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Work function and energy levels of positrons in elemental semiconductors
Materials Chemistry and Physics 44 (1996) 267-272
Work function and energy levels of positrons in elemental semiconductors N. Bouarissa a, B. Abbar b, J.P. Dufour c, N. Amrane b, H. Aourag b,* aPhysics Department, University OJ Wetly,Setif 19000, Algeria b Computational Materials Science Laboratory, Physics Department, University of Sidi-Bel-Abbes, Sldi-Bel-Abbes 22000, Algeria ’ Laboratoire de Physique du Solide, Universitt de Bourgogne, Dqon 21000, France Received 14 November 1994; accepted 13 September 1995
Abstract We have studied the behaviour of the positron in diamond, silicon and germanium by calculating its energy levels at different points of the reciprocal space using the pseudopotential approach coupled with the independent particle approximation. These energies determine quantities, such as the positron and positronium work functions and the deformation potentials, which are important parameters in slow-positron-beam experiments. WC have tentatively estimated the positron diffusion constant in these semiconductors. The results are compared to values extracted from experiments. Kqvwords: Positrons;
Elemental
semiconductors;
Work functions
1. Introduction The slow-positron-beam technique is a powerful new tool for studying solid surfaces and defect profiles as a function of the distance from the surface [I]. The positron slowing-down process and the ensuing implantation profile can be readily understood on the basis of the Monte Carlo simulations [2]. The positron work function is of interest in the study of the behaviour of positrons at surfaces [3], and the positron deformation potential is related to positron diffusion in metals and semiconductors [4]. The experimental measurements of these quantities can also be used as a test of theoretical calculations of the electron and positron chemical potentials and surface dipole potential of materials. The purpose of the present work is to highlight positron diffusion and surface emission in C, Si and Ge. The electron and positron band structure, and in particular the bottom of the lowest-energy band, are calculated. The positron and electron energy levels in perfect
* Corresponding
author.
0254-0584/96/$15.00 0 1996 Elsevier Science S.A. All rights reserved .SSDT0254-0584(95)01682-6
bulk germanium give the positronium work function. If the deformation potential and the electron and positron chemical potentials are known, the positron work function does not, therefore, involve the surface dipole potential, which, being a surface property, is more difficult to calculate accurately. Moreover, the volume dependences of these levels can be used to estimate the positron diffusion constant. The basis of the present calculations is the empirical pseudopotential method (EPM) [5]. It is first used in determining the electron structure. The calculation of the delocalized positron Bloch states depends on the independent particle model (IPM). In practice, the electron and positron states are calculated by the linear-muffin-tinorbital (LMTO) method within the atomic spheres approximation (ASA) [6,7] . The LMTO-ASA is one of most efficient methods for electron-structure calculations. It has also been used for positron states in bulk metals in order to detemine the momentum distribution of the annihilating positron-electron pairs [8]. However, those calculations are technically difficult and computationally time consuming. On the other hand, some other calculations of the positron states in semiconductors, based on the density functional theory (DFT) [9],
268
N. Bouarissa et al. /Materials
Chemistry and Physics 44 (1996) 267-272
were used. It is well known that electronic band structures of semiconductors based on the DFT underestimate the band gaps by as much as 50-100%. The empirical methods [lo], while simple in nature, and with the drawback that a large number of fitting parameters are required, are at present probably the most accurate way of calculating the band gaps, and produce energy levels that are in good agreement with measured values. We remark, at this point, that while a positron in a solid state is a part of the system with important many-body interactions, the quantum independent particle model is often very useful. In particular, in elemental semiconductors [ 1 1- 131, at least at low temperature, particle-phonon interactions (which are the main scattering processes in high purity materials) are weak [ 141. In this situation, it is assumed that the positron essentially retains its IPM character and that it propagates in Bloch states, coupled only to a small lattice distortion [ 14,151. These factors have prompted us to make such calculations, based on the EPM coupled with the IPM, in diamond, silicon and germanium.
2. Calculations
6 = {Xci,n[AE”J)]2/(m - N)}n2
should be minimal. AE(‘,j) = Et;;’ _ EF;i,’ where Ef$ and E$l’,’ are the observed and calculated LSs between the ith state at the wave vetor k = ki and the jth at k = kj, respectively, in the m chosen pairs (i,j). N is the number of EP parameters. The calculated energies given by solving the EP secular equation depend nonlinearly on the EP parameters. The starting values of the parameters are improved step by step by iterations until 6 is minimized. Let us denote the parameters by P, (u = 1,2, . . . , N) and write them as P,(n + 1) + AP,, where P,(n) is the value at the nth iteration. These corrections AP, are determined simultaneously by solving a system of linear equations
=Z&,[E:;;’ - E:t&)](Q:t
- e’,,)
u’ = 1, 2, . , , N (2)
where E~;ic)(n) is the value at the nth iteration, and QU is given by e’, = 2’4,,,[C~(ki)I*(aH(ki)/aP”),,,C~,(ki)
The electronic structure of Ge has been calculated using the empirical pseudopotential method (EPM). Briefly, the fundamental concept involved in pseudopotential calculation is that the ion core can be omitted. Computationally this is crucial for it means that the deep ion core has been removed and a simple planewave basis will yield rapid convergence. Let us define our empirical pseudopotential (EP) parameters of a semiconductor as a superposition of the pseudo-atomic potential of the form V(r) = V,(r) + V,,(r), where V,_and VN,_are local and nonlocal parts, respectively. It has been shown [5] that the local approach is sufficient to describe with great precision the different experimental energy levels. However, for the calculations of the exact charge densities (the accuracy of the wave functions), the local form is not sufficient and the inclusion of the nonlocal part is therefore necessary. Since we are mainly concerned with the calculations of the energy spacings, we have omitted the nonlocal part in our calculations. We regard the Fourier components of V,(r) as the EP local parameters. We detemine the EP parameters by the nonlinear least squares method, in which all the parameters are simultaneously optimized under a defined criterion of minimizing the root-mean square (rms) deviation. The data for the experimental electronic band structure at normal and low pressure are used. Our nonlinear least squares method requires that the rms deviation of the calculated level spacings (LSs) from the experimental ones, defined by
(1)
(3)
H(ki) is the pseudo-Hamiltonian matrix at k = ki in the plane wave representation, and the ith pseudo-wave function at k = ki is expanded as $:i(~) = X,Ci(ki)
exp[i(k, + k,)r]
(4)
k, being the reciprocal lattice vector. Eq. (2) shows that all of the parameters are determined automatically in an interdependent way. We follow the approach of Aourag et al. [ 16,171 for evaluating the positron wave function. With the assumption that there is only one positron for many electrons, there is no exchange part because there is no positron-positron interaction. The positron potential is purely Coulombic in nature. There is a repulsive ion core potential and an attractive Hartree potential. In addition to these two there is a third part which comes from the electron-positron correlation. The total positron potential can thus be expressed as v,(r) = K(r) + v,(r) + v,,(r)
(5)
where Vi(r), V,(r) and V,,(r) are the ionic, Coulomb, and electron-positron correlation potentials, respectively. The electron-positron potential is a slow function of the electron density. It is generally flat in the interstitial region swamped by the Vi(r) and V,(r) in the ion core region. Hence, it is not considered here. Since v(r) is periodic and the diamond structure involves non-primitive lattice translations it is expressed as K(r) = %, Rj CR, o;(r - Rn - Rj”)
(6)
N. Bouarissa et al. /Materials
Chemistry and Physics 44 (1996) 267-272
269
Vacuum
Crystal
Fermi
Bottom
YJJ/
zero
level
Bottom of the lowest band
: JL
of valence band
Potential
Potential
(b)
(a)
Fig. 1. Schematic view of (a) the electron energy levels and (b) the positron energy levels, in a solid. A is the dipole potential on the surface. VB is the width of the valence band, & and C#I + are the electron and positron work functions, respectively, p_ and p+ are the electron and positron chemical potentials, respectively.
where RR denotes the set of all Bravais lattice vectors and Rja is a non-primitive vector of a two-atom basis. In the point core approximation V:(r) = Z,e’/r
(7)
(2, is the number of valence electrons). On the other hand, the electron-positron potential is expressed as V,(r) = -2
s
[p(r’)/lr -r’(]d3r
Coulomb
(8)
The solution does not require a large number of plane waves because it has no wiggles in the ion core region. We can represent the wave function by plane waves. The wave function for the positron becomes Y+(r) = (l/Q)“’
& A(G)eiG’
(9)
The coefficients ,4(G) are found by solving the secular equation for the positron. The important energy levels for an electron and a positron in a solid are shown schematically in Fig. 1. The potential is fixed to the crystal zero. It is essential to note that the crystal zero is common for electrons and positrons. The thermalized positrons are near the bottom of the lowest energy band at k = 0. The surface dipole potential for positrons is equal in magnitude to that for electrons but opposite in sign. Therefore the vacuum level is below the crystal zero. Usually, the bottom of the lowest band is rather near the vacuum level. If the vacuum level is the lower one the positron work function 4, is negative, as in Fig. 1. The positron chemical potential p+ is defined as the difference between the bottom of the lowest band and the crystal zero level and is usually negative. If the solid is a heterostructure formed by different materials, the requirement that the Fermi level is the same everywhere in the solid determines the heights of the dipole steps at the interfaces. The differences in the
positron affinities for different layers of the structure (band offsets) can therefore be defined as the distances between the bulk positron energy levels. For the materials A and B in contact, the positron energy difference is thus AE~,B=E~-EB,=~A-~~$~LP;-~~
(10)
This equation can also be used for predicting the affinity of positrons for precipitate alloys [ 181. The important quantities measured in slow-positronbeam experiments are closely related to the energy levels described above. The most directly related quantity is the maximum kinetic energy of positronium (Ps) atoms ejected into a vacuum from the sample. The negative of this energy is called the Ps work function 4%. The work function $_ of a semiconductor can be expressed as [19] &=A-p_
(11)
where d is the surface dipole potential barrier against electron escapes, and ,u_ is the internal electron chemical potential. Since the positron experiences just the negative of this potential barrier upon emission from the surface, so its work function is given by 4, = -A --p?
(12)
The emission of the Ps atom from the surface can also be described as $+=$++4--0.5Ry
(13)
which is less than the positron work function because the electron work function is usually less than 0.5 Ry. The internal electron chemical potential p_ and the positron chemical potential p+ are purely bulk properties that can be obtained from a band structure calculation. The quantity ,L_ is equal to the Fermi energy and p+ is the lowest energy of the positron energy band.
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270
et al. /Materials
Chemistry and Physics 44 (1996) 267-272
Table 1 The calculated form factors V(G’), and the calculated and observed fundamental band gaps in eV for C, Si and Ge V(3) (RY)
C Si Ge
6.740 10.26 10.67
V(8) (RY)
-0.773 -0.257 -0.256
V(ll) (RY)
0.195 -0.203 0.040
f15-f25
0.077 - 0.040 0.032
r25.-x,
r25.-LI
Cal.
Obs.
Cal.
Obs.
Cal.
Obs.
8.64 3.41 1.79
7.73” 3.4b 3.2’
5.74 1.24 1.09
5.43” 1.15b 1.4’
4.84 2.11 0.79
5.20” 2.4b 0.84’
a From Ref. [27]. b From Ref. [28]. ’ From Ref. [29].
We may eliminate A from Eq. (11) by means of the electron work function $_ as qb+=-p+-A=-/L+-p--4_
(14)
In this work we will use the experimental values for 4_. in order to determine 4,. The deformation potential model [ 191 is used in the treatment of the contribution to the relaxation time by positron-acoustic-phonon scattering, which is a quantity that is needed to calculate the positron diffu-
sion constant [20]. The deformation potential Ed+ is defined as Q aE+la!d, where E+(Q) is the energy of the lowest positron Bloch state at the crystal volume Q, provided we take the local variation of the electron density into account. Bergersen et al. [4] express Ed’ as the sum of three contributions. The first two come from the zero-point energy E, and the electron-positron correlation energy EC,,, . In terms of these quantities, the positron chemical potential is given by The third contribution to the deformap+ = E, + EC,,,. tion potential comes from a charge transfer between regions of different density. This generates an electrostatic potential equivalent to a surface dipole term that compensates the shift in p_ due to crystal dilation, therefore maintaining a constant electron chemical potential throughout the crystal. Consequently, we can calculate the deformation potential from Ed’ = Q d/dSJ(p+ + CL_)= -a
(4
r
xw
L
l-
K
x
waveVector
&#&X2
(15)
The deformation potential and the Ps work function depend on p+ + p_, and therefore do not involve the surface dipole potential A, which, being a surface property, is more difficult to calculate accurately. The volume derivatives of the chemical potentials are determined by performing EPM and IPM electron structure and positron calculations respectively, for a few slightly different lattice constants. The deformation potential theory gives the positron diffusion constant due to acoustic-phonon scattering as D, = [87~/9]“~(h~(c~~))/[m*~‘~(k~T)~‘~Ed+~]
(16)
Table 2 Calculated chemical potentials and their derivatives for electrons and positrons in eV compared with those of the LDA calculation [9] P-
Fig. 2. (a) Electronic band structure, and (b) positronic band structure, of Ge.
C Si Ge
P+
V?k
V%
av
av
Cal.
LDA
Cal.
LDA
Cal.
LDA
Cal. LDA
-3.0 -1.7 -0.5
-0.54 -0.60
-6.0 -6.51 -4.93
-6.41 -6.19
-11.22 -7.93 -7.93
-7.51 -7.64
1.77 3.34 +1.32 1.16 +I.02
N. Bouarissa et al. 1 Materials Chemistry and Physics 44 (1996) 267-272
Table 3 Comparison of the theoretical Ps and positron work functions in eV with the experimental ones [22,23], and with those of the LDA calculation [9]
Table 4 Positron diffusion constants at 300 K, the deformation potential Ei, and the average elastic constant (cii) calculated with some experimental data and other calculated values Ed’ (eV)
C Si Ge
+ 10.0 +4.14 +4.80
-0.49 +2.37 +0.63
+2.21 +1.98
>o -1
+2.71 +0.31 - 1.37
-0.02 0.15
where m* is the positron effective mass, T the absolute temperature, and (cij ) the elastic constant associated with longitudinal waves averaged over the directions of propagation. The calculation of the (ci, ) is a complicated numerical task and therefore we approximate it for the ( 110) plane by (Cii > = ( 1/2)(c, 1 +
Cl2
+
h‘d
(17)
3. Results The calculated electron and positron band structures in Ge are shown in Fig. 2. The positron band structure is rather free-particle-like and resembles closely the lowest valence electron band. Also, the positron effective mass at point I is near the free particle mass m* < 1.1. The lattice constant, the pseudopotential form factors used, and the calculated and measured principal energy levels for C, Si and Ge are shown in Table 1. A good agreement is observed between the calculated and measured energy gaps. The calculated chemical potentials for electrons and positrons are shown in Table 2. The electron chemical potential, i.e., the position of the Fermi level, depends on the details of the electron band structure. The values of the electron and the positron chemical potentials and their pressure derivatives shown in Table 2 are in agreement with the results of Boev et al. [9]. From the calculated chemical potentials the Ps work function bps can be directly deduced via Eq. (13). The values obtained are shown in Table 3 and compared with the available experimental data from Ps time-offlight measurement [ 2 11. It is interesting to note that the theory predicts a slightly positive Ps work function for Si. This theoretical value is calculated with the electron chemical potential coinciding with the top of the valence band; the Ps work function becomes negative. Thus, Ps emission should be sensitive to doping. Also in Table 3, the theoretical positron work function, calculated by means of the experimental electron work function via Eq. (14), is given. Our results are compared with the experimental ones measured directly in the slow-positron-beam experiments [22,23]. The theory again predicts that the
211
(Ci, > (Mbar)
Cal.
LDA
C Si Ge
-9.45 -4.49 -6.71
-6.19 -6.62
a From ’ From ’ From d From
Ref. Ref. Ref. Ref.
11.82d 1.67d 1.58d
D, (cm2 s-‘) Cal.
LDA
Exp.
7.95 4.99 2.07
3.05 2.13
2.7”, 1.56b 0.5’, 0.55b
[24]. [22]. [25]. [26].
positron work function is clearly positive whereas the experimental one is - 1.OeV. Si may be a difficult case for the slow-positron-beam experiment due to the changing of the surface causing an extra field; and impurities in the surface may also hinder the accurate determination of the work function. In the case of germanium the experiment predicts that the positron work function is positive only, and this is consistent with our values. However, our result seems to be lower than the value obtained through LDA calculation [9]. In the absence of accurate experimentation, no comment could be ascribed to these discrepancies. In Table 3 the first calculated positron and Ps work functions for diamond are also presented. The chemical potentials shown in Table 2 can be directly used in estimating the positron affinity different materials in heterostructures. For example, for Si and Ge in contact, the positron favors Si with the positron energy difference AEF ” = 0.72 eV The lower chemical potential for electrons in Si than in Ge reflects the smaller d-band width of Si. The calculated positron deformation potential and diffusion constants are shown in Table 4. The deformation potential for Ge with 0.5 eV agrees with the value estimated by Boev et al. [9], whereas large discrepancies are seen for Si; we may attribute this to the fact that in the LDA formalism of Boev et al. [9], the Fermi level is underestimated in comparison with EPM calculations. The deformation potential and diffusion constants agree within 1 eV with those previously estimated by Boev et al. [9]. The positron diffusion constants are calculated using the deformation potential model via Eq. (16). The value of 1.5 was used for the positron effective mass m* for all hosts. However, the calculated magnitudes of the diffusion constants are larger than the experimental ones. The larger value for Si and the smaller value for Ge are from mobility measurement
N. Bouarissa et al.
212
1Materials Chemistry and Physics 44 (1996) 267-272
[22], in which a bias electric field is applied across a semiconductor detector. However, we are tempted to believe that the difference between the calculated values and the measured ones could be due essentially to the uncertainty in the effective mass. We may conclude on the basis of the overall trends that theory and experiment are in qualitative agreement, and that the diffusion constant increases as the gap of the semiconductors increases (Ge, Si and C). In conclusion, the scheme presented, being simple and capable of predicting quantities directly measurable by the slow-beam technique, is a useful tool to be employed as a support to the experiments.
References [II W. Brandt and A. Dupasquier (ed.), Positron Solid State Physics, North-Holland, Amsterdam, 1983. I21 S. Valkealahti and R.M. Nieminen, Appl. Phys. A, 35 (1984) 51. [31 R.M. Nieminen and C.H. Hodges, Solid State Commun., 18 (1976) 1111. I41 B. Bergersen, E. Pajanne, P. Kubica, M.J. Stott and C.H. Hodges, Solid State Commun., 15 (1974) 1377. [51 J.R. Chelikowsky and M.L. Cohen, Phys. Rev. B., 14 (1976) 376.
161 O.K. Andersen, 0. Jepsen and D. GlBtzel, in F. Bassani, F. Fumi and M.P. Tosi (eds.), Highlights of Condensed-Matter Theory, North-Holland, Amsterdam, 1985. [71 H.C. Skriver, The LMTO Method, Springer, New York, 1983. PI A.K. Singh and T. Jarlborg, J. Phys. F, 15 (1985) 727. [91 O.V. Boev, R.M. Nieminen and M.J. Puska, Phys. Reo. B, 36 (1987) 7786.
[lOI Y.F. Tsay, S.S. Mitra and B. Bendow, Phys. Rev. B, 10 (1974) 1476. [Ill C. Pannetta, Solid State Commun., 77 (1991) 159. [ 121 H. Aourag, A. Belaidi, T. Kobayasi, R.N. West and B. Khelifa, Phys. Status Solidi B, 156 (1989) 497.
iI31 H. Aourag, A. Belaidi, T. Kobayasi, R.N. West and B. Khelifa, Phys. Status Solidi B, 15.5 (1989) 191.
iI41 A. Seeger, in P.C. Jain, R.M. Singru and K.P. Gopianathan (eds.), Positron Annihilation, World Scientific, Singapore, 1985, p. 48. r151 T. McMullen, in P.C. Jain, R.M. Singru and K.P. Gopianathan (eds.), Positron Annihilation, World Scientific, Singapore, 1985, p. 657. [W H. Aourag, B. Khelifa, A. Belaidi and Z. Belarbi, Phys. Status Solidi B, 160 (1990) 201.
[I71 H. Aourag, B. Soudini, B. Khelifa and A. Belaidi, Phys. Status Solidi B, 161 (1990) 685.
M.J. Stott and P. Kubica, Phys. Rev. B, 11 (1975) 1. C.H. Hodges and M.J. Stott, Phys. Reu. B, 7 (1973) 73. J. Bardeen and W. Schokley, Phys. Rev., 80 (1950) 72. R.H. Howell, I.J. Rosenberg, M.J. Fluss, R.E. Goldberg and R.B. Laughlin, Phys. Rev. B, 35 (1987) 5303. PI A.P. Mills, in W. Brandt and A. Dupasquier (eds.), Positron Solid State Physics, North-Holland, Amsterdam, 1983, p. 432. [231 H. Huomo, A. Vehanen, M.D. Bentzon and P. Hautojlrvi, iI81 [I91 WI Pll
Phys. Rev. B, 35 (1987) 8252.
1241 B. Nielsen, K.G. Lynn, A. Vehanen and P.J. Schultz, Phys. Rev. B, 32 (1985) 2296.
[251 H. H. Jorch, K.G. Lynn and I.K. MacKenzie, Phys. Rev. Let?., 47 (1981) 362.
WI H.J. McSkimin and P. Andreatch, Jr., J. Appl. Phys., 35 (1969) 3312. [271 F.J. Himpsel, J.F. Van Der Veen and D.E. Eastman, Phys. Rev. B, 22 (1980) 1967. WI B.O. Seraphin and N. Bottka, Phys. Rev. Lett., 15 (1965) 104. v91 J. Fisher, Proc. 10th Conf Semi. and Phys., Cambridge, MA, USA. 1970, p. 427.
Work function and energy levels of positrons in elemental semiconductors N. Bouarissa a, B. Abbar b, J.P. Dufour c, N. Amrane b, H. Aourag b,* aPhysics Department, University OJ Wetly,Setif 19000, Algeria b Computational Materials Science Laboratory, Physics Department, University of Sidi-Bel-Abbes, Sldi-Bel-Abbes 22000, Algeria ’ Laboratoire de Physique du Solide, Universitt de Bourgogne, Dqon 21000, France Received 14 November 1994; accepted 13 September 1995
Abstract We have studied the behaviour of the positron in diamond, silicon and germanium by calculating its energy levels at different points of the reciprocal space using the pseudopotential approach coupled with the independent particle approximation. These energies determine quantities, such as the positron and positronium work functions and the deformation potentials, which are important parameters in slow-positron-beam experiments. WC have tentatively estimated the positron diffusion constant in these semiconductors. The results are compared to values extracted from experiments. Kqvwords: Positrons;
Elemental
semiconductors;
Work functions
1. Introduction The slow-positron-beam technique is a powerful new tool for studying solid surfaces and defect profiles as a function of the distance from the surface [I]. The positron slowing-down process and the ensuing implantation profile can be readily understood on the basis of the Monte Carlo simulations [2]. The positron work function is of interest in the study of the behaviour of positrons at surfaces [3], and the positron deformation potential is related to positron diffusion in metals and semiconductors [4]. The experimental measurements of these quantities can also be used as a test of theoretical calculations of the electron and positron chemical potentials and surface dipole potential of materials. The purpose of the present work is to highlight positron diffusion and surface emission in C, Si and Ge. The electron and positron band structure, and in particular the bottom of the lowest-energy band, are calculated. The positron and electron energy levels in perfect
* Corresponding
author.
0254-0584/96/$15.00 0 1996 Elsevier Science S.A. All rights reserved .SSDT0254-0584(95)01682-6
bulk germanium give the positronium work function. If the deformation potential and the electron and positron chemical potentials are known, the positron work function does not, therefore, involve the surface dipole potential, which, being a surface property, is more difficult to calculate accurately. Moreover, the volume dependences of these levels can be used to estimate the positron diffusion constant. The basis of the present calculations is the empirical pseudopotential method (EPM) [5]. It is first used in determining the electron structure. The calculation of the delocalized positron Bloch states depends on the independent particle model (IPM). In practice, the electron and positron states are calculated by the linear-muffin-tinorbital (LMTO) method within the atomic spheres approximation (ASA) [6,7] . The LMTO-ASA is one of most efficient methods for electron-structure calculations. It has also been used for positron states in bulk metals in order to detemine the momentum distribution of the annihilating positron-electron pairs [8]. However, those calculations are technically difficult and computationally time consuming. On the other hand, some other calculations of the positron states in semiconductors, based on the density functional theory (DFT) [9],
268
N. Bouarissa et al. /Materials
Chemistry and Physics 44 (1996) 267-272
were used. It is well known that electronic band structures of semiconductors based on the DFT underestimate the band gaps by as much as 50-100%. The empirical methods [lo], while simple in nature, and with the drawback that a large number of fitting parameters are required, are at present probably the most accurate way of calculating the band gaps, and produce energy levels that are in good agreement with measured values. We remark, at this point, that while a positron in a solid state is a part of the system with important many-body interactions, the quantum independent particle model is often very useful. In particular, in elemental semiconductors [ 1 1- 131, at least at low temperature, particle-phonon interactions (which are the main scattering processes in high purity materials) are weak [ 141. In this situation, it is assumed that the positron essentially retains its IPM character and that it propagates in Bloch states, coupled only to a small lattice distortion [ 14,151. These factors have prompted us to make such calculations, based on the EPM coupled with the IPM, in diamond, silicon and germanium.
2. Calculations
6 = {Xci,n[AE”J)]2/(m - N)}n2
should be minimal. AE(‘,j) = Et;;’ _ EF;i,’ where Ef$ and E$l’,’ are the observed and calculated LSs between the ith state at the wave vetor k = ki and the jth at k = kj, respectively, in the m chosen pairs (i,j). N is the number of EP parameters. The calculated energies given by solving the EP secular equation depend nonlinearly on the EP parameters. The starting values of the parameters are improved step by step by iterations until 6 is minimized. Let us denote the parameters by P, (u = 1,2, . . . , N) and write them as P,(n + 1) + AP,, where P,(n) is the value at the nth iteration. These corrections AP, are determined simultaneously by solving a system of linear equations
=Z&,[E:;;’ - E:t&)](Q:t
- e’,,)
u’ = 1, 2, . , , N (2)
where E~;ic)(n) is the value at the nth iteration, and QU is given by e’, = 2’4,,,[C~(ki)I*(aH(ki)/aP”),,,C~,(ki)
The electronic structure of Ge has been calculated using the empirical pseudopotential method (EPM). Briefly, the fundamental concept involved in pseudopotential calculation is that the ion core can be omitted. Computationally this is crucial for it means that the deep ion core has been removed and a simple planewave basis will yield rapid convergence. Let us define our empirical pseudopotential (EP) parameters of a semiconductor as a superposition of the pseudo-atomic potential of the form V(r) = V,(r) + V,,(r), where V,_and VN,_are local and nonlocal parts, respectively. It has been shown [5] that the local approach is sufficient to describe with great precision the different experimental energy levels. However, for the calculations of the exact charge densities (the accuracy of the wave functions), the local form is not sufficient and the inclusion of the nonlocal part is therefore necessary. Since we are mainly concerned with the calculations of the energy spacings, we have omitted the nonlocal part in our calculations. We regard the Fourier components of V,(r) as the EP local parameters. We detemine the EP parameters by the nonlinear least squares method, in which all the parameters are simultaneously optimized under a defined criterion of minimizing the root-mean square (rms) deviation. The data for the experimental electronic band structure at normal and low pressure are used. Our nonlinear least squares method requires that the rms deviation of the calculated level spacings (LSs) from the experimental ones, defined by
(1)
(3)
H(ki) is the pseudo-Hamiltonian matrix at k = ki in the plane wave representation, and the ith pseudo-wave function at k = ki is expanded as $:i(~) = X,Ci(ki)
exp[i(k, + k,)r]
(4)
k, being the reciprocal lattice vector. Eq. (2) shows that all of the parameters are determined automatically in an interdependent way. We follow the approach of Aourag et al. [ 16,171 for evaluating the positron wave function. With the assumption that there is only one positron for many electrons, there is no exchange part because there is no positron-positron interaction. The positron potential is purely Coulombic in nature. There is a repulsive ion core potential and an attractive Hartree potential. In addition to these two there is a third part which comes from the electron-positron correlation. The total positron potential can thus be expressed as v,(r) = K(r) + v,(r) + v,,(r)
(5)
where Vi(r), V,(r) and V,,(r) are the ionic, Coulomb, and electron-positron correlation potentials, respectively. The electron-positron potential is a slow function of the electron density. It is generally flat in the interstitial region swamped by the Vi(r) and V,(r) in the ion core region. Hence, it is not considered here. Since v(r) is periodic and the diamond structure involves non-primitive lattice translations it is expressed as K(r) = %, Rj CR, o;(r - Rn - Rj”)
(6)
N. Bouarissa et al. /Materials
Chemistry and Physics 44 (1996) 267-272
269
Vacuum
Crystal
Fermi
Bottom
YJJ/
zero
level
Bottom of the lowest band
: JL
of valence band
Potential
Potential
(b)
(a)
Fig. 1. Schematic view of (a) the electron energy levels and (b) the positron energy levels, in a solid. A is the dipole potential on the surface. VB is the width of the valence band, & and C#I + are the electron and positron work functions, respectively, p_ and p+ are the electron and positron chemical potentials, respectively.
where RR denotes the set of all Bravais lattice vectors and Rja is a non-primitive vector of a two-atom basis. In the point core approximation V:(r) = Z,e’/r
(7)
(2, is the number of valence electrons). On the other hand, the electron-positron potential is expressed as V,(r) = -2
s
[p(r’)/lr -r’(]d3r
Coulomb
(8)
The solution does not require a large number of plane waves because it has no wiggles in the ion core region. We can represent the wave function by plane waves. The wave function for the positron becomes Y+(r) = (l/Q)“’
& A(G)eiG’
(9)
The coefficients ,4(G) are found by solving the secular equation for the positron. The important energy levels for an electron and a positron in a solid are shown schematically in Fig. 1. The potential is fixed to the crystal zero. It is essential to note that the crystal zero is common for electrons and positrons. The thermalized positrons are near the bottom of the lowest energy band at k = 0. The surface dipole potential for positrons is equal in magnitude to that for electrons but opposite in sign. Therefore the vacuum level is below the crystal zero. Usually, the bottom of the lowest band is rather near the vacuum level. If the vacuum level is the lower one the positron work function 4, is negative, as in Fig. 1. The positron chemical potential p+ is defined as the difference between the bottom of the lowest band and the crystal zero level and is usually negative. If the solid is a heterostructure formed by different materials, the requirement that the Fermi level is the same everywhere in the solid determines the heights of the dipole steps at the interfaces. The differences in the
positron affinities for different layers of the structure (band offsets) can therefore be defined as the distances between the bulk positron energy levels. For the materials A and B in contact, the positron energy difference is thus AE~,B=E~-EB,=~A-~~$~LP;-~~
(10)
This equation can also be used for predicting the affinity of positrons for precipitate alloys [ 181. The important quantities measured in slow-positronbeam experiments are closely related to the energy levels described above. The most directly related quantity is the maximum kinetic energy of positronium (Ps) atoms ejected into a vacuum from the sample. The negative of this energy is called the Ps work function 4%. The work function $_ of a semiconductor can be expressed as [19] &=A-p_
(11)
where d is the surface dipole potential barrier against electron escapes, and ,u_ is the internal electron chemical potential. Since the positron experiences just the negative of this potential barrier upon emission from the surface, so its work function is given by 4, = -A --p?
(12)
The emission of the Ps atom from the surface can also be described as $+=$++4--0.5Ry
(13)
which is less than the positron work function because the electron work function is usually less than 0.5 Ry. The internal electron chemical potential p_ and the positron chemical potential p+ are purely bulk properties that can be obtained from a band structure calculation. The quantity ,L_ is equal to the Fermi energy and p+ is the lowest energy of the positron energy band.
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et al. /Materials
Chemistry and Physics 44 (1996) 267-272
Table 1 The calculated form factors V(G’), and the calculated and observed fundamental band gaps in eV for C, Si and Ge V(3) (RY)
C Si Ge
6.740 10.26 10.67
V(8) (RY)
-0.773 -0.257 -0.256
V(ll) (RY)
0.195 -0.203 0.040
f15-f25
0.077 - 0.040 0.032
r25.-x,
r25.-LI
Cal.
Obs.
Cal.
Obs.
Cal.
Obs.
8.64 3.41 1.79
7.73” 3.4b 3.2’
5.74 1.24 1.09
5.43” 1.15b 1.4’
4.84 2.11 0.79
5.20” 2.4b 0.84’
a From Ref. [27]. b From Ref. [28]. ’ From Ref. [29].
We may eliminate A from Eq. (11) by means of the electron work function $_ as qb+=-p+-A=-/L+-p--4_
(14)
In this work we will use the experimental values for 4_. in order to determine 4,. The deformation potential model [ 191 is used in the treatment of the contribution to the relaxation time by positron-acoustic-phonon scattering, which is a quantity that is needed to calculate the positron diffu-
sion constant [20]. The deformation potential Ed+ is defined as Q aE+la!d, where E+(Q) is the energy of the lowest positron Bloch state at the crystal volume Q, provided we take the local variation of the electron density into account. Bergersen et al. [4] express Ed’ as the sum of three contributions. The first two come from the zero-point energy E, and the electron-positron correlation energy EC,,, . In terms of these quantities, the positron chemical potential is given by The third contribution to the deformap+ = E, + EC,,,. tion potential comes from a charge transfer between regions of different density. This generates an electrostatic potential equivalent to a surface dipole term that compensates the shift in p_ due to crystal dilation, therefore maintaining a constant electron chemical potential throughout the crystal. Consequently, we can calculate the deformation potential from Ed’ = Q d/dSJ(p+ + CL_)= -a
(4
r
xw
L
l-
K
x
waveVector
&#&X2
(15)
The deformation potential and the Ps work function depend on p+ + p_, and therefore do not involve the surface dipole potential A, which, being a surface property, is more difficult to calculate accurately. The volume derivatives of the chemical potentials are determined by performing EPM and IPM electron structure and positron calculations respectively, for a few slightly different lattice constants. The deformation potential theory gives the positron diffusion constant due to acoustic-phonon scattering as D, = [87~/9]“~(h~(c~~))/[m*~‘~(k~T)~‘~Ed+~]
(16)
Table 2 Calculated chemical potentials and their derivatives for electrons and positrons in eV compared with those of the LDA calculation [9] P-
Fig. 2. (a) Electronic band structure, and (b) positronic band structure, of Ge.
C Si Ge
P+
V?k
V%
av
av
Cal.
LDA
Cal.
LDA
Cal.
LDA
Cal. LDA
-3.0 -1.7 -0.5
-0.54 -0.60
-6.0 -6.51 -4.93
-6.41 -6.19
-11.22 -7.93 -7.93
-7.51 -7.64
1.77 3.34 +1.32 1.16 +I.02
N. Bouarissa et al. 1 Materials Chemistry and Physics 44 (1996) 267-272
Table 3 Comparison of the theoretical Ps and positron work functions in eV with the experimental ones [22,23], and with those of the LDA calculation [9]
Table 4 Positron diffusion constants at 300 K, the deformation potential Ei, and the average elastic constant (cii) calculated with some experimental data and other calculated values Ed’ (eV)
C Si Ge
+ 10.0 +4.14 +4.80
-0.49 +2.37 +0.63
+2.21 +1.98
>o -1
+2.71 +0.31 - 1.37
-0.02 0.15
where m* is the positron effective mass, T the absolute temperature, and (cij ) the elastic constant associated with longitudinal waves averaged over the directions of propagation. The calculation of the (ci, ) is a complicated numerical task and therefore we approximate it for the ( 110) plane by (Cii > = ( 1/2)(c, 1 +
Cl2
+
h‘d
(17)
3. Results The calculated electron and positron band structures in Ge are shown in Fig. 2. The positron band structure is rather free-particle-like and resembles closely the lowest valence electron band. Also, the positron effective mass at point I is near the free particle mass m* < 1.1. The lattice constant, the pseudopotential form factors used, and the calculated and measured principal energy levels for C, Si and Ge are shown in Table 1. A good agreement is observed between the calculated and measured energy gaps. The calculated chemical potentials for electrons and positrons are shown in Table 2. The electron chemical potential, i.e., the position of the Fermi level, depends on the details of the electron band structure. The values of the electron and the positron chemical potentials and their pressure derivatives shown in Table 2 are in agreement with the results of Boev et al. [9]. From the calculated chemical potentials the Ps work function bps can be directly deduced via Eq. (13). The values obtained are shown in Table 3 and compared with the available experimental data from Ps time-offlight measurement [ 2 11. It is interesting to note that the theory predicts a slightly positive Ps work function for Si. This theoretical value is calculated with the electron chemical potential coinciding with the top of the valence band; the Ps work function becomes negative. Thus, Ps emission should be sensitive to doping. Also in Table 3, the theoretical positron work function, calculated by means of the experimental electron work function via Eq. (14), is given. Our results are compared with the experimental ones measured directly in the slow-positron-beam experiments [22,23]. The theory again predicts that the
211
(Ci, > (Mbar)
Cal.
LDA
C Si Ge
-9.45 -4.49 -6.71
-6.19 -6.62
a From ’ From ’ From d From
Ref. Ref. Ref. Ref.
11.82d 1.67d 1.58d
D, (cm2 s-‘) Cal.
LDA
Exp.
7.95 4.99 2.07
3.05 2.13
2.7”, 1.56b 0.5’, 0.55b
[24]. [22]. [25]. [26].
positron work function is clearly positive whereas the experimental one is - 1.OeV. Si may be a difficult case for the slow-positron-beam experiment due to the changing of the surface causing an extra field; and impurities in the surface may also hinder the accurate determination of the work function. In the case of germanium the experiment predicts that the positron work function is positive only, and this is consistent with our values. However, our result seems to be lower than the value obtained through LDA calculation [9]. In the absence of accurate experimentation, no comment could be ascribed to these discrepancies. In Table 3 the first calculated positron and Ps work functions for diamond are also presented. The chemical potentials shown in Table 2 can be directly used in estimating the positron affinity different materials in heterostructures. For example, for Si and Ge in contact, the positron favors Si with the positron energy difference AEF ” = 0.72 eV The lower chemical potential for electrons in Si than in Ge reflects the smaller d-band width of Si. The calculated positron deformation potential and diffusion constants are shown in Table 4. The deformation potential for Ge with 0.5 eV agrees with the value estimated by Boev et al. [9], whereas large discrepancies are seen for Si; we may attribute this to the fact that in the LDA formalism of Boev et al. [9], the Fermi level is underestimated in comparison with EPM calculations. The deformation potential and diffusion constants agree within 1 eV with those previously estimated by Boev et al. [9]. The positron diffusion constants are calculated using the deformation potential model via Eq. (16). The value of 1.5 was used for the positron effective mass m* for all hosts. However, the calculated magnitudes of the diffusion constants are larger than the experimental ones. The larger value for Si and the smaller value for Ge are from mobility measurement
N. Bouarissa et al.
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1Materials Chemistry and Physics 44 (1996) 267-272
[22], in which a bias electric field is applied across a semiconductor detector. However, we are tempted to believe that the difference between the calculated values and the measured ones could be due essentially to the uncertainty in the effective mass. We may conclude on the basis of the overall trends that theory and experiment are in qualitative agreement, and that the diffusion constant increases as the gap of the semiconductors increases (Ge, Si and C). In conclusion, the scheme presented, being simple and capable of predicting quantities directly measurable by the slow-beam technique, is a useful tool to be employed as a support to the experiments.
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