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Tight-bonding calculation of acceptor energies in germanium and silicon
J. Phys. Chem. Solids.
Pergamon
Press 1957. Vol. 2. pp. 115-118.
TIGHT-BONDING ENERGIES
CALCULATION
IN GERMANIUM
OF ACCEPTOR AND
SILICON
R. G. SHULMAN Bell Telephone
Laboratories,
(Received
Murray Hill, New Jersey
8 September
1956)
Abstract-The differences of hole ionization energies among the group-III elements acting as acceptors in silicon indicate the necessity of considering the specific contribution of the acceptor atom. Energies of two electron bonds are calculated using Morse curves to approximate interatomic potentials. The additional binding energy increases sharply from gallium to indium in accord with experiment. Arguments are presented to show why varying the acceptor element is less important in germanium.
INTRODUCTION
RECENT experiments have shown that the Group III elements acting as acceptors in silicon, have levels located at different energies in the forbidden gap.(l) Previous theoretical estimates of acceptor energies@) have followed the effective mass approximation. In this approximation the acceptor is considered to be an electrostatic point charge and the hole a mobile charge free to move around the .acceptor. The strong dependence of the hole activation energy upon the chemical element acting as acceptor indicates the necessity of considering the specific contribution of the acceptor atom. We shall calculate acceptor activation energies using a model where the bound hole is assumed to be immobilized at the acceptor atom. After deriving values of ionization energy for this tightly bound model we discuss how this might be incorporated in an effective mass calculation.
Table 1.” Energy Levels of Group III Acceptors in Silicon =Energy Level
ls-continuum 2p-continuum 3p-continuum 4p-continuum
Al r= 1.26 a
Ga ?-= 1.26 A
In += 1.44 A
eV
eV
eV
eV
0.046
0.067
0.071
0.156
0.0460
0.012
0.012
0.013
0.013
0~0115
0.006
0.008
0.011
0.008
0~0051
0.003
0.003
0.003
0.003
0.0029
B 0r8sil
__
Hydrogenie Model
-__* From BURSTEIN,P~cus, HENVIS, and WALLIS J. Phys. Chem. Solids 1, 65 (1956).
able that the penetrating s-orbitals should be affected by perturbations at the core while the non-penetrating p-type excited states are relatively unaffected. In order to understand this perturbation we consider the tight-bonding model of Fig. 1. A trapped hole is represented in Fig. l(b) by an unpaired silicon-acceptor bond while in the body of the crystal a particular silicon-silicon bond is paired. Fig. l(a) shows the pertinent sections of the crystal after an electron has been removed from the Si-Si bond and placed in the Si-Ac bond. The change between these two states is the ionization of
Tight-bonding model The experimental results include not only the thermal activation energy, i.e. the energy required to free a hole from its hydrogenic 1s ground state, but also more recent optical datac3) which measure the activation energy of excited states as well. The energies of these states are presented in Table 1. The 2p, 3p and 4p levels are relatively independent of the acceptor atom while the 1s state varies considerably with the acceptor element. It is reason115
R.
116
G.
SHULMAN
one hole. For our purposes we here have divided the ionization energy of a hole into two parts : The coulombic energy, including kinetic and potential energy terms, required to free a hole from the environment of an acceptor, and the additional bonding energy which arises because the silicon-acceptor Si
Si
p
(al
//B
Si
(61
FIG. 1. Tight
II II =Si -_Si =
II
4%
Si
E(d) = -
Si
Si \; Si
binding model
=Si=Si II of acceptors
II
=
II in silicon.
bond does not have the same energy as a siliconsilicon bond. This latter energy is calculated subject to the following assumptions: (1) No appreciable distortion of the lattice exists around the acceptors and no nuclear rearrangements occur during the ionization. (2) Morse curves describe the bond energies. The Morse curves for different bonds will be compared and the energies discussed with respect to infinite nuclear separation. Now we are really comparing the energy of the bond not with infinite separation but rather with the energy of the one electron bond which exists when we break the covalent bond. Since the one electron bond energies are small it is reasonable to assume that their variations are negligible. (3) The silicon-acceptor dissociation energies, required for the Morse curves, are computed by using a relation of PAULING’S(~) which states that for similar electronic configurations the dissociation energies are inversely proportional to the internuclear distance. The main justification for applying this formula comes from the excellent agreement between theory and experiment for the diamond, silicon, germanium and grey tin series. The bond energy is given by E(r) = D[,-2a(r-r~)_2e-a(r-7,)] where
r is the internuclear
distance
(1) maintained
the crystal, re is the sum of silicon and acceptor tetrahedral radii, D is the dissociation energy and a is a constant to be determined. Electron energies are used. Expanding about the point Y = I,, substituting Dsi_si g 1 eV and using PAULING’S relation between D and T, eq. (1) becomes
in
2.34 -[-l+a2d2-a3L13+ ( re
. . .]+l
(2)
>
where E(d) is the additional energy above the top of the valence band that the tightly bound hole possesses and where A = r-y,. Optical measurements indicate that boron in silicon has excited states which are very close to a hydrogen-like model while in the other acceptors the states differ from a hydrogenic model. Therefore, we suggest that the boron-silicon bond has the same energy at 2.34 A as the Si-Si bond at this distance, or that E(A) for boron equals zero. This was used to calculate a value for the constant a in eq. (2) and the value of 1 a65 A-l obtained was used for the other acceptors. The results of these numerical calculations are presented in Table 2. Table 2
Element
B Al Ga In Tl
Thermal Ionization Energy 0.045(8) 0.057(S) 0.065(a) 0.16 (*) 0.26 (c)
Optical Ionization Energy
_
I
0.046tb)
0.067tb) 0.071(b) 0.156tb)
I
r&v
td)
Calculated
E(A)
Tetrahedral Radii
0.0 0.06 0.06 0.36 0.43
0.88 1.26 1.26 144 1.47
-
(8) MORIN, MAITA, SHULMAN,and Hhxwhy Phys. Rev. 96, 833 (1954). (b) BURSTEIN, PICUS, HENVIS, and LAX Phys. Rev. 98, 1536 (1955). (c) SHULMANR. G. and WYLUDA B. J. (Unpublished results). td) PAULING L. Nature of the Chemical Bond, Cornell Univ. Press (1942).
The second column lists the ionization energies determined by Hall measurements, while the third column presents the same quantities measured optically. Although these two methods do not agree perfectly the differences are not important
TIGHT-BONDING
CALCULATION
OF ACCEPTOR
for our considerations. In the fourth column we list the values of E(d) calculated from eq. (2). Now E(d) is the bonding energy assuming the process pictured in Fig. 1. From what has been said above it can be seen that the true ionization energy of a hole from its ground state is its coulombic energy modified by some contribution from E(O). Table 2 illustrates the qualitative agreement of E(d) with experiment in that both show a sharp increase going from gallium to indium. Furthermore qualitatively this increase does not depend upon any assumed parameters but is a consequence of assuming that the potential forces between nuclei rise more steeply at distances shorter than equilibrium than they do at distances longer than equilibrium. Another qualitative accomplishment of the above calculations is seen in the ionization energy of thallium in silicon which is reported here for the first time. According to the tight-bonding calculations the ionization energy for thallium should be somewhat greater than that of indium while the main discontinuity should be between gallium and indium. That this is the case can be seen from Table 2 where the thallium ionization energy is listed. The previous difficulty in determining this energy was growing a crystal containing thallium since thallium has an appreciable vapor pressure at the boiling point of silicon and it rapidly vaporizes. This crystal was grown by E. BUEHLER by starting with high purity n-type silicon and dropping in successive thallium pellets. The p-type region was
2
300
350
1
400
450
500
f
FIG. 2. Conductivity of 50 Q-cm p-type thallium doped silicon as a function of reciprocal temperature, H
ENERGIES
IN Ge AND
Si
117
isolated and a bridge cut from it. Potentiometric resistivity measurements made as a function of temperature are shown in Fig. 2 where the slope determines the ionization energy as 0.26 eV.* In addition to explaining the variations in acceptor energies the above model accounts for the relatively constant values of donor ionization energies. These values as determined by Hall measurements? are : phosphorous, 0.044 eV; arsenic, 0.049 eV; antimony, 0.039 eV and bismuth, 0.067. When a donor electron is ionized no bonds are broken or formed and the effects of different covalent radii are negligible. In germanium the acceptor ionization energies have been shownc5) to be quite insensitive to the acceptor element. The difference between silicon and germanium can be understood by considering our calculated energies as perturbations upon a simplified effective mass calculation. Starting with a hydrogenic Is function ‘p = (u3rr)-*e-r/a where (I = Kh21me2 is the effective Bohr radius. and K the dielectric constant we consider a perturbing potential v(‘(d) assumed to be constant out to a radius b. The additional ionization energy is
s
4” V(A)+ dr = 4 b3 ~----+... 3 a3
b 4r2 -e-2rla
s 0 a3
dr
2b4 a4
(3)
Since b is of the order of 2.4A in both silicon and germanium and &-& -2asi -34A the above expansion is justified and only the first term need be retained. We see that the effect of perturbations at the core of an acceptor in silicon will be at least an order of magnitude more effective than similar perturbations in germanium. These ideas have been used to explain comparative values of ionization energies. However, it has not been possible to find order of magnitude agreement with the absolute values of observed energies by considering these tight-bonding energies as perturbations upon the effective mass Hamiltonian. This is not * This value has been confirmed by H. BRIDGERS,Jr. who has made Hall measurements on the same sample down to 170°K. t These values are from ref. (1) but include F. J. MORIN’S revised value for phosphorous and a previously unreported value for bismuth.
118
R.
G.
SHULMAN
surprising considering that the effective mass approximation is not correct close to the acceptor __ since both dielectric constant and potential are varying rapidly. The final solution must incorporate a perturbation of the acceptor atom with an effective mass approach in a manner which is presently difficult to indicate.
particular, N. B. HANNAY, G. H. WANNIER and H. REISS have given generously their time and advice. REFERENCES 1. 2. 3.
Acknowledgements-It is a pleasure to acknowledge the suggestions of many colleagues at Bell Laboratories during the course of this work. In
4, 5.
MORIN, MAITA, SHULMAN,and HANNAY Phys. Rev. 96, 833 (1954). KOHN W. and SCHECHTERD. Phys. Rev. 99, 1903, (1955). BURSTEIN, PICUS, HENVIS, and LAX Phys. Rev. 98, 1536 (1955). PAULINGL. J. Phys. Chem. 58, 662 (1954). GEBALLE T. H. and MORIN F. J. Phys. Rev. 95, 1085 (1954).
Pergamon
Press 1957. Vol. 2. pp. 115-118.
TIGHT-BONDING ENERGIES
CALCULATION
IN GERMANIUM
OF ACCEPTOR AND
SILICON
R. G. SHULMAN Bell Telephone
Laboratories,
(Received
Murray Hill, New Jersey
8 September
1956)
Abstract-The differences of hole ionization energies among the group-III elements acting as acceptors in silicon indicate the necessity of considering the specific contribution of the acceptor atom. Energies of two electron bonds are calculated using Morse curves to approximate interatomic potentials. The additional binding energy increases sharply from gallium to indium in accord with experiment. Arguments are presented to show why varying the acceptor element is less important in germanium.
INTRODUCTION
RECENT experiments have shown that the Group III elements acting as acceptors in silicon, have levels located at different energies in the forbidden gap.(l) Previous theoretical estimates of acceptor energies@) have followed the effective mass approximation. In this approximation the acceptor is considered to be an electrostatic point charge and the hole a mobile charge free to move around the .acceptor. The strong dependence of the hole activation energy upon the chemical element acting as acceptor indicates the necessity of considering the specific contribution of the acceptor atom. We shall calculate acceptor activation energies using a model where the bound hole is assumed to be immobilized at the acceptor atom. After deriving values of ionization energy for this tightly bound model we discuss how this might be incorporated in an effective mass calculation.
Table 1.” Energy Levels of Group III Acceptors in Silicon =Energy Level
ls-continuum 2p-continuum 3p-continuum 4p-continuum
Al r= 1.26 a
Ga ?-= 1.26 A
In += 1.44 A
eV
eV
eV
eV
0.046
0.067
0.071
0.156
0.0460
0.012
0.012
0.013
0.013
0~0115
0.006
0.008
0.011
0.008
0~0051
0.003
0.003
0.003
0.003
0.0029
B 0r8sil
__
Hydrogenie Model
-__* From BURSTEIN,P~cus, HENVIS, and WALLIS J. Phys. Chem. Solids 1, 65 (1956).
able that the penetrating s-orbitals should be affected by perturbations at the core while the non-penetrating p-type excited states are relatively unaffected. In order to understand this perturbation we consider the tight-bonding model of Fig. 1. A trapped hole is represented in Fig. l(b) by an unpaired silicon-acceptor bond while in the body of the crystal a particular silicon-silicon bond is paired. Fig. l(a) shows the pertinent sections of the crystal after an electron has been removed from the Si-Si bond and placed in the Si-Ac bond. The change between these two states is the ionization of
Tight-bonding model The experimental results include not only the thermal activation energy, i.e. the energy required to free a hole from its hydrogenic 1s ground state, but also more recent optical datac3) which measure the activation energy of excited states as well. The energies of these states are presented in Table 1. The 2p, 3p and 4p levels are relatively independent of the acceptor atom while the 1s state varies considerably with the acceptor element. It is reason115
R.
116
G.
SHULMAN
one hole. For our purposes we here have divided the ionization energy of a hole into two parts : The coulombic energy, including kinetic and potential energy terms, required to free a hole from the environment of an acceptor, and the additional bonding energy which arises because the silicon-acceptor Si
Si
p
(al
//B
Si
(61
FIG. 1. Tight
II II =Si -_Si =
II
4%
Si
E(d) = -
Si
Si \; Si
binding model
=Si=Si II of acceptors
II
=
II in silicon.
bond does not have the same energy as a siliconsilicon bond. This latter energy is calculated subject to the following assumptions: (1) No appreciable distortion of the lattice exists around the acceptors and no nuclear rearrangements occur during the ionization. (2) Morse curves describe the bond energies. The Morse curves for different bonds will be compared and the energies discussed with respect to infinite nuclear separation. Now we are really comparing the energy of the bond not with infinite separation but rather with the energy of the one electron bond which exists when we break the covalent bond. Since the one electron bond energies are small it is reasonable to assume that their variations are negligible. (3) The silicon-acceptor dissociation energies, required for the Morse curves, are computed by using a relation of PAULING’S(~) which states that for similar electronic configurations the dissociation energies are inversely proportional to the internuclear distance. The main justification for applying this formula comes from the excellent agreement between theory and experiment for the diamond, silicon, germanium and grey tin series. The bond energy is given by E(r) = D[,-2a(r-r~)_2e-a(r-7,)] where
r is the internuclear
distance
(1) maintained
the crystal, re is the sum of silicon and acceptor tetrahedral radii, D is the dissociation energy and a is a constant to be determined. Electron energies are used. Expanding about the point Y = I,, substituting Dsi_si g 1 eV and using PAULING’S relation between D and T, eq. (1) becomes
in
2.34 -[-l+a2d2-a3L13+ ( re
. . .]+l
(2)
>
where E(d) is the additional energy above the top of the valence band that the tightly bound hole possesses and where A = r-y,. Optical measurements indicate that boron in silicon has excited states which are very close to a hydrogen-like model while in the other acceptors the states differ from a hydrogenic model. Therefore, we suggest that the boron-silicon bond has the same energy at 2.34 A as the Si-Si bond at this distance, or that E(A) for boron equals zero. This was used to calculate a value for the constant a in eq. (2) and the value of 1 a65 A-l obtained was used for the other acceptors. The results of these numerical calculations are presented in Table 2. Table 2
Element
B Al Ga In Tl
Thermal Ionization Energy 0.045(8) 0.057(S) 0.065(a) 0.16 (*) 0.26 (c)
Optical Ionization Energy
_
I
0.046tb)
0.067tb) 0.071(b) 0.156tb)
I
r&v
td)
Calculated
E(A)
Tetrahedral Radii
0.0 0.06 0.06 0.36 0.43
0.88 1.26 1.26 144 1.47
-
(8) MORIN, MAITA, SHULMAN,and Hhxwhy Phys. Rev. 96, 833 (1954). (b) BURSTEIN, PICUS, HENVIS, and LAX Phys. Rev. 98, 1536 (1955). (c) SHULMANR. G. and WYLUDA B. J. (Unpublished results). td) PAULING L. Nature of the Chemical Bond, Cornell Univ. Press (1942).
The second column lists the ionization energies determined by Hall measurements, while the third column presents the same quantities measured optically. Although these two methods do not agree perfectly the differences are not important
TIGHT-BONDING
CALCULATION
OF ACCEPTOR
for our considerations. In the fourth column we list the values of E(d) calculated from eq. (2). Now E(d) is the bonding energy assuming the process pictured in Fig. 1. From what has been said above it can be seen that the true ionization energy of a hole from its ground state is its coulombic energy modified by some contribution from E(O). Table 2 illustrates the qualitative agreement of E(d) with experiment in that both show a sharp increase going from gallium to indium. Furthermore qualitatively this increase does not depend upon any assumed parameters but is a consequence of assuming that the potential forces between nuclei rise more steeply at distances shorter than equilibrium than they do at distances longer than equilibrium. Another qualitative accomplishment of the above calculations is seen in the ionization energy of thallium in silicon which is reported here for the first time. According to the tight-bonding calculations the ionization energy for thallium should be somewhat greater than that of indium while the main discontinuity should be between gallium and indium. That this is the case can be seen from Table 2 where the thallium ionization energy is listed. The previous difficulty in determining this energy was growing a crystal containing thallium since thallium has an appreciable vapor pressure at the boiling point of silicon and it rapidly vaporizes. This crystal was grown by E. BUEHLER by starting with high purity n-type silicon and dropping in successive thallium pellets. The p-type region was
2
300
350
1
400
450
500
f
FIG. 2. Conductivity of 50 Q-cm p-type thallium doped silicon as a function of reciprocal temperature, H
ENERGIES
IN Ge AND
Si
117
isolated and a bridge cut from it. Potentiometric resistivity measurements made as a function of temperature are shown in Fig. 2 where the slope determines the ionization energy as 0.26 eV.* In addition to explaining the variations in acceptor energies the above model accounts for the relatively constant values of donor ionization energies. These values as determined by Hall measurements? are : phosphorous, 0.044 eV; arsenic, 0.049 eV; antimony, 0.039 eV and bismuth, 0.067. When a donor electron is ionized no bonds are broken or formed and the effects of different covalent radii are negligible. In germanium the acceptor ionization energies have been shownc5) to be quite insensitive to the acceptor element. The difference between silicon and germanium can be understood by considering our calculated energies as perturbations upon a simplified effective mass calculation. Starting with a hydrogenic Is function ‘p = (u3rr)-*e-r/a where (I = Kh21me2 is the effective Bohr radius. and K the dielectric constant we consider a perturbing potential v(‘(d) assumed to be constant out to a radius b. The additional ionization energy is
s
4” V(A)+ dr = 4 b3 ~----+... 3 a3
b 4r2 -e-2rla
s 0 a3
dr
2b4 a4
(3)
Since b is of the order of 2.4A in both silicon and germanium and &-& -2asi -34A the above expansion is justified and only the first term need be retained. We see that the effect of perturbations at the core of an acceptor in silicon will be at least an order of magnitude more effective than similar perturbations in germanium. These ideas have been used to explain comparative values of ionization energies. However, it has not been possible to find order of magnitude agreement with the absolute values of observed energies by considering these tight-bonding energies as perturbations upon the effective mass Hamiltonian. This is not * This value has been confirmed by H. BRIDGERS,Jr. who has made Hall measurements on the same sample down to 170°K. t These values are from ref. (1) but include F. J. MORIN’S revised value for phosphorous and a previously unreported value for bismuth.
118
R.
G.
SHULMAN
surprising considering that the effective mass approximation is not correct close to the acceptor __ since both dielectric constant and potential are varying rapidly. The final solution must incorporate a perturbation of the acceptor atom with an effective mass approach in a manner which is presently difficult to indicate.
particular, N. B. HANNAY, G. H. WANNIER and H. REISS have given generously their time and advice. REFERENCES 1. 2. 3.
Acknowledgements-It is a pleasure to acknowledge the suggestions of many colleagues at Bell Laboratories during the course of this work. In
4, 5.
MORIN, MAITA, SHULMAN,and HANNAY Phys. Rev. 96, 833 (1954). KOHN W. and SCHECHTERD. Phys. Rev. 99, 1903, (1955). BURSTEIN, PICUS, HENVIS, and LAX Phys. Rev. 98, 1536 (1955). PAULINGL. J. Phys. Chem. 58, 662 (1954). GEBALLE T. H. and MORIN F. J. Phys. Rev. 95, 1085 (1954).