Загрузил Олена Чернікова

Ab Initio Studies of the Unreconstructed Polar СdTe

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Journal of ELECTRONIC MATERIALS, Vol. 41, No. 10, 2012
DOI: 10.1007/s11664-012-1924-x
2012 TMS
Ab Initio Studies of the Unreconstructed Polar
CdTe (111) Surface
JIN LI,1,3,4 JACOB GAYLES,1 NICHOLAS KIOUSSIS,1 Z. ZHANG,2
C. GREIN,2 and F. AQARIDEN2
1.—Department of Physics, California State University, Northridge, CA 91330-8268, USA.
2.—Sivananthan Laboratories, 590 Territorial Dr, Unit H, Bolingbrook, IL 60440, USA.
3.—e-mail: jinli@csun.edu. 4.—e-mail: nick.kioussis@csun.edu
Ab initio electronic structure calculations were carried out for bulk cadmium
telluride (CdTe) and the unreconstructed CdTe polar (111) Cd-terminated and
1
1)
Te-terminated surfaces. The hybrid functional for the exchange and
(1
correlation potential improves the overall description of the electronic structure of bulk CdTe, by lowering Cd 4d states and hence reducing the Cd 4d–Te
5p hybridization. The Cd–Te interlayer distance of the Cd-terminated surface
exhibits a dramatic contraction in contrast with the expansion of the
Te-terminated surface, and the surface relaxations decrease as the slab
thickness increases. The underlying mechanism of the convergence of the
electrostatic potential energy, work function, and electric dipole moment of the
polar surfaces as a function of slab thickness is surface electron rearrangement leading to charge transfer from the Te- to the Cd-terminated surfaces.
The surface electric polarization induces an internal electric field in the slab,
which in turn tilts the bands of the slab double layers, thus rendering the
surface layers metallic. The electric field decreases with increasing slab
thickness due to convergence of the difference of electrostatic potentials
between the Cd- and Te-terminated surfaces.
Key words: Ab initio calculation, CdTe, x-ray and gamma-ray detector,
surface properties
INTRODUCTION
Cadmium telluride (CdTe) and cadmium zinc
telluride (CZT) are known to have great potential in
room-temperature x-ray and gamma-ray semiconductor detector applications.1–4 These include
industrial monitoring, gauging and imaging, medical imaging, nuclear safeguards and nonproliferation, and transportation security and safety. The
high atomic number and density of these compounds provide strong absorption and high detection efficiency of high-energy photons. Their wide
bandgaps allow fabrication of highly resistive
devices enabling large depletion depths and low
leakage currents when the material is brought into
(Received October 11, 2011; accepted January 11, 2012;
published online February 25, 2012)
the semi-insulating state with electrical compensation
techniques. The moderately high mobility and lifetime of charge carriers (particularly electrons) allow
good charge transport in devices depleted to many
millimeters or even centimeters in thickness.3
However, development of CdTe-based detectors
has been plagued by material problems caused by
severe microcrystallinity, high defect densities,
impurities, and stoichiometric imbalances. Besides
the quality of the bulk crystal, the properties of the
surfaces and metal/semiconductor interfaces are
often dominant factors influencing detector performance.3 Various surface treatments, such as sputtering, annealing, and mechanical and chemical
processing, change the surface morphology and
induce various defects, affecting in turn the electronic
structure and hence the transport properties; For
example, depending on the surface stoichiometry
and the Cd chemical potential, the Cd-terminated
2745
2746
Li, Gayles, Kioussis, Zhang, Grein, and Aqariden
CdTe/CZTp(111)
(A surface) undergoes a
ffiffiffi psurface
ffiffiffi
(1 9 1),
3 3 R30 or (2 9 2) reconstruction,5–11 while the Te-terminated
pffiffiffi (B surface)
pffiffiffi surface
exhibits (1 9 1), (2 9 2), 2 3 2 3 R30 , and
c(8 9 4) reconstructions.9–16
To our knowledge, no comprehensive ab initio
calculations of the polar CdTe (111) surface have
been reported so far, which can provide parameterfree analysis of the structural and electronic properties. Furthermore, understanding the properties
of the unreconstructed surface serves as the first
step for further ab initio studies, currently underway, of the effects of surface reconstruction, surface
defects, and CdTe (CZT)/metal interfaces. Also,
previous theoretical studies showed that the bandgap of bulk CdTe from density functional theory
(DFT) calculations is much smaller than the
experimental value,17–20 due to the poor treatment
of the p–d hybridization, and it is found that the
electronic properties of bulk CdTe can be greatly
improved by hybrid functional calculations. Therefore, the purpose of this work is to investigate the
structural and electronic properties of bulk CdTe
and the unreconstructed polar CdTe (111) surfaces,
and discuss the effects of various functionals for the
exchange correlation potential for bulk CdTe and
the effects of slab thickness on the structural and
electronic properties, work function, electric dipole
moment, and internal electric field.
METHOD
The ab initio calculations were carried out
employing the Vienna ab initio simulation package
(VASP)21,22 with the projected augmented wave
(PAW)23,24 approach to represent the electron–ion
interaction. We employed the local density approximation (LDA) of Ceperley and Alder,25 the Perdew
and Wang (PW91) generalized gradient approximation (GGA),26 and the GGA of Perdew, Burke,
and Ernzerhof (PBE)27 for the exchange and correlation functional. Since these approaches severely
underestimate the electronic bandgaps, we also
employed the hybrid functional of Heyd, Scuseria,
and Ernzerhof (HSE06)28,29 as implemented in
VASP,30,31 where some exact (Hartree–Fock)
exchange is mixed into the exchange and correlation
functional. The Cd 4d, 5s and Te 5s, 5p states were
included as valence states. The energy cutoff of the
plane-wave expansion of the basis functions was set
to be 475 eV. For the reciprocal-space integration
we used the Monkhorst–Pack special k-point
method32 with a 11 9 11 9 11 grid. The above
parameters yield total energy convergence to less
than 0.1 meV/atom.
The CdTe (111) surface was modeled employing
the slab supercell approach, where the number of
(111) atomic layers (ALs) was varied from 2 to 30 to
assure convergence of the work function and the
electric dipole moment. We used a 16-Å-thick vacuum region separating the periodic slabs to avoid
interactions between images. All atomic positions
were fully relaxed using the conjugate gradient
algorithm until all interatomic forces were smaller
than 0.01 eV/nm. Since the CdTe (111) surface is
polar, the dipole correction was taken into account
in the calculations. For the reciprocal-space integration we used the Monkhorst–Pack special
k-point scheme with a 11 9 11 9 1 grid.
BULK CdTe
The zincblende structure (F 43m)
of CdTe and the
slab geometry of the (111) surface are shown in
Fig. 1a and b, respectively. In Table I we list the
calculated values of the equilibrium lattice constant, a0, bulk modulus B, cohesive energy, and
bandgap of bulk CdTe, employing the LDA,25
PW9126, and PBE27 exchange-correlation functionals, respectively, and compare them with the corresponding experimental values. The LDA, PW91,
and PBE yield a lattice constant of 6.421 Å, 6.618 Å,
and 6.629 Å, respectively, compared with the
experimental value of 6.481 Å.33 The cohesive
energy of LDA, PW91, and PBE was 4.2 eV,
4.134 eV, and 5.472 eV, respectively. This is consistent with the fact that the LDA (GGA) underestimates (overestimates) the lattice constant, and
hence yields larger (lower) cohesive energy. Compared with experiments, it is found that LDA can
give bulk structural properties in better agreement
with the experiment than GGA, and the results of
LDA are also in good agreement with previous
theoretical calculations.17–20
The LDA band structure of bulk CdTe is plotted
along the symmetry axes of the Brillouin zone (BZ)
in Fig. 2a. It exhibits a direct bandgap of 0.6 eV at
the C point. Figure 2b shows the corresponding
total density of states (DOS) and the orbital- and
atom-resolved partial DOS. One can see that the
core states about 11 eV below the Fermi energy, EF,
are primarily of Te 5s character, and the semicore
states about 8 eV below EF are primarily of Cd 4d
character. The valence states below EF are of Cd 5s
and Te 5p character, whereas the lowest conduction
band states are primarily of Cd 5s character.
In Table II we summarize the calculated results
for the energy gap, ECg and ELg , at the C and L
symmetry points in the BZ, the energy of the 4d Cd
semicore states at C, ed, relative to EF, and the
energies of selected transitions, Lc6 Lv6 and Cv8 – Cv7
at the C and L symmetry points, employing the
LDA, PW91, PBE, and HSE06 without and with
spin–orbit coupling (SOC), respectively. For comparison we also list in the table the experimental
values34–36 and those employing the accurate GW
random phase approximation (GW-RPA).34 It is
obvious that the LDA, PW91, and PBE of DFT
strongly underbind the semicore Cd 4d states and
consequently underestimate the energy gap between occupied and empty bands. One can see that
the LDA, PW91, and PBE ed values of 8.00 eV,
Ab Initio Studies of the Unreconstructed Polar CdTe (111) Surface
2747
(b)
(a)
A surface
d1
d2
Te
Cd
B surface
Fig. 1. (a) The zincblende structure of CdTe. (b) Side view of the CdTe (111) surface showing the Cd-terminated (A) and Te-terminated (B)
surfaces, respectively (color figure online).
Table I. Lattice constant, relative error of lattice
constant, bulk modulus, cohesive energy, and
bandgap of bulk CdTe, employing the LDA, PW91,
and
PBE
exchange-correlation
functionals,
respectively
PW91
PBE
LDA
Exp33
a0
(Å)
Error
(%)
B
(GPa)
Ec
(eV)
Eg
(eV)
6.618
6.629
6.421
6.481
2.11
2.28
0.93
35.5
34.9
45.9
44.5
4.2
4.134
5.472
0.58
0.59
0.53
1.6
7.81 eV, and 7.79 eV, respectively, are much
higher the experimental value of 10.5 eV. The SOC
lowers the LDA value of ed to 8.37 eV, in agreement
with previous DFT calculations.37 Furthermore, the
bandgap values of about 0.6 eV (0.3 eV) without
(with) SOC are much lower than the experimental
value of 1.6 eV. These long-standing problems associated with the local or semilocal forms of the
exchange-correlation functional are due to the poor
treatment of the p–d hybridization,37 which in turn
shifts the Te 5p states to higher energies, thus
decreasing the bandgap.37–39
Recent electronic structure calculations28–30,39
have shown that a significant improvement in the
description of solids having bandgaps can be generally achieved by using hybrid functionals, in
which some nonlocal exact exchange is mixed into
the exchange and correlation functional. Even
though these calculations are computationally more
costly than those employing local or semilocal
functional, they are considerably less intense than
the GW approach.40,41 Thus, in Table II we also list
the corresponding values of the HSE06 hybrid
functional29 without and with SOC, respectively,
where we have used both the LDA and PBE equilibrium lattice constants. It is important to note that
the HSE06 values for ECg , ELg , ed, and the transition
energies, Lc6 Lv6 and Cv8 – Cv7 are in good agreement with both experiment34–36 and the GW
method,34 especially for the LDA equilibrium lattice
constant. This is due to the fact that the hybrid
functional treats more accurately the semicore Cd
4d states, lowering their energy, ed, by 1 eV and
hence reducing the Cd 4d–Te 5p hybridization.
The HSE06 and HSE06 + SOC band structures
employing the LDA equilibrium structure are plotted
in Fig. 3a and b, respectively.
CdTe (111) POLAR SURFACE
We employed the LDA exchange-correlation
functional for the CdTe (111) slab calculations,
2748
Li, Gayles, Kioussis, Zhang, Grein, and Aqariden
(a)
(b) 4
9
DOS (states/(eV cell))
3
E-EF (eV)
s
p
d
Te
6
0
-3
LDA
-6
-9
2
0
Cd
0
L
W
Γ
X
W
K
Total DOS
5
0
-12
s
p
d
2
-10
-5
0
5
10
E-EF (eV)
Fig. 2. (a) Band structure of bulk CdTe along the principal symmetry axes in the BZ, employing the LDA without spin–orbit coupling.
(b) Corresponding orbital- and atom-resolved partial DOS and total DOS of bulk CdTe (color figure online).
Table II. LDA, PW91-GGA, PBE-GGA, and HSE06
values of the energy gap, ECg and EL
g , at the C and L
symmetry points in the BZ, the energy, ed, of the Cd
4d semicore states at C relative to EF, and the
energies of the Lc6 2 Lv6 and Cv8 – Cv7 transitions,
without and with SOC, respectively
LDA
LAD + SOC
PW91
PW91 + SOC
PBE
PBE + SOC
HSE06a
HSE06 + SOCa
HSE06b
HSE06 + SOCb
GW-RPA34
Exp34–36
ECg
ed
ECg
0.63
0.34
0.58
0.31
0.59
0.32
1.42
1.10
1.74
1.44
1.26
1.60
8.00
8.37
7.81
8.12
7.79
8.10
9.07
9.40
9.24
9.69
9.43
10.50
2.52
2.26
2.44
2.19
2.42
2.19
3.26
3.06
3.56
3.28
3.14
3.46
Lc6 2 Lv6
Cv8 – Cv7
2.79
0.87
2.7
0.83
2.7
0.83
3.58
0.75
3.85
3.70
4.03
0.92
0.9
All energies are in eV. Superscript ‘‘a’’ and ‘‘b’’ denote calculations
using the lattice constants of PBE (6.629 Å) and LDA (6.421 Å),
respectively.
because the evaluation of the nonlocal exact exchange
term in the hybrid functional is computationally
expensive. Furthermore, the LDA gave bulk structural properties in better agreement with experiment
compared with the GGA. To investigate the effect of
slab thickness, we carried out systematic calculations
from 1 double layer (DL) to 15 DLs, i.e., from 2 to 30
ALs. The equilibrium surface structure was obtained
by fully relaxing all atoms in the slab.
Structural Properties
The crystal surfaces of zincblende structures
exhibit crystallographic polarity; that is, the (111) and
1
1)
planes are nonequivalent, as shown in Fig. 1b.
(1
Conventionally, the close-packed plane terminated
by metal atoms (cadmium) is referred to as the (111)
or A face, whereas that terminated by nonmetal
1
1)
or B
atoms (tellurium) is referred to as the (1
face. As a result, many properties of polar {111}
surfaces in CdTe differ considerably; For example,
differences in the growth rate and quality of epitaxial films, as well as etch pit geometries, are
apparent.42
In Table III we list values of the unrelaxed and
relaxed interlayer distances, dB and d, respectively,
their difference, Dd = d dB, and the percentage
interlayer relaxation, dd = 100% 9 Dd/dB, for the
case of a nine-DL slab. Xn (X = Cd, Te) denotes the
atomic species on the nth DL. The bulk interlayer
Cd–Te distance in the same DL, d1, is 0.927 Å, while
the bulk interlayer distance of adjacent DLs, d2, is
2.78 Å. Interestingly, the Cd-terminated A surface
(Cd1) exhibits a large inward relaxation of 0.138 Å
while the Te subsurface (Te1) in the top DL relaxes
outward by 0.079 Å, leading to a dramatic
(23.41%) contraction of the Cd1-Te1 interlayer
spacing from 0.927 Å to 0.710 Å. In sharp contrast,
both the Te-terminated (Te9) and Cd subsurface
(Cd9) of the B surface exhibit small outward relaxation of 0.050 Å and 0.019 Å, respectively, leading
to a 3.34% expansion of the Te–Cd interlayer spacing. This outward relaxation of the Te-terminated B
surface was also observed by x-ray photoelectron
diffraction analysis of the reconstructions of CdTe
(111) B surface.12 The dominant relaxations occur
mainly in the Cd- and Te-terminated DLs, and the
d1 and d2 interlayer distances converge quickly to
about 0.9 Å and 2.81 Å, respectively, below the
surface. We found that the surface relaxation
decreases as the slab thickness increases; For example, for a 15-DL slab thickness, the interlayer distances for the first A and B surfaces are 0.73 Å and
0.930 Å, respectively, corresponding to 21.25%
and 0.32% contraction and expansion relative to the
bulk values.
Since the (111) CdTe surface is polar, the slab
possesses a macroscopic electrostatic field perpendicular to the surface, and consequently the cleavage energy of the surface, on the basis of the ionic
model,43,44 diverges with respect to slab thickness.
Ab Initio Studies of the Unreconstructed Polar CdTe (111) Surface
(a)
(b) 10
10
Lc1
5
0
L3
Lc6
Γ1c
E-EF (eV)
5
E-EF (eV)
2749
Γ15
-5
Γ c6
L 4,5
0
L6
Γ8
Γ7
-5
HSE06
HSE06+SOC
-10
-10
-15
W
L
Γ
-15
X
W
K
W
L
Γ
X
W
K
Fig. 3. Band structure of bulk CdTe using the HSE06 hybrid exchange correlation functional based on the LDA equilibrium structure (a) without
SOC and (b) with SOC (color figure online).
Table III. Values of the unrelaxed and relaxed
interlayer distances, dB and d, respectively,
Dd = d 2 dB, and the percentage interlayer
relaxation, dd for the case of a nine-DL slab
Cd1-Te1
Te1-Cd2
Cd2-Te2
Te2-Cd3
Cd3-Te3
Te3-Cd4
Cd4-Te4
Te4-Cd5
Cd5-Te5
Te5-Cd6
Cd6-Te6
Te6-Cd7
Cd7-Te7
Te7-Cd8
Cd8-Te8
Te4-Cd5
Cd9-Te9
dB (Å)
d (Å)
Dd (Å)
dd (%)
0.927
2.78
0.927
2.78
0.927
2.78
0.927
2.78
0.927
2.78
0.927
2.78
0.927
2.78
0.927
2.78
0.927
0.710
2.859
0.893
2.819
0.897
2.812
0.899
2.809
0.899
2.809
0.898
2.808
0.896
2.811
0.895
2.823
0.958
0.217
0.079
0.034
0.039
0.03
0.032
0.028
0.029
0.028
0.029
0.029
0.028
0.031
0.031
0.032
0.043
0.031
23.41
2.84
3.67
1.40
3.24
1.15
3.02
1.04
3.02
1.04
3.13
1.01
3.34
1.12
3.45
1.55
3.34
However, the ionic model does not take into account
surface charge redistribution between the cationand anion-terminated surfaces, which as discussed
in detail in the next subsection, leads to convergent
cleavage energy, work function, and electric dipole
moment as a function of slab thickness. Since the
interlayer distance d2 in Fig. 1b between DLs is
longer than that d1 within a single DL, one expects
that cleavage will preferentially take place across
DLs. The cleavage energy, Ecl, is
Ecl ¼ ðEslab mEbulk Þ=A;
(1)
where Eslab is the total energy of the slab containing
m CdTe formula units, Ebulk is the energy of the
bulk CdTe formula unit cell, and A is the slab surface area. For nonpolar slabs, the surface energy is
Fig. 4. Cleavage energy of the CdTe (111) polar surface versus the
number of atomic slab layers N (color figure online).
simply half the cleavage energy, since the slab has
two identical surfaces. However, for polar slabs the
two surfaces are different, and hence no unique
surface energy can be defined. The cleavage energy
of the fully relaxed slab as a function of the number
of ALs, N, is shown in Fig. 4. One can see that Ecl
initially increases with slab thickness and converges to the value of 0.104 eV/Å2 (1.66 J/m2) at
about 14 ALs.
Work Function, Dipole Moment, and Internal
Electric Field
The surface work function W, defined as the
minimum energy required to remove an electron
from the bulk of the material to the vacuum, can be
calculated as
W ¼ Vð1Þ EF ;
(2)
where EF is the Fermi energy, and V(1) is the
in-plane averaged electrostatic potential in the vacuum region. Figure 5 shows the in-plane averaged
electrostatic potential as a function of distance z
2750
Li, Gayles, Kioussis, Zhang, Grein, and Aqariden
Electrostatic potential (eV)
8
4
WTe
WCd
0
EF
-4
-8
-12
Te1 Cd1
Te9 Cd
9
-16
0
5
10
15
20
25
30
35
40
45
Z (Å)
Fig. 5. In-plane averaged electrostatic potential as a function of
distance z from the bottom to the top vacuum region of the slab for
nine DLs (color figure online).
from the bottom vacuum to the top vacuum region of
the slab for the case of nine DLs. The nine local
minima correspond to each DL, while the eight local
saddle points correspond to the regions between
adjacent DLs. The two flat electrostatic potentials,
VCd(1) and VTe(1), corresponding to the Cd- and
Te-terminated vacuum regions, lead to two different
work functions: WCd and WTe, respectively. Note,
that except for the Cd- and Te-terminated surfaces,
the electrostatic potential minima decrease linearly
across the slab (as indicated by the red line), thus
giving rise to an electric field in the slab interior
from the A to the B surface, similar to other polar
surfaces.43–46 The deviation of the electrostatic
potential minima at the Cd- and Te-terminated
surfaces from linearity is due to charge transfer
between the two surfaces, discussed in more detail
below.
Figure 6 displays the variation of the work functions with slab thickness for the Cd-terminated and
Te-terminated surfaces, and their difference,
DW = WTe WCd. We find that WCd, WTe, and DW
converge to 6.3 eV, 3.2 eV, and 3.1 eV, respectively,
for N ‡ 18 ALs. Thus, throughout the remaining
calculations we employ an 18-AL slab thickness.
The experimental47 values of the work function of
the (111) A and B surfaces of Cd0.9Zn0.1Te are 5.5 eV
and 5.3 eV, respectively. The difference between the
ab initio and experimental values presumably
arises from the Zn alloying effect. We also show in
Fig. 6 the difference in electrostatic potential minima, DU, between the Te- and Cd-terminated surfaces in Fig. 5 as a function of the number of ALs N.
We find that DU converges to about 3.1 eV for
18 ALs (nine DLs), similar to the variation of DW.
As demonstrated below, the convergence of both DW
and DU with slab thickness is due to electron charge
transfer from the Te- to the Cd-terminated surface.
Thus, the internal electric field in the central slab
(excluding the A and B DL surfaces), Eint = (DU/
D) fi 0 as the slab thickness D fi 0, analogous to
Fig. 6. Work function for Cd and Te termination, and their difference,
DW = WTe WCd, and the difference, DU, in the electrostatic
potential minimum in Fig. 5 between the Te- and Cd-terminated
surfaces versus the number of atomic layers N (color figure online).
the results of previous studies of other polar
surfaces.43,44,48,49
There are two effects on the electric dipole
moment of the CdTe (111) polar surface. The first is
associated with atomic structural relaxation, while
the second is electronic relaxation associated with
electronic rearrangement between the cation and
anion surfaces. In Fig. 7 we plot the electric dipole
moment of the unrelaxed and fully relaxed slabs
versus slab thickness. We find that lrel fi 0.30 eÅ
and lunrel fi 0.28 eÅ, respectively, at N 18,
indicating that the structural effect on the slab
electric dipole moment, lstr = lrel lunrel = 0.02
eÅ, is small, and that the electronic rearrangement
plays a crucial role in stabilizing the properties of
the polar surface. We also plot in Fig. 7 the difference in work function, DW = WTe –WCd, as a function of slab thickness. Interestingly the relaxed slab
dipole moment lrel and DW exhibit similar dependence on slab thickness, which can be quantitatively
expressed as
el
(3)
DW ¼ rel ;
e0 A
where e0 is the vacuum permittivity. Using the
converged value of lrel = 0.3 eÅ in Eq. 3, we find
that DW = 3.0 eV, in excellent agreement with our
ab initio value of 3.1 eV.
To elucidate the underlying mechanism of electronic relaxation, as a first step we employed the
simple ionic model43,44 that does not take into
account electronic rearrangement of the A and B
polar surfaces. Within this model, shown in Fig. 8a,
the unrelaxed slab can be viewed as a set of n = N/2
DLs with charges +Q and Q for the Cd and Te ALs,
and with an electric dipole moment, li = Qd1, for
the ith DL, where d1 is the interlayer distance between the A and B layers of the DL. Thus, the net
slab dipole moment is
Ab Initio Studies of the Unreconstructed Polar CdTe (111) Surface
2751
Using the unrelaxed values of d1 = 0.927 Å and
d2 = 2.78 Å in Eqs. 7 and 8, we find that d = 0.107 e
and ltot = 0.297 eÅ. The latter value is in excellent
agreement with the ab initio calculated value of the
electric dipole moment of the unrelaxed slab of
0.28 eÅ. Furthermore, Bader population analysis
gives a charge of ±0.33 e (±0.43 e) for the surface
(bulk) Cd and Te atoms, and hence a charge transfer
d = 0.1 e, which is also in excellent agreement with
the value of 0.107 e obtained from Eq. 7. These
results demonstrate that the dominant mechanism
responsible for stabilizing the polar surface is
indeed the surface electronic rearrangement.
Fig. 7. Electric dipole moment of the unrelaxed and relaxed slab
(left-hand ordinate) and the difference in work function, DW =
WTe WCd (right-hand ordinate), versus slab thickness (color figure
online).
Fig. 8. Schematic of the ionic model of the CdTe (111) polar surface
without (left) and with (right) surface charge transfer, d, necessary to
stabilize the polar surface (color figure online).
ltot ¼ nli ¼ nQd1 ;
(4)
which diverges with slab thickness (n fi 1), in
contrast to our ab initio calculations that yield a
convergent ltot = 0.28 eÅ for N ‡ 18. Figure 8b
shows a generalization of the ionic model that takes
into account the effect of electron rearrangement
solely at the two surfaces; namely, a charge transfer, d, takes place between the cation- and anionterminated surfaces, rendering the corresponding
surface charges +(Q d) and (Q d), respectively, with a net slab dipole moment given by
ltot ¼ nQd1 dðn 1Þðd1 þ d2 Þ dd1
¼ n½Qd dðd1 þ d2 Þ þ dd2 :
(5)
For ltot to be independent of n, the condition
Qd1 dðd1 þ d2 Þ ¼ 0
(6)
must be satisfied. This is turn yields
d ¼ Qd1 =ðd1 þ d2 Þ
(7)
ltot ¼ dd2 :
(8)
and
Electronic Properties
Figure 9a shows the DL-resolved local DOS
(LDOS) for the 18-AL slab, where the Cd- and
Te-projected DOS for each DL are denoted with
black and red curves, respectively. One can see that
the DL-projected DOS of the upper (lower) half of
the slab shifts rigidly towards lower (higher) energies relative to those of the slab center, due to the
internal electric field, Ein, formed between the cation
Cd-terminated and anion Te-terminated polar (111)
surfaces. When the electric field is large enough, the
bottom (top) of the conduction (valence) band of the
Cd- (Te-)terminated surfaces shift below (above) EF,
leading to metallization of the polar surfaces43,44,48
and surface charge transfer, d, from the Te- to the
Cd-terminated surface. This charge transfer, mainly
occurring near the surface, further shifts the bottom
(top) edge of the conduction (valence) band of the
Cd- (Te-)terminated surfaces, and is responsible for
their deviation from the linear behavior of the other
DLs (see also Fig. 5). Note that the DL in the slab
center remains semiconducting with a bandgap of
0.8 eV. This electric field-induced ‘‘band tilt’’ and
concomitant surface charge rearrangement is shown
schematically in Fig. 9b.
The band structure of the CdTe (111) polar surface is shown in Fig. 10a along the symmetry
directions in the two-dimensional BZ. Consistent
with the DOS in Fig. 9a, the band structure shows
metallic behavior with both valence and conduction
states crossing EF. To elucidate the origin of these
bands we show in Fig. 10b the band structure in a
narrow energy of approximately ±1 eV about EF,
where the blue (red) symbols denote the surface
states from the Cd- (Te-)terminated surfaces. Note
that the conduction band minimum at the
Cd-terminated DL is lower in energy than the upper
edge of the valence band at the Te-terminated DL,
consistent with the results of DOS. Furthermore,
one can clearly see in Fig. 10a the electric fieldinduced shift of the bands; For example, the nine
dispersionless split bands in the energy range from
13 eV to 10 eV correspond to the Te 5s core
states of each DL which are shifted linearly on going
from the Cd-terminated to the Te-terminated DL
surface, which is also consistent with the shift of the
2752
Li, Gayles, Kioussis, Zhang, Grein, and Aqariden
(a)
(b)
LDOS (states/(eV cell))
0.6
0.3
0.0
0.3
Cd-terminated
conduction
band
0.0
0.3
0.0
0.3
0.0
0.3
electrons
Local VBM
EF
0.0
0.3
0.0
0.3
Local CBM
0.0
0.3
Cd
0.0
0.3
Te
Valence band
Te-terminated
0.0
-14
-12
-10
-8
-6
-4
-2
0
2
4
E-EF (eV)
Fig. 9. (a) Double-layer-resolved LDOS for the 18-AL slab (nine DLs), where the Cd- and Te-projected DOS for each DL are denoted with black
and red curves, respectively. The Fermi energy, denoted by the vertical line, is at 0 eV. (b) Schematic shift of the local valence band maximum
(VBM) and local conduction band minimum (CBM) across the slab. The bottom (top) of the conduction (valence) band near the
Cd- (Te-)terminated surfaces shifts below (above) EF (denoted by the vertical line), leading to their metallization and to surface electron charge
transfer from the Te- to the Cd-terminated surfaces (color figure online).
(a)
2
(b)
0
1
-2
E-EF (eV)
E-E F (eV)
-4
-6
0
-8
-10
-1
-12
K
Γ
M
K
Γ
M
Fig. 10. (a) Band structure of the (111) CdTe surface slab consisting of nine DLs along the symmetry directions in the two-dimensional
Brillouin zone. (b) Band structure in a narrow energy range around EF, where the blue (red) symbols denote the surface states from the
Cd- (Te-)terminated surfaces (color figure online).
DOS in Fig. 9a. It can be seen that the energy shift
is almost uniform for each DL excluding the A and B
DL surfaces, owing to the charge transfer being
confined solely near the two surfaces.
CONCLUSIONS
We have carried out first-principles electronic
structure calculations for bulk CdTe and the ideal
CdTe (111) polar surface. The hybrid exchange
functional lowers the energy of the semicore Cd 4d
states by about 1 eV and reduces in turn the Cd 4d–
Te 5p hybridization, thus shifting the Te 5p states to
lower energies and increasing the bandgap.
The Cd-terminated surface exhibits a large inward relaxation while the Te subsurface of the DL
relaxes outward, leading to a dramatic contraction
of about 23% of the Cd–Te interlayer in a nine-DL
slab. In sharp contrast, the Te–Cd interlayer spacing of the Te-terminated surface contracts by about
3%, and the surface relaxations decrease as the slab
thickness increases. The cleavage energy and work
Ab Initio Studies of the Unreconstructed Polar CdTe (111) Surface
function of the Te- and Cd-terminated surfaces
converge to 0.104 eV/Å2, 6.3 eV, and 3.2 eV,
respectively, for slab thickness N ‡ 18. The underlying origin of the convergent dipole moment, difference of work function, and difference in
electrostatic potential energy between the Cd- and
Te-terminated surfaces is the electron charge
transfer of about 0.1 e from the B to A surfaces. The
electric polarization induces an internal electric
field in the slab region, which in turn shifts the
bands of the slab layers, thus rendering the surface
layers metallic.
ACKNOWLEDGEMENTS
The research at California State University
Northridge and Sivanathan Laboratories was supported by Grant No. HDTRA1-10-1-0113. JG was
also supported by the NSF-PREM Grant DMR00116566.
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