Significant Enhancement of HoleMobility in [110] Silicon NanowiresCompared to Electrons and Bulk Silicon
A. K. Buin,*
,†
A. Verma,
‡
A. Svizhenko,
§
and M. P. Anantram
†
Nanotechnology Program, Uni
V
ersity of Waterloo, 200 Uni
V
ersity A
V
enue West,Ontario N2T 1P8, Canada, Department of Electrical Engineering, Texas A&M Uni
V
ersity

Kings
V
ille, Texas 78363, and Sil
V
aco Data Systems Inc.,4701 Patrick Henry Dri
V
e, Santa Clara, California 95054
Received October 22, 2007; Revised Manuscript Received December 19, 2007
ABSTRACT
Utilizing sp3d5s* tightbinding band structure and wave functions for electrons and holes we show that acoustic phonon limited hole mobilityin [110] grown silicon nanowires (SiNWs) is greater than electron mobility. The room temperature acoustically limited hole mobility for theSiNWs considered can be as high as 2500 cm
2
/V s, which is nearly three times larger than the bulk acoustically limited silicon hole mobility.It is also shown that the electron and hole mobility for [110] grown SiNWs exceed those of similar diameter [100] SiNWs, with nearly 2 ordersof magnitude difference for hole mobility. Since small diameter SiNWs have been seen to grow primarily along the [110] direction, resultsstrongly suggest that these SiNWs may be useful in future electronics. Our results are also relevant to recent experiments measuring SiNWmobility.
With the semiconductor industry fabricating devices withfeature sizes in the tens of nanometers, there is a potentialfor silicon nanowire (SiNW) devices to play an importantrole in future electronics, sensors, and photovoltaic applications. Since the band structure of SiNWs varies vastly withtheir physical structures,
1

4
it provides a tantalizing possibility of utilizing different SiNWs within the same application to achieve optimum performance. For example, adifference in the physical structure of the SiNWs results ina difference in sensing properties.
5
This implies that, in orderto bring the promise of SiNWs to fruition, it is necessary tocharacterize a vast number of SiNWs with different crosssectional shapes and sizes, and axis orientations, for theirelectronic properties. In particular, one of the most importantproperties is the lowfield mobility. An accurate knowledgeof the lowfield mobility is important in order to select theright SiNW for a particular application. The lowfieldmobility is strongly influenced by acoustic phonons,
6
surfaceroughness scattering,
7
and impurity scattering.
8
Surfaceroughness and impurity scattering are to a large extentcontrollable parameters. Phonon scattering on the other handis intrinsic and, hence, places an upper bound on the expectedcarrier mobility. Recent results suggest that electron mobilityin SiNWs can be significantly increased by coating themwith acoustically hardened materials to clamp the boundary,
8
or by inducing a strain.
9
This in turn also makes a strongcase for investigating the performance limitations of freestanding SiNWs.Utilizing various boundary conditions, many researchershave computed the electron

phonon scattering rates innanowires based on an effective mass equation.
10

16
Reference 17 is an exception that considered a tightbindingHamiltonian for the electron, albeit with bulklike confinedphonons, where bulk quantized transverse phonon vectorsemulate confined phonon behavior. Further, the analysis hasbeen limited to consideration of the lowest conductionsubband. It is without doubt that these methods make it easierto compute scattering rates. However, they come with apossibility of a compromise in accuracy, in particular, forholes where the valence subbands are closely spaced togetherand intersubband scattering may be significant. Accountingfor intersubband hole scattering is one reason it is challengingto theoretically investigate hole transport in these dimensionally reduced structures, a task all the more imperative becauseof the recently reported very high hole mobility
18
inexperiments.It is clear, therefore, that, in order to reproduce an accuratedescription of the physical effects taking place in theseconfined structures and evaluate charge transport properties,
* To whom correspondence should be addressed. Email: abuin@ecemail.uwaterloo.ca.
†
University of Waterloo.
‡
Texas A&M University

Kingsville.
§
Silvaco Data Systems Inc.
NANOLETTERS
2008Vol. 8, No. 2760

765
10.1021/nl0727314 CCC: $40.75 © 2008 American Chemical Society
Published on Web 01/19/2008
it is important to take into account both charge and phononconfinement. Concomitantly, it is also important to investigate how these properties compare to bulkphononvalues with a more detailed treatment of these threedimensional phonons. Bulk phonons not only provide an easeof modeling but also make it easier to incorporate intersubband carrier scattering, which, as our results show, is veryimportant even in many small diameter SiNWs. Moreimportantly, consideration of bulk phonons address thescenario of a SiNW encapsulated within an acousticallysimilar material.Here we present results on a detailed computation of electron and hole lowfield mobility for [110] axially orientedfreestanding SiNWs (henceforth referred to as [110] SiNWsin this paper) with diameters up to 3.1 nm and at varioustemperatures, where the principal charge scattering mechanism is through acoustic phonons. Both confined and bulk phonons are considered. The band structure for these SiNWsis determined by using an
sp
3
d
5
s*
tightbinding (TB) scheme,and the confined acoustic phonon dispersion for each SiNWare obtained by solving the elastic continuum wave equation.Bulk phonon dispersion is assumed to be linear, and a Debyecutoff energy is used to define the domain of bulk phononwave vectors. Electron and holeacoustic phonon momentumrelaxation rates are calculated from the firstorder perturbation theory and deformation potential scattering. In computing the charge momentum relaxation rates for a SiNW fromconfined phonons, all acoustic phonon modes up to 70meV
19

21
are taken into account, and TB electron and holewavefunctions are incorporated. Finally, lowfield mobilityvalues are determined through momentum relaxation timeapproximation and verified for electronconfined phononinteraction through ensemble Monte Carlo (EMC) simulations.Figure 1 shows a typical wire cross section for the [110]SiNW. The periodicity of the SiNW along the lattice is givenby the lattice constant
a
/
2, where
a
is the lattice constantfor bulk Si. To eliminate unphysical surface states, H atomsare used to passivate the Si dangling bonds at the SiNWedge. Within the
sp
3
d
5
s*
TB scheme, the Si

Si, and Si

Hparameters are obtained from refs 22 and 23, respectively.Figure 2 shows the first few conduction and valencesubbands for a 2.4 nm diameter [110] SiNW. For the largestdiameter SiNW considered in our work, 3.1 nm, the energyseparation between the lowest two conduction subbands is18 meV, while it is 17 meV for the highest two valencesubbands. As is expected, as the SiNW diameter increases,the band gap and intersubband spacing decrease. However,for [110] SiNWs this does not occur in the simple mannergiven by an effective mass Hamiltonian.The electron and hole wavefunctions within the TB schemefor the [110] SiNWs are given bywhere
ν
is the subband index,
k
z
is the electron or hole wavevector along the
z
axis,
r
is the radius vector,
N
is the numberof unit cells, and
τ
m
is the basis vectors within the unit cell
n
.
φ
m
are orthonormal Slatertype (Lo¨wdin)
24
atomic orbitals,
c
ν
,
m
are expansion coefficients whose values are obtainedwithin the TB scheme,
25
and
e
z
is the unit vector along theSiNW axis.Different types of confined phonon modes such asdilatational, torsional, and flexural can exist within a SiNW.
26
Due to symmetry considerations, only dilatational phononmodes may contribute to intrasubband electron

phononscattering.
27,28
Dilatational phonon modes correspond to amixed nature of axial

radial atomic vibrations. Dispersionrelationship for a coupled axial

radial dilatational mode isobtained from the elastic wave equation and given by thePochhammer

Chree equation,
26
where,
r
is the radius of the phonon box and
J
0
and
J
1
areBessel functions of the first kind. Also,
k
t
,
l
2
)
ω
2
/
V
t,l
2

q
2
,where
V
represents the bulk acoustic velocity with subscripts
t
and
l
standing for transverse and longitudinal, respectively.Equation 2 is solved numerically to obtain
ω
(
q
) by consider
Figure 1.
Cross section of
d
)
1.16 nm nanowire. Grey balls, Siatoms; white balls, H atoms.
Figure 2.
Band structure of 2.4 nm dimater [110] SiNW.
ψ
ν
(
k
z
,
r
)
)
1
N
∑
n
,
m
c
ν
,
m
(
k
z
)
e
ik
z
na
/
(2)
φ
m
(
r

(
τ
m
(
e
z
na
2
))
(1)2
k
l
r
(
q
2
+
k
t
2
)
J
1
(
k
l
r
)
J
1
(
k
t
r
)

(
k
t
2

q
2
)
2
J
0
(
k
l
r
)
J
1
(
k
t
r
)

4
q
2
k
l
k
t
J
1
(
k
l
r
)
J
0
(
k
t
r
)
)
0 (2)
Nano Lett.,
Vol. 8, No. 2,
2008 761
ing the SiNW embedded within an equivalent circularphonon box with radius
r.
Figure 3 depicts the confinedacoustic phonon dispersion for the 2.4 nm diameter [110]SiNW with the band structure given in Figure 2The electron and hole

acoustic phonon scattering ratesare calculated from the Fermi’s golden rule and deformationpotential approximation
29
aswhere
k
z
and
k
′
z
are the initial and final crystal momentumrespectively,
N
q
is the phonon equilibrium Bose

Einsteinoccupation number,
µ
and
ν
are initial and final subbands,respectively,
δ
is the Delta Dirac energy conserving function,and
q
is the phonon wave vector.
H
q
,
e

ph
is the electron

phonon interaction Hamiltonian. One therefore obtains themomentum relaxation rates from confined acoustic phononsaswhere
µ
,
ν
indicate initial and final subbands, respectively.
JDOS
ν
(
q
,
k
,
q
p
)
)

∂
(
E
ν
(
k
z
(
q
p
)
(
p
ω
n
(
q
p
))/
∂
q


1
is theelectron or hole

phonon joint density of states,
n
indicatesa summation over phonon modes,
p
indicates summation overthe roots of the equation
∆
E
µ
,
ν
(
k
z
(
q
)
(
p
ω
n
(
q
)
)
0 (where
∆
E
µ
,
ν
)
E
ν

E
µ
), and

corresponds to absorption/emission.
E
a
is the deformation potential,
F
is the SiNW mass density,
γ
is the normalization constant,
30
and
k
l
2
)
ω
2
/v
l
2

q
2
.
S
µ
,
ν
,the overlap factor between subband
µ
and subband
ν
, is givenbywhere
c
µ
,
m
and
c
ν
,
m
/
are defined through eq 1.In computing the momentum relaxation rates betweenelectrons/holes and bulk acoustic phonons, the phonondispersion is taken to be linear within the Debye approximation. The phonon dispersion is
ω
(

q

))
) V
s

q

, where thelongitudinal sound velocity
V
s
)
9.01
×
10
3
m/s.
31
With theaid of the energy conserving Dirac Delta function and thediscrete momentum conserving Delta function along theSiNW axis, one obtains the charge momentum relaxationrates through bulk phonons, from initial subband
µ
to finalsubband
ν
, asIn the above equation,
q
n
)
((
∆
E
µ
,
ν
(
k
z
(
q
))/(
p
v
s
))
2

q
2
,and
S
1,
µ
,
ν
)
1/(2
π
)
∫
02
π

S
µ
,
ν
(
q
)

2
d
φ
is the overlap factorintegrated over the phonon angular part in the confinementplane. The bulk Debye energy
E
D
)
55 meV
32
is utilized todefine the domain of integration,
Ω
. In general,
Ω
)
min,max(1
st
BZ,
q
n
2
+
q
2
e
q
D
), where
q
D
is the Debye wavevector. In obtaining the above momentum relaxation ratesprescription, we have neglected Umklapp processes.
11
Furthersimplifications allow us to obtainwhere
A
m
,
m
′
)
c
µ
,
m
(
k
z
)
c
ν
,
m
/
(
k
′
z
)
c
µ
,
m
′
/
(
k
z
)
c
ν
,
m
′
(
k
′
z
), and
∆
j
m
,
m
′
)
(
j
m

j
m
′
), and where
j
is
x
,
y
, or
z
.In evaluating the momentum relaxation rates using theabove equations, we use a value of 9.5 eV for the electrondeformation potential and 5 eV for the hole deformationpotential.
29
Electron and hole lowfield mobility values areestimated based on relaxation time approximation withmomentum relaxation time approximation (MRTA) andgiven by
Figure 3.
Dispersion curve for first 11 “dilatational” phonon modesfor a 2.4 nm diameter [110] SiNW.
W
ν
,
m
(
k
z
,
k
′
z
,
q
)
)
2
π
p

〈
ψ
ν
(
k
′
z
),
N

q

+
12
(
12

H
q
,
e

ph

ψ
µ
(
k
z
),
N

q

〉

2
δ
(
E
ν
(
k
′
z
)

E
µ
(
k
z
)
(
p
ω
(

q

)) (3)
W
µ
,
ν
(
k
z
)
)
∑
k
′
z
W
µ
,
ν
(
k
′
z
,
k
z
)
(
1

k
′
z
k
z
)
)
E
a
2
4
π
p
3
FV
s4
∑
n
∑
p
q
p
k
z
γ

1
[
N
(
E
ph
(
q
p
))
+
12
(
12
]

S
µ
,
ν
(
q
p
)

2
E
ph
3
(
q
p
)
JDOS
ν
(
k
z
,
q
p
) (4)
S
µ
,
ν
)
∑
m
c
µ
,
m
(
k
z
)
c
ν
,
m
/
(
k
′
z
)
J
0
(
k
l
F
m
)
e
iqz
m
(5)
W
µ
,
ν
(
k
z
)
)
∑
k
′
z
W
µ
,
ν
(
k
′
z
,
k
z
)
(
1

k
′
z
k
z
)
)
E
a
2
4
π
p
3
FV
s4
∫
Ω
∆
E
µ
,
ν
(
k
z
(
q
)
2
S
1,
µ
,
ν
(
q
n
,k
z
(
q
)
(
N

q

+
12
(
12
)
qk
z
d
q
(6)
S
1,
µ
,
ν
)
∑
m

c
µ
,
m
(
k
z
)

2

c
ν
,
m
(
k
′
z
)

2
+
∑
m
,
m
′
,
m
*
m
′
A
m
,
m
′
J
0
(
q
t
∆
x
m
,
m
′
2
+
∆
y
m
,
m
′
2
)
e

iq
∆
z
m,m
′
(7)
µ
)
1
∑
i
n
i
∑
i
µ
i
n
i
(8)
762
Nano Lett.,
Vol. 8, No. 2,
2008
where
n
i
is the charge carrier population in subband
i
, andfor each subband
i
, where
W
i
(
k
z
)
)
∑
ν
W
i
,
ν
(
k
z
) and
f
0
(
k
z
)
≈
e

E
i
/
k
B
T
.
E
i
is the electron/hole energy in subband
i
, measuredwith respect to the lowest/highest conduction/valence subband. Additionally, we verify lowfield mobility valuesobtained using eq 9 for electronconfined phonons scatteringby solving the Boltzmann Transport Equation using ensembleMonte Carlo (EMC) simulations.
29
EMC simulations utilizetabulated values of the electron band structure and scatteringrates computed using results above. In performing EMCsimulations, we consider SiNWs to be infinitely long anddefectfree. The temperature and electric field are assumeduniform. Given the nature of the phonons considered, onlyintrasubband scattering, restricted to the lowest conductionsubband, is considered, and electron energies are restrictedto the bottom of the next higher subband. The bandstructurewithin that small region is divided into 12 000 grid pointsto minimize statistical noise at low electric fields.The electron lowfield acoustic confined phonon limitedmobility computed using eqs 4 and 8 are shown in Figure 4.At room temperature, we find the mobility to scale approximately with diameter (
d
) and effective mass (
m
eff
) asAt low temperatures, we find that the scaling of mobilitywith effective mass remains the same, while the scaling withdiameter is significantly weaker than
d
2
.Bulk phonon limited electron lowfield mobility is alsoshown in Figure 4. We find that bulk phonons yield anelectron mobility that is slightly larger than that of confinedphonons. For the smallest diameter (1.27 nm) SiNWconsidered, the confined phonon mobility is about 38% lowerthan the bulk phonon mobility, at room temperature. Moresignificantly, this difference decreases to only 25% for theSiNW with a diameter of 3.1 nm. The diminished importanceof phonon confinement at larger diameters is intuitivelyunderstood by noting that the phonon confinement isrelatively weaker at 3.1 nm. As a result, confined phononsat 3.1 nm more closely resemble bulk phonons and thereforeyield more comparable momentum relaxation rates whencompared to the 1.27 nm diameter SiNW. We note that thebulk phonon scaling of mobility is qualitatively similar tothat of confined phonons. The scaling of mobility with bulk phonons is consistent with the analytical results for scatteringrates in reference 10. We remark that the electron mobilityfor the [110] SiNW is about four times larger than themobility of [100] SiNW considered in ref 17. We alsocompare our electron mobility to ref 8, which found amobility of around 336 cm
2
/V s for a 4 nm diameternanowire. This value is about 66% smaller than our mobilityfor the 3.1 nm diameter [110] SiNW due to smaller effectivemass shown in Figure 5. However, when scaled for thedifferent diameters, effective masses, deformation potentialand sound velocity, our results agree to within 15%.The situation is somewhat different and more interestingfor holes. Figure 6 depicts the hole bulk acoustic phononlimited lowfield mobility versus diameter for [110] SiNWs,at temperatures ranging from 77 to 300 K. As with the casefor electrons, hole mobility reduces as temperature increasesbecause of an increase in momentum relaxation rates.Importantly, however, two things stand out. First, irrespectiveof the temperature, we find hole mobility to be significantlyhigher than electron mobility for a given SiNW. This is incontrast to bulkSi where electron mobility is higher. Thiscan also be seen in Figure 7, which shows the variation of electron and hole mobility with temperature for an approximately 1.93 nm diameter [110] SiNW. Second, for thelarger diameter SiNWs, we also find that hole mobility ishigher than the bulkSi acoustic phonon limited hole mobilityvalue by nearly a multiplicative factor of three
33
at 300 K.These observed values of mobility are in line with experimentally observed results,
18
where a peak hole mobility as
Figure 4.
Electron mobility (with bulk and confined phonons)versus [110] SiNW diameter for temperatures ranging from 77 to300 K.
µ
i
)
2
ek
B
Tm
eff
∫
1
st
BZ
E
i
(
k
z
)
W
i
(
k
z
)

1
f
0
(
k
z
) d
k
z
∫
1
st
BZ
f
0
(
k
z
) d
k
z
(9)
µ
e
∝
d
2
m
eff

1.5
(10)
Figure 5.
Electron effective mass versus [110] SiNW diameter.“C1” is first conduction band.
Nano Lett.,
Vol. 8, No. 2,
2008 763
high as 5000 cm
2
/V s is observed and is attributed to reducedroughness scattering. Interestingly, by comparing results inFigures 4 and 6 with reported bulk Si values at 77 K, onecan readily observe that the hole mobility value (8486 cm
2
/Vs) is 55% lower than the bulk value (11 481 cm
2
/V s)
33
forthe largest diameter SiNWs considered, while the electronbulkSi mobility value (23 000 cm
2
/V s)
34
is about eight timeslarger. Therefore, a reduction in temperature does not provideany mobility advantage over bulkSi for these SiNWs.Additionally at 300 K, hole mobility values in Figure 6are nearly 2 orders of magnitude greater than the reportedhole mobility for similar diameter [100] SiNWs.
17
This isprimarily attributed to very heavy hole effective masses
35
in[100] SiNWs in contrast to [110] SiNWs. Figure 8 depictsthe hole effective mass for the top three valence subbandsversus diameter for the [110] SiNW.Unlike electrons for the SiNWs under consideration,developing a “ruleofthumb” for the hole lowfield mobilityversus diameter is not straightforward. As can be seen inFigure 6, this is due to the behavior of hole mobility for the
∼
1.93 nm diameter SiNW. This behavior has two primaryreasons. For the 1.93 nm diameter [110] SiNW, the highesttwo valence subbands are separated by an energy of approximately 15 meV, resulting in relatively high intersubband scattering. However, for the larger diameter SiNWsconsidered, these two subbands become nearly degenerate(energy separation is
=
1 meV), thereby significantlyreducing intersubband scattering due to phonon emissionfrom the higher to the lower subband. Moreover, looking atFigure 8, we also readily observe that the hole effective massfor the second subband decreases from approximately 0.25for the 1.93 nm SiNW to 0.19 for the 2.4 nm SiNW. Thesefactors result in a relatively sharp increase in hole lowfieldmobility for the larger diameter SiNWs compared to the 1.93nm SiNW. Importantly, these results clearly demonstrate theimportance of hole intersubband scattering because of thesmall energy differences between valence subbands. Therefore, while we have also calculated the hole mobility valuesdue to the scattering arising from the dilatational modes of confined phonons, we do not report them because theygrossly exaggerate the values of hole mobility since thedilatational modes are not sufficient to describe intersubbandscattering.In conclusion, we have evaluated electron and hole lowfield mobility for [110] axially aligned SiNWs ranging indiameter from approximately 1 to 3.1 nm, utilizing bulk andconfined acoustic phonons for electrons, and bulk acousticphonons for holes, and including intersubband scattering.Hole acoustic phonon limited mobility for the 3.1 nmdiameter SiNW is found to be nearly 3 times the bulkSiacoustically limited mobility value at room temperature.Additionally, hole mobility for the SiNWs considered in ourwork is also found to be higher than electron mobility, whichis in contrast to bulkSi, where electron mobility is higher.Moreover, both electron and hole mobility for the [110]SiNWs are greater than the reported [100] SiNW mobilityvalues for similar diameters. Importantly, while electron andhole mobility values reported in this work help place an upperbound on the expected lowfield mobility, they also clearlyhave technological implications for SiNW electronics for twoimportant reasons. They point toward a preferential SiNWaxis, [110], which is all the more important because small
Figure 6.
Hole Mobility with bulk phonons versus versus [110]SiNW diameter for temperatures ranging from 77 to 300 K.
Figure 7.
Temperature dependence of mobility of 1.93 nm diameter[110] SiNW.
Figure 8.
Hole effective mass versus [110] SiNW diameter. “V1”is the topmost valence band, “V2” is the next to the topmost valenceband, and “V3” is the lowest among 3.
764
Nano Lett.,
Vol. 8, No. 2,
2008