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. REFERENCES 1. Z. Zanio, Cadmium Telluride, Semiconductors and Semimetals, Vol. 13 (New York: Academic Press, 1978). 2. R. Triboulet, Y. Marfaing, A. Cornet, and P. Siffert, J. Appl. Phys. 45, 2759 (1974). 3. C. Szeles, Phys. Status Solidi B 241, 783 (2004). 4. T.E. Schlesinger, J.E. Toney, H. Yoon, E.Y. Lee, B.A. Brunett, L. Franks, and R.B. James, Mater. Sci. Eng. R 32, 103 (2001). 5. Y.S. Wu, C.R. Becker, A. Waag, K. von Schierstedt, R.N. Ricknell-Tassius, and G. Landwehr, Appl. Phys. Lett. 62, 1510 (1993). 6. J. Gordon, P. Morgen, H. Shechter, and M. Folman, Phys. Rev. B 42, 1852 (1995). 7. B.J. Kowalski, B.A. Orłowski, and J. Ghijsen, Appl. Surf. Sci. 166, 237 (2000). 8. G.Q. Zha, W.Q. Jie, T.T. Tan, P.S. Li, W.H. Zhang, and F.Q. Xu, Chem. Phys. Lett. 427, 197 (2006). 9. C. Hsu, S. Sivananthan, X. Chu, and J.P. Faurie, Appl. Phys. Lett. 48, 908 (1986). 10. Y.S. Wu, C.R. Becker, A. Waag, M.M. Kraus, R.N. BicknellTassius, and G. Landwehr, Phys. Rev. B 44, 8904 (1991). 11. C.K. Egan, Q.Z. Jiang, and A.W. Brinkman, J. Vac. Sci. Technol. A 29, 011021 (2011). 12. R. Duszak, S. Tatarenko, J. Cibert, K. Saminadayar, and C. Deshayer, J. Vac. Sci. Technol. A 9, 3025 (1991). 13. R. Duszak, S. Tatarenko, J. Cibert, N. Magnéa, H. Mariette, and K. Saminadayar, Surf. Sci. 251/252, 511 (1991). 14. S. Tatarenko, B. Daudin, D. Brun, V.H. Etgens, and M.B. Veron, Phys. Rev. B 50, 18479 (1994). 15. S. Rujirawat, Y. Xin, N.D. Browning, S. Sivananthan, D.J. Smith, S.-C.Y. Tsen, and Y.P. Chen, Appl. Phys. Lett. 74, 2346 (1999). 16. G.Q. Zha, W.Q. Jie, T.T. Tan, X.X. Bai, L. Fu, W.H. Zhang, and F.Q. Xu, J. Mater. Res. 24, 1639 (2009). 17. A.E. Merad, M.B. Kanoun, G. Merad, J. Cibert, and H. Aourag, Mater. Chem. Phys. 92, 333 (2005). 2753 18. J. Heyd, J.E. Peralta, G.E. Scuseria, and R.L. Martin, J. Chem. Phys. 123, 174101 (2005). 19. J.H. Yang, S.Y. Chen, W.J. Yin, X.G. Gong, A. Walsh, and S.H. Wei, Phys. Rev. B 79, 245202 (2009). 20. A. Carvalho, A.K. Tagantsev, S. Öberg, P.R. Briddon, and N. Setter, Phys. Rev. B 81, 075215 (2010). 21. G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996). 22. G. Kresse and J. Furthmüller, Comput. Mater. Sci. 6, 15 (1996). 23. P.E. Blöchl, Phys. Rev. B 50, 17953 (1994). 24. G. Kresse and J. Joubert, Phys. Rev. B 59, 1758 (1999). 25. D.M. Ceperley and B.J. Alder, Phys. Rev. Lett. 45, 566 (1980). 26. J.P. Perdew, J.A. Chevary, S.H. Vosko, K.A. Jackson, M.R. Pederson, D.J. Singh, and C. Fiolhais, Phys. Rev. B 46, 6671 (1992). 27. J.P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996). 28. J. Heyd, G.E. Scuseria, and M. Ernzerhof, J. Chem. Phys. 118, 8207 (2003). 29. J. Heyd, G.E. Scuseria, and M. Ernzerhof, J. Chem. Phys. 124, 219906 (2006). 30. J. Paier, R. Hirschl, M. Marsman, and G. Kresse, J. Chem. Phys. 112, 234102 (2005). 31. A.V. Krukau, O.A. Vydrov, A.F. Izmaylov, and G.E. Scuseria, J. Chem. Phys. 125, 224106 (2006). 32. H.J. Monkhorst and J.D. Pack, Phys. Rev. B 13, 5188 (1976). 33. O. Madelung (ed.), Numerical Data and Functional Relationships in Science and Technology, Vol. 17, Parts a and b, 1982; Vol. 22, Part a, 1987, Landolt-B\’’{o}rnstein, New Series, Group III (Berlin: Springer-Verlag). 34. R. Triboulet and P. Siffert, CdTe and Related Compounds; Physics, Defects, Hetero- and Nano-structures, Crystal Growth, Surfaces and Applications (Cambridge, UK: Cambridge University Press, 1998). 35. D.J. Chadi, J.P. Walter, and M.L. Cohen, Phys. Rev. B 5, 3058 (1972). 36. D.T.F. Marple and H. Ehrenreich, Phys. Rev. Lett. 8, 87 (1962). 37. S.H. Wei and A. Zunger, Phys. Rev. B 37, 8958 (1988). 38. O. Zakharov, A. Rubio, X. Blase, M.L. Cohen, and S.G. Louie, Phys. Rev. B 50, 10780 (1994). 39. X.F. Wu, E.J. Walter, A.M. Rappe, R. Car, and A. Selloni, Phys. Rev. B 80, 115201 (2009). 40. M.S. Hybertsen and S.G. Louie, Phys. Rev. Lett. 55, 1418 (1985). 41. M.S. Hybertsen and S.G. Louie, Phys. Rev. B 34, 5390 (1986). 42. T.H. Myers, J.F. Schetzina, T.J. Magee, and R.B. Ormond, J. Vac. Sci. Technol. A 1, 1598 (1983). 43. P.W. Tasker, J. Phys. C 12, 4977 (1979). 44. B. Meyer and D. Marx, Phys. Rev. B 67, 035403 (2003). 45. J.H. Song, T. Akiyama, and A.J. Freeman, Phys. Rev. B 77, 035332 (2008). 46. P. Kempisty, S. Krukowski, P. Strak, and K. Sakowski, J. Appl. Phys. 106, 054901 (2009). 47. G.Q. Zha, W.Q. Jie, X.X. Bai, T. Wang, L. Fu, W.H. Zhang, J.F. Zhu, and F.Q. Xu, J. Appl. Phys. 106, 053714 (2009). 48. M.H. Tsai and S.K. Dey, Eur. Phys. J. Appl. Phys. 36, 125 (2006). 49. J. Soltys, J. Piechota, M. Łopuszyński, and S. Krukowsik, New J. Phys. 12, 043024 (2010).