- Email: [email protected]
Dependence of lattice constants and bulk moduli on pseudopotential properties
Volume 100A, number 6
PHYSICS LETTERS
6 February 1984
DEPENDENCE OF LATTICE CONSTANTS AND BULK MODULI ON PSEUDOPOTENTIAL PROPERTIES PUi K. LAM i and Marvin L. COHEN
Department of Physics, University of California, Berkeley, CA 94720, USA and Materials and Molecular Research Division, Lawrence Berkeley Laboratory, Berkeley, CA 94720, USA Received 19 August 1983 Revised manuscript received 5 December 1983
Using a nearly-free-electron model and model pseudopotentials, an analytical relation between the Wigner-Seitz radius for a metal, rws, and the core radius of the model pseudopotential, rc, is derived. In addition, a semi-empiricalformula for the bulk moduli is obtained.
The structural properties of metals are governed by the interactions between the valence electrons and the atomic core (nucleus plus core electrons) and the self-consistent interactions among the valence electrons. It has been demonstrated [1,2] that the local density formalism [3] gives a good description for the self-consistent interactions of the valence electrons for ground state properties of metals. In addition, pseudopotentials [4] adequately describe the interaction between the valence electrons and the core. We explore here the relation of metallic sizes and bulk moduli to pseudopotential properties. The aim of this investigation is to simplify the formalism so that the relationship between structural properties and pseudopotential properties is apparent. We concentrate on simple (sp) metals because the dose-packed structure and the nearly-free electron character of the wave functions for these metals allow simple approximations. A few metals from the beginning of transition metal series are also included in our analysis. Rydberg atomic units will be used throughout, i.e. e 2 -- 2, m = 1/2, and ~ = 1. Starting with a free-electron model and replacing the Wigner-Seitz cell by a sphere, Heine and Weaire [5,6] derived an expression for the crystal energy 1 Present address: National Bureau of Standards, Washington, DC 20234, USA. 0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North.Holland Physics Publishing Division)
using a real space representation. Their expression for the crystal energy is similar to the one given by eq. (3) below. We present an alternate derivation here which starts with the exact expression in momentum space and then apply the approximations. The advantage of this derivation is that it shows dearly which terms are neglected in the model. Within the local density formalism and pseudopotential approach the crystal energy can be written in a momentum space representation [7],
Nk E/atom --'N~ 1 ~ e(k) _ 1~D, a G~O VH(G) p(G) k
+ r~a o~ [exc(G)- rxc(G)] + [~ + Vxc(O)] Z + "/Ewald,
(1)
where N ~ l~l~k e(k) is the mean eigenvalue averaged over all the occupied states; the eigenvalues are calculated with the average potential set to zero [7]. ~a is the atomic volume, VH (G) is the Hartree potential, exc(G) and Vxc(G) are the exchange-correlation energy density and potential, p(G) is the charge density, ct is a pseudopotential term, Z is the number of valence electrons, and ")'Ewald is the Ewald energy. The pseudopotential term, a, is defined to be
293
Volume 100A, number 6
~=K2a 1
PHYSICS LETTERS
f[Vion(r)+2Z/r] d3r-~Vps(O)/K2a,
(2)
where Vion(r) is the ionic pseudopotential. The a term represents only the integrated effect of the pseudopotential. Other effects of the pseudopotential are implicitly contained in [e(k)] and p(G). For metals, the following approximations are made. The charge density is assumed to be uniform, i.e., we neglect all the G 4= 0 terms in p(G), and e(k) is assumed to be given by the free electron dispersion, i.e. e(k) = k 2 . The terms neglected in these approximations are often grouped together and called the band structure term [5]. With the above approximations, the crystal energy is given by
-0.88Z(rws/Z 113 + 7.8) -1 (3)
where rws is the Wigner-Seitz radius defined by 4
3
-~rt rws = ~a"
(4)
The first term in eq. (3) is the eigenvalue or kinetic energy term in this case because the potential is zero, the second term is the exchange term, the third term is the correlation term (Wigner's correlation formula is used [8]), the fourth term is the pseudopotential term, and the last term is the Ewald term. Approximate numerical values have been used for ~ (97r/4) 2/3 2.21, (3/2~r)(9zr/4) 1/3 ~ 0.916, and 3/4~r ~ 0.24 in eq. (3). The first three terms in eq. (3) give the energy for a jeUinm model. The Ewald term accounts for the fact that the ionic cores are discrete entities rather than a smeared out jellium. The pseudopotential term accounts for the f'mite size of the core. The Ewald constant, Fs, for the fcc, bcc, and hcp structures are given by [5] F fcc= 1.79175,
Fsbcc= 1.79186,
Fsbcp= 1.79168.
(5) If the Wigner-Seitz unit cell were replaced by a sphere then the Ewald constant is F ws= 1.8. Since F s for the three metallic structures, fcc, bcc, 294
and hcp, is very close to 1.8, F s = 1.8 is used in our calculation. The first and second derivative of the correlation term with respect to rws are very small compared to the derivatives of the other terms. The contribution to the equilibrium volume and bulk modulus from the correlation term will therefore be neglected. Differentiating eq. (3) to obtain the equilibrium rw0s, one gets
rOs = [2B + (4B 2 + 12AC)l/2]/2A,
(7)
where
A = 0.916Z 4/3 + 1.8Z 2,
B = 2.21Z 5/3,
C = 0.24 Vps(0) Z.
(8)
The term A is the sum of the exchange and the Ewald term, B is the kinetic energy term, and C is the pseudopotential term. For the values we are considering, 1ZAC~ 4B 2,
Z/atom = 2.21ZS/3/r2~ - 0.916Z 4/31rws
+ 0.24 Vps(O)Z/r3s - FsZ2/rws ,
6 February 1984
(6)
rOs ~ B/A + (3C/A) 1/2,
(9)
that is, the equilibrium rw0s is determined by two effects: one from the competition between the kinetic energy and the exchange plus Ewald energy [first term in eq. (9)], and one from the finite size of the core represented by Vps(0) [second term in eq. (9)]. Using the empty core or Ashcroft form for the pseudopotential [9], Vion(r) = 0,
r < rc,
=-2Z/r,
r>~rc,
(10)
the pseudopotential term, Vps(0), is given by Vps(0) =
.
(11)
Substituting this value of Vps(0) in eqs. (8) and then (7) gives
rOs ~ B(Z)/A (Z) + {3Z/[A (Z)] 1/2 } rc "
(12)
i.e. there is linear relation between rOs and r c. In fig. 1, the dependence of r°s on r e given by eq. (7) is shown. This linear relation between metallic sizes and pseudopotential core radii has been empirically correlated [10]. An interesting feature to be noted is that the slopes of rOs versus r c are almost independent of Z (fig. 1). The linear relation is obtained because Vps(0) o: r 2. This fact is not a property peculiar only to the empty core potential. The pseudopotential
Volume 100A, number 6
PHYSICS LETTERS
6 February 1984
Table 1 Ratio ofBc0 to B°e.
6
4 ¢O
Z=1
~n t.,
o o ac/ae
y
Li Na K Rb Cs
1.23 1.03 0.99 0.79 0.91
0.99 ± 0.16
1.0
Be Mg Ca Zn Sr Cd
1.61 1.30 1.39 1.29 1.31 1.08
1.31 ± 0.15
1.3
Ba
1.29
Hg
1.19
B
2.17 2.10 2.27 2.17 1.75 2.10 2.29 2.15
2.13 ± 0.17
2.2
3 2
= =
1
Z=3
0
i
0
i
i
1
i
2
Z=2
r c (a.u.) Fig. 1. Relation between r ° s and r c for various valence
charges, Z.
term, l,'ps(0), for the Heine-Abarenkov potential [11] ,1 and the Sharma-Kachhava linearized potential [ 12] are 5IrZr c " 2 and 21rZr 2 , respectively. The Zr 2 dependence of Vps(0 ) is probably quite general. The term ct has the dimension of energy i.e. Z/r, hence from the definition of Vps(0), eq. (2), Vps(0 ) must be
ocZ r 2" We now choose the pseudopotential parameter, rc, such that it will give the experimental rws and see how well this pseudopotential can be used to predict the bulk modulus. The expression for the bulk modulus at the equilibrium volume is given by
B 0 = ~[A/rw s 2 0 _ B/(rOs )21 ~ a 1 .
(13)
The ratios of of the calculated bulk modulus, B 0, to the experimental bulk modulus, B 0, are shown in table 1. For the Z = 1 case, B 0 and B 0 are very close. As Z i n c r eases, t he ratio o f B 0c to B e deviates from unity. The deviation is expected because, for higher Z, the pseudopotential is more attractive and there is more perturbation of the uniform charge density. In other words, the band structure term becomes more important. Kachhava and Sharma [ 13] calculated the contribution to the bulk modulus from the band structure term and found that this term is negligible for small Z (a few % for Z = 1), but it increases rapidly for higher Z (~100% in magnitude for Z = 3). Moruzzi et al. [14] have demonstrated that the bulk ~:1 The depth of the pseudopotential is set equal to -2Z/r c for obtaining the expression for Fps(0) in the text.
Z--3
A1 Sc Ga Y In La T1
modulus of a metal depends more strongly on the interstitial charge density than the average density. As Z increases, the pseudopotential pulls more charge inward making the interstitial charge density lower; therefore, B 0 is lower than B 0 . In a calculation by Chelikowsky [15], the profiles of the valence charge density for various metals are evaluated. The interstitial charge density is very close to the average charge density for Z = 1 but is substantially lower than the average density for higher Z. Hence the reason why the bulk modulus is overestimated using a uniform charge density approximation can be understood qualitatively. The striking feature is that the ratio o f B 0 to B 0 appears to depend on Z only (table 1). This dependence implies that there is a scaling relation for metals with the same Z. This scaling arises from the relation between rOs and r e. If we plot the pseudopotentials in normalized units, Vion(r)/(Z/rOs) versus r/rOs, the pseudopotentials almost coincide with one another because r ffrOs ~ constant. 295
Volume 100A, number 6
PHYSICS LETTERS
Empirically, BO/B 0 is approximately given by
BOc/BO e ~ 0.3(Z - 1) 2 + 1 -- 7.
(14)
We do not have a derivation of the functional form given in eq. (14), however this form leads to an empirical formula for the bulk modulus
B 0 - 2[A/rO s
B/(rOs)2l~2a 1 [0.3(Z - l ) 2 + 11-1,
(15) where A and B are given by eq. (8). The work was supported by National Science F o u n d a t i o n Grant No. DMR7822465 and by the Director, Office of Energy Research, Office of Basic Energy Sciences, Material Sciences Division of the U.S. Department of energy under Contract No. DEACO3-76SF00098.
References [1] P.K. Lam and M.L. Cohen, Phys. Rev. B26 (1983) 5986. [2] M.Y. Chou, P.K. Lain and M.L. Cohen, Solid State Commun. 42 (1982) 861.
296
6 February 1984
[3] P. Hohenberg and W. Kohn, Phys. Rev. 136 (1964) B864; W. Kohn and L.J. Sham, Phys. Rev. 140 (1965) Al133. [4] M.L. Cohen and V. Heine, in: Solid state physics, Vol. 24, eds. H. Ehrenreich, F. Seitz and D. Turnbull (Academic Press, New York, 1970) p. 37. [5] V. Heine and D. Weaire, in: Solid state physics, Vol. 24, eds. H. Ehrenreich, F. Seitz and D. Turnbull (Academic Press, New York, 1970) p. 249. [6] W.A. Harrison, in: Pseudopotentials in the theory of metals (Benjamin, New York, 1966) p. 289. [7] J. lhm, A. Zunger and M.L. Cohen, J. Phys. C12 (1979) 4409. [8] E. Wigner, Phys. Rev. 46 (1934) 1002. [9] N.W. Ashcroft, Phys. Lett. 23 (1966)48. [10] A. Zunger, Phys. Rev. B22 (1980) 5839. [11] I.V. Abarenkov and V. Heine, Philos. Mag. 12 (1965) 529. [12] K.S. Sharma and C.M. Kachhava, Solid State Commun. 30 (1979) 749. [13] C.M. Kachhava and K.S. Sharma, J. Phys. F10 (1980) 827. [14] L. Moruzzi, J.F. Janak and A.K. WiUiams,in: Calculated electronic properties of metals (Pergamon, New York, 1978) p. 23. [151 J.R. Chelikowsky, Phys. Rev. B21 (1980) 3074.
PHYSICS LETTERS
6 February 1984
DEPENDENCE OF LATTICE CONSTANTS AND BULK MODULI ON PSEUDOPOTENTIAL PROPERTIES PUi K. LAM i and Marvin L. COHEN
Department of Physics, University of California, Berkeley, CA 94720, USA and Materials and Molecular Research Division, Lawrence Berkeley Laboratory, Berkeley, CA 94720, USA Received 19 August 1983 Revised manuscript received 5 December 1983
Using a nearly-free-electron model and model pseudopotentials, an analytical relation between the Wigner-Seitz radius for a metal, rws, and the core radius of the model pseudopotential, rc, is derived. In addition, a semi-empiricalformula for the bulk moduli is obtained.
The structural properties of metals are governed by the interactions between the valence electrons and the atomic core (nucleus plus core electrons) and the self-consistent interactions among the valence electrons. It has been demonstrated [1,2] that the local density formalism [3] gives a good description for the self-consistent interactions of the valence electrons for ground state properties of metals. In addition, pseudopotentials [4] adequately describe the interaction between the valence electrons and the core. We explore here the relation of metallic sizes and bulk moduli to pseudopotential properties. The aim of this investigation is to simplify the formalism so that the relationship between structural properties and pseudopotential properties is apparent. We concentrate on simple (sp) metals because the dose-packed structure and the nearly-free electron character of the wave functions for these metals allow simple approximations. A few metals from the beginning of transition metal series are also included in our analysis. Rydberg atomic units will be used throughout, i.e. e 2 -- 2, m = 1/2, and ~ = 1. Starting with a free-electron model and replacing the Wigner-Seitz cell by a sphere, Heine and Weaire [5,6] derived an expression for the crystal energy 1 Present address: National Bureau of Standards, Washington, DC 20234, USA. 0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North.Holland Physics Publishing Division)
using a real space representation. Their expression for the crystal energy is similar to the one given by eq. (3) below. We present an alternate derivation here which starts with the exact expression in momentum space and then apply the approximations. The advantage of this derivation is that it shows dearly which terms are neglected in the model. Within the local density formalism and pseudopotential approach the crystal energy can be written in a momentum space representation [7],
Nk E/atom --'N~ 1 ~ e(k) _ 1~D, a G~O VH(G) p(G) k
+ r~a o~ [exc(G)- rxc(G)] + [~ + Vxc(O)] Z + "/Ewald,
(1)
where N ~ l~l~k e(k) is the mean eigenvalue averaged over all the occupied states; the eigenvalues are calculated with the average potential set to zero [7]. ~a is the atomic volume, VH (G) is the Hartree potential, exc(G) and Vxc(G) are the exchange-correlation energy density and potential, p(G) is the charge density, ct is a pseudopotential term, Z is the number of valence electrons, and ")'Ewald is the Ewald energy. The pseudopotential term, a, is defined to be
293
Volume 100A, number 6
~=K2a 1
PHYSICS LETTERS
f[Vion(r)+2Z/r] d3r-~Vps(O)/K2a,
(2)
where Vion(r) is the ionic pseudopotential. The a term represents only the integrated effect of the pseudopotential. Other effects of the pseudopotential are implicitly contained in [e(k)] and p(G). For metals, the following approximations are made. The charge density is assumed to be uniform, i.e., we neglect all the G 4= 0 terms in p(G), and e(k) is assumed to be given by the free electron dispersion, i.e. e(k) = k 2 . The terms neglected in these approximations are often grouped together and called the band structure term [5]. With the above approximations, the crystal energy is given by
-0.88Z(rws/Z 113 + 7.8) -1 (3)
where rws is the Wigner-Seitz radius defined by 4
3
-~rt rws = ~a"
(4)
The first term in eq. (3) is the eigenvalue or kinetic energy term in this case because the potential is zero, the second term is the exchange term, the third term is the correlation term (Wigner's correlation formula is used [8]), the fourth term is the pseudopotential term, and the last term is the Ewald term. Approximate numerical values have been used for ~ (97r/4) 2/3 2.21, (3/2~r)(9zr/4) 1/3 ~ 0.916, and 3/4~r ~ 0.24 in eq. (3). The first three terms in eq. (3) give the energy for a jeUinm model. The Ewald term accounts for the fact that the ionic cores are discrete entities rather than a smeared out jellium. The pseudopotential term accounts for the f'mite size of the core. The Ewald constant, Fs, for the fcc, bcc, and hcp structures are given by [5] F fcc= 1.79175,
Fsbcc= 1.79186,
Fsbcp= 1.79168.
(5) If the Wigner-Seitz unit cell were replaced by a sphere then the Ewald constant is F ws= 1.8. Since F s for the three metallic structures, fcc, bcc, 294
and hcp, is very close to 1.8, F s = 1.8 is used in our calculation. The first and second derivative of the correlation term with respect to rws are very small compared to the derivatives of the other terms. The contribution to the equilibrium volume and bulk modulus from the correlation term will therefore be neglected. Differentiating eq. (3) to obtain the equilibrium rw0s, one gets
rOs = [2B + (4B 2 + 12AC)l/2]/2A,
(7)
where
A = 0.916Z 4/3 + 1.8Z 2,
B = 2.21Z 5/3,
C = 0.24 Vps(0) Z.
(8)
The term A is the sum of the exchange and the Ewald term, B is the kinetic energy term, and C is the pseudopotential term. For the values we are considering, 1ZAC~ 4B 2,
Z/atom = 2.21ZS/3/r2~ - 0.916Z 4/31rws
+ 0.24 Vps(O)Z/r3s - FsZ2/rws ,
6 February 1984
(6)
rOs ~ B/A + (3C/A) 1/2,
(9)
that is, the equilibrium rw0s is determined by two effects: one from the competition between the kinetic energy and the exchange plus Ewald energy [first term in eq. (9)], and one from the finite size of the core represented by Vps(0) [second term in eq. (9)]. Using the empty core or Ashcroft form for the pseudopotential [9], Vion(r) = 0,
r < rc,
=-2Z/r,
r>~rc,
(10)
the pseudopotential term, Vps(0), is given by Vps(0) =
.
(11)
Substituting this value of Vps(0) in eqs. (8) and then (7) gives
rOs ~ B(Z)/A (Z) + {3Z/[A (Z)] 1/2 } rc "
(12)
i.e. there is linear relation between rOs and r c. In fig. 1, the dependence of r°s on r e given by eq. (7) is shown. This linear relation between metallic sizes and pseudopotential core radii has been empirically correlated [10]. An interesting feature to be noted is that the slopes of rOs versus r c are almost independent of Z (fig. 1). The linear relation is obtained because Vps(0) o: r 2. This fact is not a property peculiar only to the empty core potential. The pseudopotential
Volume 100A, number 6
PHYSICS LETTERS
6 February 1984
Table 1 Ratio ofBc0 to B°e.
6
4 ¢O
Z=1
~n t.,
o o ac/ae
y
Li Na K Rb Cs
1.23 1.03 0.99 0.79 0.91
0.99 ± 0.16
1.0
Be Mg Ca Zn Sr Cd
1.61 1.30 1.39 1.29 1.31 1.08
1.31 ± 0.15
1.3
Ba
1.29
Hg
1.19
B
2.17 2.10 2.27 2.17 1.75 2.10 2.29 2.15
2.13 ± 0.17
2.2
3 2
= =
1
Z=3
0
i
0
i
i
1
i
2
Z=2
r c (a.u.) Fig. 1. Relation between r ° s and r c for various valence
charges, Z.
term, l,'ps(0), for the Heine-Abarenkov potential [11] ,1 and the Sharma-Kachhava linearized potential [ 12] are 5IrZr c " 2 and 21rZr 2 , respectively. The Zr 2 dependence of Vps(0 ) is probably quite general. The term ct has the dimension of energy i.e. Z/r, hence from the definition of Vps(0), eq. (2), Vps(0 ) must be
ocZ r 2" We now choose the pseudopotential parameter, rc, such that it will give the experimental rws and see how well this pseudopotential can be used to predict the bulk modulus. The expression for the bulk modulus at the equilibrium volume is given by
B 0 = ~[A/rw s 2 0 _ B/(rOs )21 ~ a 1 .
(13)
The ratios of of the calculated bulk modulus, B 0, to the experimental bulk modulus, B 0, are shown in table 1. For the Z = 1 case, B 0 and B 0 are very close. As Z i n c r eases, t he ratio o f B 0c to B e deviates from unity. The deviation is expected because, for higher Z, the pseudopotential is more attractive and there is more perturbation of the uniform charge density. In other words, the band structure term becomes more important. Kachhava and Sharma [ 13] calculated the contribution to the bulk modulus from the band structure term and found that this term is negligible for small Z (a few % for Z = 1), but it increases rapidly for higher Z (~100% in magnitude for Z = 3). Moruzzi et al. [14] have demonstrated that the bulk ~:1 The depth of the pseudopotential is set equal to -2Z/r c for obtaining the expression for Fps(0) in the text.
Z--3
A1 Sc Ga Y In La T1
modulus of a metal depends more strongly on the interstitial charge density than the average density. As Z increases, the pseudopotential pulls more charge inward making the interstitial charge density lower; therefore, B 0 is lower than B 0 . In a calculation by Chelikowsky [15], the profiles of the valence charge density for various metals are evaluated. The interstitial charge density is very close to the average charge density for Z = 1 but is substantially lower than the average density for higher Z. Hence the reason why the bulk modulus is overestimated using a uniform charge density approximation can be understood qualitatively. The striking feature is that the ratio o f B 0 to B 0 appears to depend on Z only (table 1). This dependence implies that there is a scaling relation for metals with the same Z. This scaling arises from the relation between rOs and r e. If we plot the pseudopotentials in normalized units, Vion(r)/(Z/rOs) versus r/rOs, the pseudopotentials almost coincide with one another because r ffrOs ~ constant. 295
Volume 100A, number 6
PHYSICS LETTERS
Empirically, BO/B 0 is approximately given by
BOc/BO e ~ 0.3(Z - 1) 2 + 1 -- 7.
(14)
We do not have a derivation of the functional form given in eq. (14), however this form leads to an empirical formula for the bulk modulus
B 0 - 2[A/rO s
B/(rOs)2l~2a 1 [0.3(Z - l ) 2 + 11-1,
(15) where A and B are given by eq. (8). The work was supported by National Science F o u n d a t i o n Grant No. DMR7822465 and by the Director, Office of Energy Research, Office of Basic Energy Sciences, Material Sciences Division of the U.S. Department of energy under Contract No. DEACO3-76SF00098.
References [1] P.K. Lam and M.L. Cohen, Phys. Rev. B26 (1983) 5986. [2] M.Y. Chou, P.K. Lain and M.L. Cohen, Solid State Commun. 42 (1982) 861.
296
6 February 1984
[3] P. Hohenberg and W. Kohn, Phys. Rev. 136 (1964) B864; W. Kohn and L.J. Sham, Phys. Rev. 140 (1965) Al133. [4] M.L. Cohen and V. Heine, in: Solid state physics, Vol. 24, eds. H. Ehrenreich, F. Seitz and D. Turnbull (Academic Press, New York, 1970) p. 37. [5] V. Heine and D. Weaire, in: Solid state physics, Vol. 24, eds. H. Ehrenreich, F. Seitz and D. Turnbull (Academic Press, New York, 1970) p. 249. [6] W.A. Harrison, in: Pseudopotentials in the theory of metals (Benjamin, New York, 1966) p. 289. [7] J. lhm, A. Zunger and M.L. Cohen, J. Phys. C12 (1979) 4409. [8] E. Wigner, Phys. Rev. 46 (1934) 1002. [9] N.W. Ashcroft, Phys. Lett. 23 (1966)48. [10] A. Zunger, Phys. Rev. B22 (1980) 5839. [11] I.V. Abarenkov and V. Heine, Philos. Mag. 12 (1965) 529. [12] K.S. Sharma and C.M. Kachhava, Solid State Commun. 30 (1979) 749. [13] C.M. Kachhava and K.S. Sharma, J. Phys. F10 (1980) 827. [14] L. Moruzzi, J.F. Janak and A.K. WiUiams,in: Calculated electronic properties of metals (Pergamon, New York, 1978) p. 23. [151 J.R. Chelikowsky, Phys. Rev. B21 (1980) 3074.