Table Of ContentThe anisotropic ultrahigh hole mobility in strain-engineering
two-dimensional penta-SiC
2
Yuanfeng Xu1, Zeyu Ning1, Hao Zhang1,3,∗, Gang Ni1, Hezhu Shao2, Bo
Peng1, Xiangchao Zhang1, Xiaoying He1, Yongyuan Zhu3 and Heyuan Zhu1
1Department of Optical Science and Engineering and Key Laboratory
7
1
0 of Micro and Nano Photonic Structures (Ministry of Education),
2
Fudan University, Shanghai 200433, China
n
a
J 2Ningbo Institute of Materials Technology and Engineering,
3
Chinese Academy of Sciences, Ningbo 315201, China
1
] 3NanjingUniversity, NationalLaboratoryofSolidStateMicrostructure, Nanjing210093, China
i
c
s
- Using the first-principles calculations based on density functional theory, we systemati-
l
r
t callyinvestigatethestrain-engineering(tensileandcompressivestrain)electronic,mechan-
m
. ical and transport properties of monolayer penta-SiC . By applying an in-plane tensile
t 2
a
m
or compressive strain, it is easy to modulate the electronic band structure of monolayer
-
d penta-SiC , which subsequently changes the effective mass of carriers. Furthermore, the
n 2
o
obtained electronic properties are predicted to change from indirectly semiconducting to
c
[
metallic. Moreinterestingly,atroomtemperature,uniaxialstraincanenhancetheholemo-
1
v bility of penta-SiC along a particular direction by almost three order in magnitude, i.e.
2
5
1 from 2.59 ×103cm2/Vs to 1.14 ×106cm2/Vs (larger than the carrier mobility of graphene,
7
3
3.5 ×105cm2/Vs), with little influence on the electron mobility. The high carrier mobil-
0
.
1 ity of monolayer penta-SiC may lead to many potential applications in high-performance
2
0
7
electronicandoptoelectronicdevices.
1
:
v
i
X
r
a
2
I. INTRODUCTION
Two-dimensional (2D) materials have attracted great attention in the fields of nanoscale ma-
terials and nanotechnology since the experimental realization of graphene1–3, which possesses
remarkable mechanical, electronic and optical properties4–6 possibly leading to many novel appli-
cationsinmicroelectronicandoptoelectronicdevices. Alargenumberofgraphene-likestructures
and other layered structures with different chemical compositions and different crystal structures
havebeenproposedandsynthesisedsincethen. Recently,Zhangetal.7,predictedanew2Dmate-
rialcomposedofcarbonpentagons,i.e.penta-graphene,withmixed sp2/sp3 orbitalhybridization,
which is dynamically and mechanically stable, and can withstand temperatures as high as 1000
K. Monolayerpenta-graphenepossessesunusualnegativePoisson’sratio,ultrahighidealstrength
outperforming graphene and other interesting electronic properties7,8, which make it a potential
candidateforwideapplicationsintheoptoelectronicsandphotovoltaicsdevices.
Many efforts have been devoted on the discovery of new 2D materials with penta strcuture
since the successful prediction of penta-graphene. The silicon counterpart of the penta structure
was proposed as well, although it is dynamically instable in the monolayer form9. Recently, a
novel pentagonal structure composed of carbon and silicon atoms in a sp2d2 hybridization, i.e.
pentagonal silicon dicarbide (penta-SiC )10, is proposed and further confirmed to be dynamically
2
stableandexhibitsenhancedsimilarelectronicpropertycomparedtopenta-graphene.
Duringtheprocessesofexperimentalrealizationof2Dmaterials,mismatchbetween2Dmate-
rial and substrate often results in the formation of crystal distortion due to strain or stress modu-
lation. Experimentalmeasurementsofphysicalpropertiesof2Dmaterialscouldbedifferentfrom
those with perfect structures. Therefore it is necessary to investigate the strain-dependence prop-
erties of 2D materials comparing experimental results. In addition, strain-engineering (external
strain and stress) is an effective way to modulate the electronic, mechanical, optical properties
and transport of 2D materials11–14. It has been reported that strain engineering can be used to
modulatetheelectronicpropertiesofMoS 15,blackphosphorene16,TiS 17 andReS 13. Thestrain-
2 3 2
engineering method was proved to be effective in increasing the band gap of pengta-SiC 10 as
2
well. However, further investigation on the carrier mobility of monolayer penta-SiC under vari-
2
ousstrainhasnotbeenaddressed.
In this work, we systematically investigate electronic, mechanical and transport properties of
monolayer penta-SiC under tensile and compressive strain by using first-principles calculations.
2
3
We find that under in-plane uniaxial strain, penta-SiC remains an indirect semiconductor until
2
22%, however, the uniaxial compressive strain can easily converted it from semiconducting into
metallicbeyond−15%. Whenapplyingabiaxialstrain,penta-SiC canchangefromsemiconduc-
2
tor to metallic material under relatively low compressive strain of −8%. In general, the electronic
band structure of penta-SiC is sensitive to tensile and compressive strain. Moreover, we have
2
quantitatively demonstrated that the carrier mobility can be enhanced from 2.59 ×103cm2/Vs to
1.14×106cm2/Vsbyapplyingauniaxialcompressivestrain.
II. METHODANDCOMPUTATIONALDETAILS
The calculations are performed using the Vienna ab-initio simulation package (VASP) based
on density functional theory18. The exchange-correlation energy is described by the generalized
gradient approximation (GGA) using the Perdew-Burke-Ernzerhof (PBE) functional. The calcu-
lationa are carried out by using the projector-augmented-wave pseudo potential method with a
plane wave basis set with a kinetic energy cutoff of 500 eV. When optimizing atomic positions,
the energy convergence value between two consecutive steps is chosen as 10−5 eV and the max-
imum Hellmann-Feynman force acting on each atom is 10−3 eV/Å. The Monkhorst-Pack scheme
is performed for the Brillouin zone integration with k-point meshes of 13×13×1 and 21×21×1
for geometry optimization and self-consistent electronic structure calculations, respectively. To
verify the results of the PBE calculations, the electronic structure of penta-SiC with zero strain
2
is calculated using hybrid Heyd-Scuseria-Ernzerhof (HSE06) functional19. HSE06 improves the
precision of band gap by reducing the localization and delocalization errors of PBE and HF func-
tionals. The mixing ratio used in the HSE06 is 25% for short-range Hartree-Fock exchange. The
screeningparameteruissettobe0.4Å−1.
III. RESULTSANDDISCUSSION
A. Electronicbandstructureofpenta-SiC
2
The top and side views of the fully optimized structure of penta-SiC are shown in Fig. 1(a)
2
and (b), respectively. The unit cell (the red dashed rectangle) contains two silicon atoms and four
carbonatoms. Theoptimizedpenta-SiC hasatetragonalcrystalstructurewiththespacegroupof
2
p42 m. As shown in Fig. 1(a) and (b), all pentagons contain four equivalent Si-C bonds and one
1
4
(a) (c) S
ection d d1 Γ X
dir 2
b- n
o
b0 directi
bi-
a0
unit cell
(b) a-direction
h
FIG. 1. (a) Top view (b) side view of the atomic structure of penta-SiC monolayer (2×2 supercell). d
2 1
and d are the bond length of Si-C and C-C in this pentagonal structures, respectively. h is the buckling
2
distance. (c)isPBEcalculatedorbital-projectedelectronicbandstructuresofmonolayerpenta-SiC along
2
therepresentshigh-symmetrydirectionsofBrillouinzone(BZ)anddensityofstates(DOS)ofpenta-SiC .
2
C-C bond with three atomic layers (two C atom layers on the top and bottom, and Si atom layer
in the middle). As shown in Table. I, the associated lattice constant of penta-SiC is a=b=4.41 Å
2
with the buckling height h=1.33 Å. The bonding lengths are d =1.91 Å and d =1.36 Å, which
1 2
areingoodagreementwithpreviouslytheoreticalresults20. Toavoidartificialinteractionbetween
atomlayers,theseparationbetweenthelayersissettobe15Åformonolayerpenta-SiC .
2
The PBE calculations show that, SiC is a semiconductor with an indirect band gap of 1.39
2
eV, and the minimum of the conduction band (VBM) is located at S point, the maximum of the
valence band (VBM) is located between S and Γ points. The corresponding HSE06 calculation
gives a larger band gap of 2.85 eV comparing to the PBE result since PBE always underestimates
thevalueofbandgapofsemiconductors. Fig.1(c)showsthePBEcalculationsoforbital-projected
electronic band structures and density of states (DOS) of penta-SiC . Analysis on the PDOS (Si-
2
3s, 3p and C-2s, 2p orbitals) of penta-SiC reveals that C-2p and Si-3p orbitals mainly dominate
2
the electronic states near the Fermi level. However, the contributions by the C-2p states to the
total DOS is larger than that by Si-3p, and in the energy range of -2 to 0 eV for valence band, the
Si-3p states contribute larger than that by C-2p, indicating the strong hybridization between the
Si 3p states and C 2p states. The analysis on the chemical bonding of pengta-SiC based on the
2
DOS is in consistency with the calculated bond lengths, which show an identical length of Si-C
5
TABLE I. Calculated structure properties of 2D penta-SiC : lattice constant(a, b), buckling distance (h),
2
Si-CandC-Cbondlength(d ,d ). ThemagnitudeofbandgapcalculatedunderPBEandHSE06approxi-
1 2
mationrespectively.
structure latticeconstant[Å] h[Å] d [Å] d [Å] E (PBE)[eV] E (HSE)[eV] Young’smodulus(GPa)
1 2 g g
penta-SiC 4.41 1.33 1.91 1.36 1.39 2.85 292
2
(a) uniaxial strain
m)150
/
N
C ( 100
50 a-direction
b-direction
0
-10 -5 0 5 10 15
Strain x( )
(b) biaxial strain
150
)
m
/100
N
C ( 50
0
b-direction
-10 -5 0 5 10 15
Strain xy( )
FIG.2. CalculatedelasticmodulusC2Dalonga-andb-directionsundervariousmechanicalstrainforpenta-
ij
SiC .
2
bonding. The four identical bond lengths and bonding angles indicate that Si atom connects to
the four neighboring C atoms via sp2d hybridization. Similar analysis on C atoms shows that C
connectstothetwoneighboringSiatomsandoneCatomvia sp3 hybridization.
B. Strain-engineeringmechanicalpropertiesofpenta-SiC
2
Wehavestudiedtheeffectsofin-planeuniaxialandbiaxialstrainsonmechanical,electronand
transport properties of penta-SiC . Here, ε , ε and ε refer to the components of relative strain
2 x y xy
alonga−,b−andbi(a+b)-directionsrespectively. Sincepenta-SiC hasnearlyisotropicphysical
2
properties along a- and b-directions respectively, only the strain ε is considered in the following.
x
6
In addition, under the in-plane biaxial strain, the physical properties are nearly the same along
a- and b-directions, so only the physical properties along the b-direction are presented here. The
positive(negative)signsofstrainsrepresenttensile(compressive)strain,andthevalueofstrainis
evaluatedasthelatticestretching(condensing)percentage.
Due to three-dimensional (3D) periodic boundary conditions, the 2D elastic constant C2D
ij
should be rescaled by multiplying the c lattice parameter corresponding to the vacuum space be-
tween2Dlayers21,i.e.,C2D=cC3D. Thecalculatedin-planeelasticconstantC2D =C2D=140 N/m
ij ij 11 22
for penta-SiC , indicates that penta-SiC has nearly isotropic mechanical property. The positive
2 2
values ofC2D andC2D mean that penta-SiC is mechanically stable, according to the the mechan-
11 22 2
ical stability criteria. The 3D Young’s modulus Y can be expressed as Y=C2D/t, where t is the
ij
effective thickness of penta structure (4.8 Å for penta-SiC ), and the calculated value of Y is 292
2
GPaforSiC ,similartothatofMoS (270±100GPa)22,whilelargerthanthatofblackphospho-
2 2
rene (experimental values: 179 GP and 55 GP along a- and b-directions respectively)23, TiS (96
3
GP and 153 GP along a- and b-directions respectively)14 and HfS (137 GPa)14. The difference
2
canbeattributedtothebondlengthandbonddensityofdifferent2Dmaterials.
Furthermore, we have calculated the elastic modulus along the a- and b-directions under var-
ious mechanical strain, as shown in Fig. 2. By stretching (compressiving) the atom-atom bond
length, it is shown that a decreased (increased) of bond energy may reduce (enlarge) the value of
elastic constant. Fig. 2(a) shows the dependence of elastic constant along a- and b-directions un-
der uniaxial (ε ) strain. Under uniaxial strain, the elastic constant decreases monotonously in the
x
range of−10% ≤ ε ≤ 10% along a-direction, while along b-direction, it has a maximumvalue at
x
atensilestrainofε =1%anddecreaselinearlyfromε =1%to10%,andundercompressivestrain,
x x
theelasticconstantdecreaseslinearlyatahighrate. WhileforbiaxialstrainasshowninFig.2(b),
thevalueofelasticmodulusoftwodirectionsdecreaseswithsimilartendencyofb-directionunder
uniaxialstrainforpenta-SiC .
2
C. Strain-engineeringelectronicpropertiesofpenta-SiC
2
Fig. 3 shows the evolution of the valence and conduction band structures as a function of
different strains, mainly from −10% to 10% of the fully relaxed structure. Fig. 3 (a) and (b) show
the dependence of theenergy bands of monolayer penta-SiC on strain along a-and bi-directions,
2
respectively. The shift of the band edge as shown in Fig. 3 (a) and (b) can be understood in terms
7
of the bonding and antibonding states24. The VBMs (V and V ) and the CBMs (C andC ) are
s Γ s s−Γ
shown in Fig. 3 (a) and (b). The monolayer penta-SiC is an indirect semiconductor described by
2
VBM of V and CBM of C . However, by applying an uniaxial tensile strain (ε ), the energies
s s−Γ x
corresponding to the V and C change greatly, which transforms penta-SiC into a new type
s s−Γ 2
of indirect semiconductor formed by VBM of V and CBM of C . Under uniaxial compressive
Γ s
strain, C decreases while V increases, leading to a shrinking indirect band gap of the strained
s−Γ s
penta-SiC . While for an external strain along bi-direction (ε ), large changes take place for the
2 xy
energies corresponding to V , V , C and C as shown in Fig. 3 (b). When ε = −8%, the
s Γ s s−Γ xy
semiconductor(indirect)-to-metal transition takes place. The strain tunable electronic structure of
monolayer SiC will significantly influence the electronic transport properties, which can be used
2
intheelectronicandoptoelectronicapplications.
Fig. 3 (c) and (d) shows the evolution of calculated band gap under various strains. When
applying the in-plane uniaxial strain for −15% ≤ ε ≤ 22%, penta-SiC remains an indirect
x 2
semiconductor with a maximum band gap of 1.481 eV at 2%, while the gap energies and the
positions of the VBM and CBM change. However, when applying the in-plane biaxial strain for
−15% ≤ ε ≤ 15%, penta-SiC also remains the indirect semiconducting feature, and it converts
xy 2
intoametalbeyond15%and−9%. However,thebandgapreachesamaximumvalueof1.483eV
atthestrainof1%.
As can be seen in Fig. 3 (c) and (d), the electronic band gap is very sensitive to the application
ofcompressiveandtensilestrain. Thereasonmaybethatpenta-SiC exhibitsapartialinversionof
2
the vertical ordering of the p-p-σ and p-p-π electronic bands, and the energy states corresponding
to the in-plane σ-orbitals are located in the vicinity of the Fermi level. Meanwhile, the in-plane
σ-bonds is sensitive to variations in the bonding length, which is the main factor leading to the
variationofbandgapupondifferentstrains10.
Inordertounderstandthecontributionofdifferentorbitalsfromdifferentatomstotheelectronic
states, we have calculated the total (DOS) and partial (PDOS) densities of states for penta-SiC
2
withtensileandcompressivestrains: ±10%strainsalonga-directionandbi-direction,respectively,
as shown in Fig. 4. By comparison, it’s obvious that the contribution from different orbits of Si
and C atoms to the valence band does not change too much when uniaxial or biaxial strain is
applied. However, when uniaxial or biaxial compressive strain larger than ε = −10% is applied,
x
the contribution from C 2p states becomes dominant. Therefore, when monolayer penta-SiC is
2
freeofstrainorundersmallstrain(lessthan10%),theconductionbandmainlyconstructsthelow
8
(a) (b)
E (eV) strain along x-direction VCss Cs-VΓΓ strain along bi-direction VCss Cs-Γ VΓ
(c) (d)
V)
e
E (
FIG.3. PBEcalculatedbandstructuresunder(a)uniaxial(a-direction)strainand(b)biaxial(bi-direction)
strain for penta-SiC . (c) and (d) show the evolution of band gap for SiC as a function of the applied
2 2
various (uniaxial and biaxial) strain, respectively. Furthermore, the crystal structure is fully relaxed(both
unitcellparametersandatomicpositionsundersymmetryconstraints).
(a) ε = 0% (b) εx = -10% (c) εx = 10% (d) εxy = -10% (e) εxy = 10%
V)
y (e
g
ner
E
FIG.4. PBEcalculatedtotalandpartialdensitiesofstatesforpenta-SiC underuniaxialandbiaxialstrain
2
withcompressiveandtensilestrength: (a)nostrain,(b)ε = −10%,(c)ε = 10%,(d)ε = −10%,and(e)
x x xy
ε = 10%,respectively.
xy
hybridization states of Si 3p and C 2p orbits, when a large strain is applied, the contribution from
C2pstatesoverwhelmsthatfromSi3pstates.
9
D. Strain-engineeringtransportpropertiesofpenta-SiC
2
Here we perform the deformation potential theory, which was proposed by Bardeen and
Shockley25toanalyzetheintrinsiccarriermobilityu,andhasbeenwidelyusedinlow-dimensional
systems15,26–28. Based on the acoustic phonon limited approach25, the charge carrier mobility of
2Dmaterialcanbewrittenas,
2e(cid:126)3C
µ = , (1)
3k T|m∗|2E2
B l
whereeistheelectroncharge,T isthetemperaturewhichisequalto300Kthroughoutthepaper.
C istheelasticmodulusofauniformlydeformedcrystalbystrains,m∗ istheeffectivemassgiven
by m∗ = (cid:126)2(∂2E(k)/∂k2)−1 (where (cid:126) is the reduced Planck’s constant, k is the magnitude of wave-
vector in momentum space, and E(k) denotes the energy corresponding to the wave vector). E is
l
theDPconstantdefinedby Ee(h) = ∆E /(δl/l),where∆E istheenergyshiftofthe
l CBM(VBM) CBM(VBM)
bandedgewithrespecttothevacuumlevelunderasmalldilationδl ofthelatticeconstantl.
According to Eq. (1), the carrier mobility of electrons and holes along certain directions for
penta-SiC are obtained, as shown in Fig. 7 and Table. III. Although the PBE calculations al-
2
ways underestimate the band gap, the calculated carrier mobilities are in good agreement with
experimental results for many 2D materials15,26,27,29,30. Firstly, we compare the a-direction uniax-
ial strain modulated charge carrier mobility of penta-SiC along a- and b-directions as shown in
2
Fig. 7. Three different physical parameters, namely carrier effective mass (m∗), DP constant (E )
l
andelasticmodulus(C),aresubjectedtochangeataparticulartemperatureunderstrain.
The band structures of strained penta-SiC change when the structure changes, which subse-
2
quently alter the effective mass of carriers determined by the curvatures of band edges near the
Fermi level. Fig. 5 demonstrates the evolution of the calculated effective masses of electrons (m∗)
e
and holes (m∗) along a- and b-directions under uniaxial strains along a-direction. The m∗ at S
h h
point is 0.29m and m∗ at S −Γ point is 0.90m without strain (m is the static electron mass). As
0 e 0 0
shown in Fig. 3(a), applied strains along a-direction ε may lead to large shifts of the positions of
x
VBMs and CBMs in the band structures. The VBM of penta-SiC shifts from V to V with the
2 s Γ
strainofε = 3%,whichresultsinadramaticincreaseofm∗ (0.30m → 0.69m alonga-direction,
x h 0 0
and 0.28m → 0.74m along b-direction) when ε increases from 2% to 3%. Meanwhile, The
0 0 x
CBM of penta-SiC shifts from C to C with the strain of ε = 8%. Therefore a dramatic
2 s−Γ s x
changeofelectroneffectivemassoccursaswellwhentheuniaxialstrainchangesfromε = 7%to
x
ε = 8%. The subsequent m∗ reduces significantly from 1.33m to 0.33m , and the b-direction m∗
x e 0 0 e
10
1.4 a-hole
a-electron
1.2
b-hole
)
e 1.0 b-electron
m
(
0.8
m
0.6
0.4
0.2
-10 -8 -6 -4 -2 0 2 4 6 8 10
Strain x ( )
Strain x ( )
FIG.5. Respectivedependenceofcalculatedcarrier(holem∗ andelectronm∗)effectivemassesalongthea-
h e
andb-directionsontheapplieduniaxialstrains.
increases from 0.94m to 1.12m , as shown in Fig. 5. Additionally, the anisotropic effective mass
0 0
under strains will lead to anisotropic carrier mobilities, which finally lead to direction-dependent
electronconductivity.
In order to calculate DP constant, a small dilation need to be applied along the direction. The
calculated DP constant for holes is 2.98 eV and 3.00 eV for electrons, respectively, which are
in good agreement with previous theoretical results20. Fig. 6 demonstrates the evolution of the
electron(E )andhole(E )DPconstantsalonga-andb-directionsunderuniaxialstrainsalong
l−e l−h
a-direction. Similar to the case of effective masses, the shifts of VBMs with a uniaxial strain of
ε = 3%andCBMswithastrainofε = 8%causedramaticchangesofDPconstantsofholesand
x x
electrons, respectively, as shown in Fig. 6. Interestingly, the obtained b- direction E of holes is
l−h
low (0.15eV → 0.40eV) in the strain range of −10% ≤ ε ≤ −6%, which subsequently results in
x
highhole-carriermobilitieslargerthan105cm2/VsalongaparticulardirectionasshowninFig.7.
The dependence of calculated carrier mobilities on the uniaxial strain is shown in Fig. 7. The
obtained hole and electron mobilities without strain are 2.59×103cm2/Vs and 2.74 ×102cm2/Vs,
respectively. For uniaxial strain along the a-direction, dramatic enhancement of the hole mobility
along the b-direction under compressive strains is observed, increasing from 2.59×103cm2/Vs to
