What causes high resistivity in CdTe

Shallow donors are often introduced into semiconductor materials to enhance n-type conductivity. However, they can sometimes also be used to obtain compensation between donors and acceptors, resulting in high resistivity in semiconductors. For example, CdTe can be made semi-insulating by shallow donor doping. This is routinely done to obtain high resistivity in CdTe-based radiation detectors. However, it is widely believed that the shallow donor alone cannot be responsible for the high resistivity in CdTe. This is based on the argument that it is practically impossible to control the shallow donor doping level so precisely that the free carrier density can be brought below the desired value suitable for radiation detection applications. Therefore, a deep native donor is usually assumed to exist in CdTe and pin the Fermi level near midgap. In this paper, we present our calculations on carrier statistics and energetics of shallow donors and native defects in CdTe and illustrate different donor-specific mechanisms for achieving carrier compensation. Our results show that the shallow donor can be used to reliably obtain high resistivity in CdTe without requiring additional deep donors. Since radiation detection applications require both high resistivity and good carrier transport, one should generally use shallow donors and shallow acceptors for carrier compensation and avoid deep centers that are effective carrier traps. This study highlights how the interaction between impurities and native defects intricately affects the Fermi level pinning in the semiconductor band gap and the associated resistivity of the material.


Introduction
Effective control over charge carriers and management of impurities and native defects is central to the reliable use of semiconductors in optoelectronic devices. Doping a semiconductor with shallow donors or acceptors is a frequently used technique to obtain an nor p-type semiconductor. On the other hand, semi-insulating materials with resistivity approaching the theoretical intrinsic limit are often obtained via doping with deep donors or acceptors. It is well known that GaAs can be made semi-insulating by promoting the formation of EL2 defect [1], a midgap deep native donor commonly believed to be As antisite [2,3]. CdTe can also be made semi-insulating via doping with deep donors such as Ge and Sn [4][5][6][7]. However, the highly resistive CdTe-based radiation detector materials are usually doped with shallow donors, such as Cl, In or Al [8][9][10][11][12][13]. The Cl doping level can be as high as ∼10 19 cm −3 while CdTe remains semi-insulating. Interestingly, despite high level of shallow donor doping, the high resistivity in CdTe is widely believed to be a result of Fermi level pinning by a deep native donor, whose identity is controversial but is usually assumed to be Te antisite [14][15][16]. In this paper, we will show that shallow donor doping can be used reliably to obtain high resistivity in CdTe without needing additional deep donors. The results will provide insights to the carrier compensation mechanisms in many semiconductors.
CdTe and its alloys, such as Cd 1−x Zn x Te (CZT), with x = 0.1-0.2, have been extensively studied for their potential applications in room temperature radiation detection [17]. A good detector material must have high resistivity (>10 9 cm), which reduces dark current and device noise, and large µτ product (mobility-lifetime product), which ensures that radiationgenerated carriers can travel through a large volume of the detector material to be collected by electrodes. The free carrier density in CdTe needs to be controlled below 10 8 cm −3 for radiation detection applications. However, the impurity concentration in CdTe is typically ∼10 15 -10 16 cm −3 . Obviously, the high resistivity is made possible by compensation among various native defects and impurities. Meanwhile, these native defects and impurities also affect carrier transport by scattering and trapping free charge carriers. Thus, a proper impurity and defect management is essential for obtaining both high resistivity and large µτ product. 3 The growth of CdTe or CZT from the melt is typically under Te-rich conditions [18,19]. The higher Cd vapor pressure than that of Te 2 causes the loss of Cd during the crystal growth. Therefore, the growth condition is Te-rich even if the starting material is stoichiometric. The resulting undoped CdTe is usually p-type with low resistivity, due primarily to the abundance of Cd vacancies (V Cd ), which are acceptors [20]. To grow semi-insulating CdTe and CZT, one may use high-pressure Bridgman (HPB) technique [8,21], which suppresses the loss of Cd during growth, or more frequently to dope CdTe with shallow donors [8][9][10][11][12][13], such as Cl, In and Al. Although the shallow donor doping is proven to be a very effective approach to obtain high resistivity in CdTe and CZT, it is widely believed that the high resistivity is due to the Fermi level pinning by a deep donor level induced by a native defect, provided that deep donor concentration, N dd , exceeds the difference of shallow acceptor, N sa , and shallow donor concentrations, N sd (N dd > N sa -N sd ) [14][15][16]. This deep donor is usually assumed to be Te antisite (Te Cd ) [4,5,22,23]. The rationale of the deep-donor-mediated carrier compensation model is that it is practically impossible to control the shallow donor concentration to almost exactly compensate the native and impurity acceptors such that the free carrier density can be less than 10 8 cm −3 [5,15]. However, the deep-donor-mediated carrier compensation model has several drawbacks, as we discuss below.
(1) We contend that it is entirely possible to compensate excess acceptors in CdTe reliably using shallow donors and obtain a free carrier density less than 10 8 cm −3 . Previous calculations on carrier statistics and resistivity in CdTe conclude that high resistivity in CdTe is attainable only through deep donor doping [15,24]. However, these calculations are all based on the incorrect assumption that the native acceptor concentration is a constant and independent of the Fermi level. This assumption has been invoked numerous times for calculations of the carrier density, the level of the hypothetical deep donor, resistivity etc. These calculations all missed an important fact that the formation energy of an acceptor defect decreases with rising Fermi level. Therefore, a higher doping level of the shallow donors in CdTe should be accompanied by an automatic increase of the native acceptor concentration that partially compensates the rise of the shallow donor concentration. This results in a smooth and relatively slow change of the Fermi level with the rise of the shallow donor concentration. The calculations that take into account the dependence of the formation energies of donors and acceptors on the Fermi level show that the high resistivity (>10 9 cm) can be maintained over a large range of the shallow donor concentration. This range is at least in the order of ∼10 16 -10 17 cm −3 and can even be up to the solid solubility of the donor (e.g. >10 19 cm −3 for Cl) as we will demonstrate in Section 3.1. (2) The deep donor model cannot be reconciled with the good µτ product in high-quality detector grade CdTe and CZT. It is well known that doping by extrinsic deep donors, such as Ge and Sn, can give rise to high resistivity [4][5][6][7]. However, the µτ product in Ge-or Sndoped CdTe and CZT (with the dopant concentration ∼10 16 cm −3 ) is significantly reduced due to the electron trapping at the deep donor level [5]. There is no reason to believe that other deep donors, such as Te Cd , will be benign in terms of electron trapping. The fact that detector-grade highly resistive CdTe and CZT have good µτ products suggests that the ionized deep donor concentration is low. It has been estimated that the concentration of a hypothetical midgap donor should be very low (in the order of 10 11 cm −3 ) based on the measured electron µτ product and the capture cross section of the deep-level defects in the high-quality detector grade CZT [24].  [4,5,22,23]. However, there is no experimental proof that Te Cd does introduce a deep donor level near midgap. Much effort has been devoted to the measurement of the deep levels in CdTe and CZT [23,[25][26][27] and it is often suggested that some observed deep levels are responsible for the high resistivity. However, the concentration of these deep-level defects is unknown and there is no proof that they control the conductivity. A recent study shows that some deep levels may be induced by dislocations [28].
It can be seen from the discussion above that the deep-donor-mediated carrier compensation model in CdTe is not well justified. Alternative explanations to the high resistivity in CdTe have been pursued. It has been suggested that the O-H complex may be important in carrier compensation in CdTe [29,30]. However, in heavily doped CdTe, such as CdTe:Cl with [Cl] ∼10 19 cm −3 , the role of O-H complex should be limited due to the much lower oxygen concentration ∼10 16 cm −3 [23,31].
First-principles calculations based on density functional theory (DFT) within local density approximation (LDA) have been used to study the defects and impurities in CdTe [27,29,30,32,33]. It is shown in [32] that Te Cd induces a (+/0) and (2 + /+) transition levels at E c -0.34 and E c -0.59 eV, respectively, in the upper half of the band gap (E c is the energy of the conduction band edge). More recent calculations [29,30,33] show that there should be a stronger Jahn-Teller distortion at neutral Te Cd than predicted in [32]. This results in a much more stable neutral Te Cd and consequently a lower donor level. The (2 + /0) transition level of Te Cd is calculated to be at E v + 0.35 eV (E v is the energy of the valence band edge) [29,30]. A combined experimental and theoretical study shows that the observed midgap levels may be deep acceptor levels caused by the Te Cd -V Cd complex [27]. Our calculations show that the Te Cd -V Cd complex indeed induces deep acceptor levels near midgap, but it has a high formation energy (as will be shown in section 3.4) and thus cannot play a significant role in carrier compensation.
In this paper, we present our calculations on carrier statistics and resistivity of CdTe. Hybrid functional calculations [34] were employed to calculate the formation energies of native defects and shallow donors. The calculated formation energies were used in the carrier statistics calculations. The hybrid functional calculations, which can partially remove the selfinteraction error and correct the band gap, have recently been applied to defect calculations and have generally shown improvement in structural, electronic, dielectric and defect properties in semiconductors [35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50]. Our results show that: (a) shallow donors can be used reliably to obtain high resistivity in CdTe without the need for any deep donors and (b) the compensation mechanism is donor-specific and depends on the binding between the extrinsic donor and the native acceptor. We illustrate this point by describing the different carrier compensation mechanisms for In-and Cl-doped CdTe. It also explains why reasonable fluctuations in the doping level during crystal growth can be tolerated and high resistivity can still be produced. These results shed important light on how the interaction between impurities and native defects intricately affects Fermi level pinning and the associated resistivity of the material.

Computational details
Density functional calculations were performed to study the native defects and shallow donor impurities in CdTe. Hybrid functionals [34], as implemented in the VASP codes [51], were used to calculate defect formation energies. The fraction of Hartree-Fock exchange was adjusted to 16.5% to correct the band gap to 1.55 eV, which is in good agreement with the experimental value of 1.61 eV [52]. The electron-ion interactions were described using projector-augmented wave potentials [53]. The wavefunctions were expanded in a plane-wave basis with cutoff energy of 275 eV. The Cd d electrons were treated as valence electrons. The lattice constant of CdTe is calculated to be 6.59 Å, in good agreement with the experimental lattice constant of 6.477 Å [54]. The bulk host material was simulated using a 64 atom cubic supercell and the Brillouin-zone sampling was performed using a 2 × 2 × 2 -centered k-mesh. The force on each atom was minimized to be less than 0.05 eV Å −1 . Corrections to image charges in supercell calculations were performed following [55]. We performed convergence tests using Perdew-Burke-Ernzerhof (PBE) [56] functionals. The formation energy of V 2− Cd was calculated using a denser k-mesh (4 × 4 × 4), a higher cutoff energy (400 eV) and a larger supercell size (216-atom cell). In each case, the change in formation energy is well below 0.05 eV.
The defect formation energy of a defect in charge state q is given by, where E Defect and E host are the total energies of the defect-containing and the host (i.e. defectfree) supercells. n i is the difference in the number of atoms for the ith atomic species between the defect-containing and defect-free supercells. µ i is a relative chemical potential for the ith atomic species, referenced to µ ref i . For Cd and Te, µ ref Cd and µ ref Te are the chemical potentials in bulk Cd and bulk Te, respectively. The third term in equation (1) represents the change in energy due to exchange of electrons between the defect and the electron reservoir. ε VBM is the energy of the valence band maximum (VBM) in the host system and ε f is the Fermi energy relative to the VBM.
The defect concentration at thermal equilibrium can be evaluated using where N site is the number of available sites for defect formation in the crystal, H is the defect formation energy defined in equation (1), k is the Boltzmann's constant and T is the temperature. The transition level, ε(q/q ), of a defect is the value of ε f at which the formation energies in charge states q and q are identical. The calculated heat of formation for CdTe, H (CdTe), is −1.13 eV, in reasonable agreement with the experimental value of -0.96 eV [54]. The stability condition for the formation of CdTe under equilibrium is given by µ Cd + µ Te = H (CdTe) = −1.13 eV. The following conditions are also applied to prevent the formation of various competing phases (i.e. CdCl 2 , TeCl 4 , Te 3 Cl 2 , In 2 Te 3 and In 2 Te 5 ) when Cl and In donor impurities are introduced 6 into CdTe: 2µ In + 3µ Te H (In 2 Te 3 ) = −1.66 eV, 2µ In + 5µ Te H (In 2 Te 5 ) = −1.99 eV.
Here, µ Cl and µ In are the relative Cl and In chemical potentials referenced to those for Cl 2 molecule and the In metal, respectively.

Carrier statistics and resistivity
The free carrier densities of electrons (n) and holes (p) as well as the Fermi level can be obtained by self-consistently solving the equations (4)-(6) below. Here, is the density of the ionized donor (acceptor) of ith type and charge state q j . Both of them are functions of q ε f , as shown in equation (1). E g is the band gap of CdTe. N C and N V are the effective densities of states of conduction and valence bands, respectively. They can be evaluated by using where m e and m h are the effective masses of the electron and the hole, respectively, and h is the Planck constant. The resistivity can be further calculated using the equation where µ n (µ p ) is the mobility of the electron (hole). We use the following values for parameters in equations (2), (5)- (7): N site = 1.47 × 10 22 cm −3 , m e = 0.11m 0 , m h = 0.73m 0 , µ n = 1000 cm 2 Vs −1 , µ p = 80 cm 2 Vs −1 and E g = 1.55 eV. The band gap is calculated by the hybrid functional calculation. A large CdTe single crystal is usually grown from the melt and subsequently cooled down to room temperature. The atoms and electrons in the crystal should attempt to equilibrate with their respective reservoirs when CdTe is cooled from the growth temperature to the room temperature. The atoms will freeze in the lattice relatively quickly when the crystal is cooled, whereas the localized charges in defects and impurities and the free carriers may equilibrate at room temperature. Therefore, the concentrations of the native defects and impurities are calculated using the growth temperature typically applied to CdTe, i.e. T = T growth = 1373 K, while the carrier density is calculated using the room temperature, i.e. T = T room = 300 K. The total electric charge for the shallow acceptor/donor is simply the local charge on the shallow acceptor/donor multiplied by the concentration of the shallow acceptor/donor calculated at T = 1373 K. Although the total number of a donor or an acceptor is determined at T = 1373 K, its charge may re-equilibrate at T = 300 K if the donor or the acceptor level is not far from the 7 Fermi level. This will change the ratio of the number of the ionized donors/acceptors to that of the neutral ones. In the carrier statistics calculations shown in section 3.1, several different types of donor and acceptor are involved, i.e. a shallow single-electron donor, a deep double-electron donor, a shallow single-electron acceptor and a shallow double-electron acceptor. We assume that all the shallow donors and the shallow acceptors are fully ionized at room temperature since we are only interested in CdTe with its Fermi level in the middle region of the band gap. For the deep double-electron donor, if the deep donor level (ε d ) is fully occupied, the deep donor would be charge neutral. Thus, the total charge of all the deep double-electron donors (Q DD ) is determined by the number of holes on the donor level at T = 300 K: where N DD is the number of the deep donors. Here, we neglect any electronic or structural relaxation upon the charge state change for the deep donor. Thus, the deep donor level position is independent on its occupation.

Compensation in CdTe: carrier statistics and resistivity
We first consider the carrier compensation between a shallow double-electron acceptor and a shallow single-electron donor. This simulates the case of shallow donor (D + ) doping of CdTe where the Cd vacancy (V 2− Cd ) is abundant. The formation energy of V 2− Cd is calculated by hybrid functional calculations, i.e. H (V 2− Cd ) = 3.063 eV − 2ε f (following equation (1)). The Cd vacancy concentration, [V 2− Cd ], is calculated at 1373 K using equation (2). When the Fermi level is at midgap (E v + 0.775 eV), [V 2− Cd ] is 4.11 × 10 16 cm −3 . This is in contrast to previous calculations where [V 2− Cd ] is fixed [15].  Figure 2(a) shows that the high resistivity (>10 9 cm) is obtained in all three cases. Figure 2(b) shows that the Fermi level rises with the deep donor concentration, [DD], but at a much slower pace when the Fermi level is above the donor level. As a result, a deeper donor usually has a larger range of [DD] that gives high resistivity, as manifested in figure 2(a). However, there could be exceptions, i.e. if the deep donor level is much lower than the midgap, ε f may not be able to rise to the midgap before the donor concentration reaches its solid solubility. Cd ) are considered in the calculation. A resistivity higher than 10 9 cm is usually required for radiation detector materials.
Although the deep donor is very effective in carrier compensation, it is also an efficient carrier trap and thus should be avoided in radiation detectors. On the other hand, the shallow donor can be reliably used to obtain high resistivity in CdTe, as manifested in figure 1. As can be seen from figure 1(b), the Fermi level rises slowly with [D + ] (linear to log[D + ]), allowing reasonable fluctuations of the donor concentration during the crystal growth without lowering the resistivity significantly. Next, we will show that the Fermi level can further be stabilized over a wider range of [D + ] by the formation of A centers and the resulting donor self-compensation.
A shallow donor, D + , can bind with a V 2− Cd to form an A center, which is a single-electron acceptor. Increasing [D + ] will induce the formation of more Cd vacancies. Some of them are isolated Cd vacancies and some others are the Cd vacancies bound in the A centers. The Cd vacancy concentration, [V 2− Cd ], mentioned in this paper always means the concentration of the isolated Cd vacancy, not including those bound in the A centers.    6 eV). However, it should be mentioned that, since the donors scatter the electron carriers, one should obtain the high resistivity using as few shallow donors as possible.
We will show in the next section how the resistivity and the Fermi level shown in this section can be related to the defect energetics.

Energetics of donors and acceptors: general picture
When ε f is near midgap, the free carrier density is negligible compared to the concentrations of the donor and the acceptor. Thus, if we plot the formation energies of V 2− Cd and D + as functions of ε f , their crossing point (ε D−V f ), where the formation energies of D + and V 2− Cd are the same, should be approximately equal to the equilibrium Fermi level of the system (see, for example, a schematic in figure 4(a)). During the crystal growth of CdTe, if more shallow donors are added, the donor chemical potential would rise and consequently its formation energy would decrease (see equation (1) Thus, a stronger binding energy leads to a lower ε D−A f with respect to the conduction band minimum (CBM).
The results in section 3.1 and the analyses performed in this section show that high resistivity (>10 9 cm) can be obtained reliably by the shallow donor doping unless ε D−A f , which is the upper limit of the Fermi level, is too low in the band gap. As long as ε D−A f is near or above the midgap, all that is required for achieving high resistivity is a high enough solid solubility for the shallow donor such that the Fermi level can be raised to the midgap region.
In the case of Fermi level pinning by ε D−V f , the tolerable uncertainty in the doping level (δ N ) is in the order of 10 16 -10 17 cm −3 based on the calculated formation energy of V 2− Cd (see 13 section 3.1). Now we will give a general assessment of δ N without relying on the calculated [V 2− Cd ]. If the shallow donor concentrations are N D1 and N D2 when ε f = ε f1 and ε f2 , respectively, it is easy to show that N D2 /N D1 = exp(2 ε f /kT ), where ε f = ε f2 − ε f1 . Using a crystal growth temperature of 1375 K and ε f < 0.275 eV for CdTe, one finds that N D2 /N D1 ranges from ∼10 0 to ∼10 2 . Therefore, an increase in [D + ] by nearly two orders of magnitude is needed to move the Fermi level from ε f1 to ε f2 . If we tighten the resistivity requirement such that ε f < 0.15 eV, N D2 /N D1 would range from 1 to 12.6, which still requires more than one order of magnitude increase in [D + ] to move the Fermi level from ε f1 to ε f2 . Therefore, the tolerable uncertainty in the shallow donor concentration can be quite large.

Energetics of shallow donors: In and Cl
The reported In concentrations in In-doped semi-insulating CZT are typically in the order of 10 16  is relatively low (e.g. ∼10 16 cm −3 ) and by Cl + Te and its A center when [Cl] is high (e.g. ∼10 19 cm −3 ); (iii) the binding energy of a Cl-A center should be larger than that of a In-A center (which may be expected as Cl Te is the nearest neighbor, while In Cd is the second-nearest neighbor to the Cd vacancy), and ε D−A f for CdTe:Cl is near midgap and lower than that for CdTe:In; and (iv) increasing In or Cl donor density to its solid solubility will result in n-type CdTe:In and semi-insulating CdTe:Cl, respectively.
We have shown in section 3.1 that high resistivity can always be obtained by the shallow donor doping, provided that the solid solubility of the shallow donor is high enough to raise the Fermi level to midgap and the A-center binding energy has a proper value so that ε D−A f is sufficiently high (near or above the midgap). Figures 6 and 7 show the formation energies of In and Cl related defects in CdTe, respectively. The maximum In and Cl chemical potentials allowed by equations (3) are used. Under the Te-rich and shallow-donor-rich limits (figures 6(a) for In and Cl satisfy the requirements for obtaining high resistivity as discussed above. The lower ε D−A f for CdTe:Cl than that for CdTe:In is due to the higher A-center binding energy calculated for Cl (1.01 eV) than for In (0.60 eV) (see equation (9)).
The experimental results showing that high resistivity is maintained with [Cl] > 10 19 cm −3 suggests that the scenario depicted in figure 5 should apply to CdTe:Cl, which requires that ε D−A f be confined within a narrow range (calculated to be from E v + 0.69 to E v + 0.88 eV for resistivity  (9)). Also, the comparison between the calculated and the measured resistivities are qualitative in nature, because the Fermi level in the real material is affected by many factors not reflected in the calculations, e.g. the non-equilibrium conditions during the crystal growth, defect kinetics during the crystal cool-down process etc.
Under the Cd-rich and shallow-donor-rich conditions, both CdTe:In and CdTe:Cl should exhibit n-type conductivity as both ε D−V f and ε D−A f are near or even above the CBM (see figures 6(b) and 7(b)).
We have also calculated the formation energies of Cl and In induced DX centers in CdTe. The structures of these DX centers can be found in [58]. For both donors, the shallow-donor and the DX-center energy lines intersect at a point above the CBM (see figures 6 and 7). Thus, the Cl and In DX centers are not stable unless the band gap can be increased by applying pressure or by alloying with Zn, as also found experimentally [59][60][61]. Our results show that the shallow donors are compensated by V Cd and A centers, not by DX centers. The experimental observation of the DX centers in Cd 0.8 Zn 0.2 Te was made after annealing the sample in Cd vapor for several days, which eliminated both V Cd and A centers [59][60][61]. Figure 8 shows the formation energies of important native defects in CdTe. The structures of these defects are similar to those reported previously [29,30]. Comparing figure 8 to figures 6 and 7 shows that, under the In and Cl rich conditions, the formation energies of the substitutional In and Cl are lower than those of the native donor defects. Thus, with proper doping levels, the shallow donor, the Cd vacancy and the A center can control the conductivity in CdTe. Compared to earlier LDA results [29,30], the hybrid functional calculations produce a lower VBM due to the partial correction of the self-interaction error. The consequence is that the deep defect levels in hybrid functional calculations are generally higher in energy relative to the VBM than in LDA calculations.

Energetics of native defects
The low-energy native donors are Cd i , V Te and Te Cd , while V Cd acts as the dominant acceptor, as shown in figure 8. Using the results in figure 8, we obtain that ε f = E v + 0.74 eV and ρ = 1.2 × 10 10 cm −3 at the Te-rich limit 2 . The calculated high resistivity for undoped CdTe seems to be inconsistent with the observation that the undoped CdTe is usually p-type with low resistivity. However, it is important to note that the high resistivity in undoped CdTe has also been obtained using the HPB technique [8,21], which suppresses the loss of Cd during crystal growth. It has been pointed out by Szeles et al [65] that even the HPB grown CdTe suffers from some Cd loss and the incorporation of large concentration of Te precipitates and Te inclusions, which indicates the Te-rich growth condition. It may be argued that the calculated defect energetics at the Te-rich limit are better compared against the HPB-grown CdTe because the particle exchange with the gas phase is significantly reduced in the HPB growth. It may be difficult to reach thermal equilibrium when the bulk CdTe attempts to equilibrate with both Te precipitates and the Cd vapor simultaneously. The constant Te chemical potential assumed in the calculations may not reflect the reality during the crystal growth when non-equilibrium conditions exist. It is possible that the Cd vacancy concentration in CdTe grown by low-pressure techniques is higher than that when the thermal equilibrium is reached, and thus a low resistivity results. Even with HPB growth methods, the Cd loss during the crystal growth still exists. This 2 The concentrations of the ionized and neutral Te Cd are calculated at high temperature of 1373 K without invoking the charge re-equilibration at the room temperature using equation (8). The reason is that charging Te Cd causes significant electronic and structural relaxation. The ground-state structures of Te 0 Cd and Te 2+ Cd are of C 3v and T d symmetries, respectively. The single-particle defect level of Te 0 Cd is below the midgap while those of Te 2+ Cd and Te + Cd are much higher than the midgap. Therefore, the carrier statistics should not be affected much by reducing the temperature from the growth temperature to the room temperature. may be related to the observation of large spatial variation of the resistivity in undoped CZT grown from HPB techniques [62], which suggests that the non-equilibrium growth conditions play an important role in the carrier compensation in CdTe. However, the creation of additional Cd vacancies due to the non-equilibrium growth condition does not pose a problem for obtaining high resistivity in CdTe:Cl and CdTe:In since the solid solubilities of Cl and In in CdTe are sufficiently high to compensate the excess Cd vacancies. The growth condition can be switched to Cd-rich by applying Cd overpressure. This will result in n-type CdTe as evidenced by the dominance of the low-energy donor defects shown in figure 8(b). Thus, controlling stoichiometry is the key to obtaining high resistivity in undoped CdTe. This, nevertheless, should be difficult, especially for obtaining uniform high resistivity in a large volume of the crystal. It is perhaps much easier to control the shallow donor concentration to be in the range of 10 16 -10 17 cm −3 than to control the native defect concentration to a similar range. Figures 6(b), 7(b) and 8(b) show that, under Cd-rich conditions, CdTe, regardless of whether doped or undoped, should exhibit n-type conductivity. Many attempts have been made to remove Te-inclusions and precipitates by annealing CdTe and CZT under Cd overpressure. This usually results in decrease in resistivity [63][64][65][66], consistent with our results. Figure 8 shows that Cd 2+ i and V 2+ Te have similar formation energies. However, our previous calculations show that Cd 2+ i has a low diffusion barrier of 0.36 eV [29,30], which means that Cd i can be mobile at low temperatures. It is likely that most of Cd 2+ i will recombine with V 2− Cd during the crystal cool-down process, which nevertheless does not affect the resistivity. Te Cd is a very deep donor with its (2 + /+) and (+/0) levels calculated at E v + 0.38 and E v + 0.58 eV, respectively. The results in figure 8(a) show that the Fermi level is pinned by Te + Cd and V 2− Cd . However, the calculated formation energies of V 2+ Te and Te 2+ Cd are very close. Removing the image charge correction will make Te Cd a negative-U center and cause the Fermi level to be pinned by V 2+ Te and V 2− Cd . Despite this uncertainty, it is clear that Te Cd is an important donor when the growth condition is close to the Te-rich limit, while V Te is important under both Te-rich and Cdrich conditions (see figure 8). Regardless of whether Te Cd or V Te is the more important donor under Te-rich conditions, the deep donor level of Te Cd will cause carrier trapping that is harmful to the detector performance. It is better to use shallow donor doping to pin the Fermi level at a somewhat higher energy than that determined by native defects, because this will significantly reduce the fraction of the ionized Te Cd that traps electrons.
Te Cd -V Cd is a deep acceptor and is also a negative-U center with its (0/2−) level located at E v + 1.02 eV (see figure 8). This defect may be related to the experimentally observed deep acceptor levels near the midgap [27]. But Te Cd -V Cd has a high formation energy and thus should not play a significant role in carrier compensation.

Conclusion
Understanding the carrier compensation mechanism and effective defect management is of the utmost importance in the use of semiconductors for technological applications. Here, we have discussed the case of highly resistive CdTe and elaborated on the different compensation mechanisms that may allow Fermi level pinning near midgap. We have showed that the carrier compensation between shallow donors (i.e. In, Cl) and acceptors (i.e. Cd vacancies, A centers) can explain the observed high resistivity and the associated shallow donor concentrations in CdTe. The carrier compensation mechanisms in Cl-doped, In-doped and undoped CdTe may be summarized as follows. (i) In CdTe:Cl, the high resistivity is robust and may be obtained with a large variation of [Cl] (10 16 -10 19 cm −3 ). The Fermi level is pinned near midgap by Cl Te and V Cd at relatively low [Cl] (e.g. 10 16 cm −3 ) and by Cl Te and the A center at relatively high [Cl] (e.g. 10 19 cm −3 ) (see figure 3(c) for an example). The self-compensation between Cl Te and its A center ensures a fixed Fermi level when [Cl] is high, up to the Cl solubility. (ii) In CdTe:In, high resistivity may be obtained with [In] ranging from 10 16 -10 17 cm −3 . The Fermi level is pinned near midgap by In Cd and V Cd in semi-insulating CdTe. The [In] range of 10 16 -10 17 cm −3 is large enough to tolerate uncertainties in the doping level during crystal growth. (iii) In undoped CdTe, the high resistivity can only be achieved if the stoichiometry can be carefully controlled.
Among many room-temperature semiconductor radiation detector materials, the best electron µτ product has been reported to be better than 10 −3 cm 2 V −1 ; e.g. in CdTe and TlBr [67,68]. Our calculations on both CdTe and TlBr show that shallow donors and acceptors, rather than deep ones, pin the Fermi level near midgap [47]. This makes considerable sense because the deep centers will reduce the electron µ product significantly and thus cannot be present in large quantities in high-quality detector-grade CdTe and TlBr. These results suggest that, although deep centers are very effective in pinning the Fermi level and increase resistivity, one should seek proper shallow donors and shallow acceptors and optimize their concentrations to obtain good carrier compensation. The deep centers should be avoided in semiconductor radiation detectors.
The general picture about carrier statistics, resistivity and the compensation mechanisms revealed for CdTe in this paper may be explored in other semiconductor materials for useful applications.