )  s  cos   E   1  s  cos   
1  f0  E 
 e 2 qd
2 
1  qs / q   q
Fv2 N v

   E  E

(S5)
v   1 ,
where f 0  (1  exp[( E   ) / kBT ]) 1 . For each mode v, there are two scattering processes to
consider. These two processes are included in the summation of Eq. (S5): one is scattering by
phonon emission (plus sign) and another one is scattering by phonon absorption (minus sign). By
expanding the summation into absorption and emission terms, we can obtain Eq. (3) in the letter
with
s  e2
Sa ( E ) 
4
1  s  cos   e
1  f (E   )
v 1 0 f  E  v Fv2 Nv  d 2k  q 1  q / q 2
 s 
0
cos    E  E   v  ,
(S6)
s  e2
Se ( E ) 
4
1  s  cos   e
1  f (E   )
v 1 0 f  E  v Fv2 Nv  d 2 k  q 1  q / q 2
 s 
0
cos    E  E   v  ,
(S7)
2 qd
2 qd
So ( E ) 
2 qd
1  s  cos    E  E 
1  f 0 ( E v ) 2  2 e

F
N
d
k


v
v 
2
1  f0  E 
q 1  qs / q 
v,
e2
4
v  .
(S8)
As can be seen from Eq. (3) in the letter, the effective scattering rate  1  E  at carrier energy E
is proportional to So(E) but decreases with increasing in-scattering contributions Sa(E) and Se(E).
The concept is illustrated in Fig. S1. The integrals in Eqs. (S6) to (S8) can be simplified; for
example, the integral in Eq. (S6) can be reduced to
2
 d k
1  s  cos   e2qd cos   E  E

2
q 1  qs / q 
v  
 2
s  v  E
e2 qd
1
d

cos

1

s

cos

,




2
vF 2  v
q
1

q
/
q


0
s
where q 
(S9)
v 2  2E   v  E  s  cos   1 / vF . Note that For hole scattering on the
2
valence band, the scattering rate can still be calculated using Eq. (S5) by changing the chemical
potential μ to   without the sign change for the carrier energy, i.e.    E      E    .
Figure S1. Illustration of in-scattering (red arrows) and out-scattering (blue arrows) processes,
and the relation among the wave vector before scattering k (dash arrow), after scattering k  (solid
arrows), and scattering phonon wave vector q (dotted arrows). Two pairs of wave vectors k  and
q are drawn for phonon absorption and emission scattering processes, respectively. Sa, Se and So
link the scattering rates/relaxation times of different carrier energies.
To relate the theoretical resistivity to the analytical fitting equation for the resistivity in the
literature,8,9 further approximation is needed for the scattering rate given in Eq. (S5). For a small
surface optical phonon energy and a high Fermi energy, i.e., EF
v , E  approximately
equals to E  v (elastic-limit approximation); the Eq. (S5) becomes
1  f 0 ( E v ) E
 (E) 
F N v
1
2 2 
4 vF v ,
1  f0  E 
v
1
e2
2
v

v
2

0
1  cos   e
d
2
2 qd
q 1  qs / q 
2
.
(S10)
This equation is identical to Eq. (6) in the letter after the integration over k  . For the SiO2
substrate, this approximation is good for the low-energy surface optical phonon mode 1 , as
studied in our previous work.10 For the high-energy surface optical phonon mode 2 , especially
when EF  2 this approximation underestimates the scattering rate contributed by
2 .
Nevertheless, the overall scattering rate  1 from the approximation (S10) is still valid above
EF  100 meV , where the maximum error of 11% occurs.10 Equation (S10) can be further
simplified by the fact that the scattering rate is mostly contributed by 90° scattering due to the
v 2  2E   v  E  / vF ;
factor 1  cos 2 ( ) . Therefore, we can set   90 , and thus q 
2
the remaining integral in (S10) is easy to evaluate, and (S10) becomes
 1 ( E ) 
Consider the case kBT
e2
4 2 vF 2
 Fv2 Nvv
v,
1  f 0 ( E v ) E
1  f0  E 
v
1
e 2 qd
q 1  qs / q 
2
.
(S11)
  EF . The resistivity ρ can be obtained from Eq. (5) in the letter with
 1 ( EF ) given by Eq. (S11):


e2
2
 1  EF 
EF


2 EFvF 2
 F   N 1  f ( E
2
v
v
v


v
0
F
v ) 
EF
v
As we have assumed low temperatures, we can further assume that
Nv 1  f0 ( EF
1
e 2 qd
q 1  qs / q 
v
2
. (S12)
kBT ; therefore,
v )  Nv . In the following, we consider the first case where a small d is
assumed such that exp(2qd )  1 for EF of a few hundred meV. Due to the fact that the
resistivity converges at EF   (Fig. 3), we can transform (S12) into a Taylor series at EF  
. The first-order term is


vF
 2
2a

2
1
EF
F
2
v
N v ,
(S13)
v
where a is a dimensionless constant given by e2 /  avg vF . The linear dependence of the
conductivity on EF in (S13) is shown in Fig. S2(a) for d  3.4 Å. When the spacing d is
increased from 3.4 Å to 12 Å, the dependence of the conductivity on EF gradually changes from
linear
dependence
to
square
dependence.
For
d  12
exp(2qd )  (1  2qd  4q 2 d 2  ...) 1  (2qd ) 1 (1/ 2qd  1  2qd  ...) 1 . Assuming EF
Å,
v , the
latter summation is fairly constant and is about 2.5 for EF of a few hundred meV. Therefore, the
first-order term of the Taylor series at EF   for (S12) can be shown as

5d


2a

2
1
EF2
 F N
2

v
.
(S14)
v
The dependence of the conductivity on EF2 in (S14) is shown in Fig. S2(b) for d  12 Å. For an
even larger d above 12 Å, more terms are needed to approximate exp(2qd ) ; the Fermi-energy
dependence of the resistivity will be of the form of EF  (  2) for EF much larger than the
surface optical phonon energies. By using the identity EF  vF kF  vF  (7.2 1014 )V , where
V is the gate voltage in unit of volt, Eq. (S14) can be related to the gate voltage as
1
 C
V


e
1
1 / kBT
F22 / F12 

,
 1 e 2 / kBT  1
(S15)
where C is given by 1.84 109 hF12 / de2 ( 2  a)2 , which is 0.046h / e 2 for graphene on SiO2
with the graphene-substrate distance d  12 Å, and F2 2 / F12  6.84 .
Figure S2. Theoretical resistivities and the corresponding conductivities calculated with Eq. (5)
using the scattering rate obtained from Eq. (3) (iterative method, blue solid curves) or from Eq.
(6) (elastic-limit approximation, blue dotted curves). The black curves in (a) are calculated with
Eq. (S13), and that in (b) is calculated with Eq. (S14). Note that the resistivities of blue solid and
dotted curves in (a) are also plotted in Fig. 2(d) in the letter.
3. Elastic scattering
In the letter, the only elastic phonon scattering considered is the longitudinal acoustic phonon
scattering as the coupling with transverse acoustic phonons is weak.11 The acoustic phonon
scattering rate is given by11
 ac1  E  
1 kBT DA2
E,
3
2
4vF2  vph
(S16)
where   7.6 107 kg/m2 is the density of graphene per layer; vph  2 104 m/s is the phonon
velocity of the longitudinal acoustic mode;12 DA is the deformation potential, which has a value
ranging from 10 eV to 30 eV11,13-16 possibly due to the difficulty in separating the contribution to
the experimental data by the acoustic phonon scattering from that by the surface optical phonon
scattering. In the letter, we choose DA  18 eV as it is the most frequently obtained value in the
experiments reported in the literature.8,9,11,14 Another elastic scattering process we consider is the
charged impurity scattering, which has a rate given by17
1
 imp
E 
ni
4
2
E

vF 
2
V2
 d
1  qs / q 
0
1  cos   ,
2
2
(S17)
where V  e2 / 2 avg q is the Fourier transform of the 2D potential energy with q  2k sin  / 2 ,
and ni is the impurity density. In past experiments, the researchers also observe “mysterious”
weak scatterers whose physical origin is still debatable. These weak scatterers can be regarded as
delta potentials, contributing a resistivity ρs of 40~100 Ω
14,18-20
that is quite independent of the
carrier density and the temperature. The scattering rate arising from weak scatterers is given by21
 s1  E  
1
2
3
4
5
6
7
8
9
10
e2

2
s E
(S18)
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