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Quantum theory of interstitial impurities: Energy spectrum, stationary states and deformation field
Solid State Communications, Vol. 53, No. 1, pp. 5 1 - 5 4 , 1985. Printed in Great Britain
0038-1098/85 $3.00 + .00 Pergamon Press Ltd.
QUANTUM THEORY OF INTERSTITIAL IMPURITIES: ENERGY SPECTRUM, STATIONARY STATES AND DEFORMATION FIELD Miguel Lagos Facultad de Ffsca, Universidad Cat61ica de Chile, Casilla 114-D, Santiago, Chile
(Received 15 March 1984 by A.A. Maradudin) The Hamiltonian of an impurity field, tightly bound to the host crystal, is diagonalized by a new transformation. The impurities couple strongly with the phonon gas by terms which are linear and quadratic in the ionic displacements of the host. Closed-form expressions for the exact stationary states and energy spectrum follow. The analytical calculation of a number of effects like phonon-assisted tunneling and optical absorption, valid for massive impurities and high concentrations, follow as direct applications.
A NUMBER OF IMPORTANT phenomena in solid state physics are originated by point defects in crystal lattices. The dynamical study of crystals strongly coupled to a defect field is therefore of general interest in solid state theory [ 1 - 4 ] . Point defects in crystals give rise to a variety of effects having dissimilar appearance: lattice deformation, relaxation energy and self-trapping, lattice-mediated interaction between defects, frequency shifts of vibrational modes, optical and localized modes, phononassisted diffusion, absorption of light, luminescence, non-radiative decay of excited defects, etc . . . . . All these phenomena have been extensively investigated and theoretical methods and approximations, which are specific to the study of each effect, have been developed. A unified treatment is at present lacking [1, 2]. In studying any phenomenon caused by lattice imperfections one has to deal with a field of localized defects strongly interacting with the crystal and, eventually, small interaction terms causing transitions between lattice and defect states. Therefore, the first step for a general dynamic theory of crystal defects, eventually coupled with electromagnetic radiation, is to find the exact tight-binding eigenvectors and eigenvalues of a defect field strongly coupled with the host crystal. To solve the problem of tightly bound defects is the main difficulty since terms causing transitions between localized states of the defects (diffusion, optical excitations, luminescence, e t c . , . . . ) can be handled perturbatively. The strong coupling between imperfections and crystal vibrations is the cornerstone of any theory of defects. The small polaron transformation gives the exact eigenvectors of a crystal with tightly bound defects of arbitrary mass [5]. Nevertheless, this transformation, which is the main competitor of the method described here [6], assumes a potential energy
of interaction between the defects and the host crystal which is linear in the ionic displacements. Since defects couple strongly with the crystal they in general cause severe lattice deformations. Therefore, the quadratic terms in the expansion of the defect-host crystal interaction in powers of the ionic displacements cannot be dropped in the general case. On the other hand, important effects like the shift of vibrational frequencies and many-body terms of the lattice-mediated interaction between defects, are due to the quadratic coupling terms [1, 7]. The attempt to include quadratic coupling in the context of the theory of the small polaron has resulted in a very complicated formalism [8]. In this paper I propose a new transformation which diagonalizes the tight-binding Hamiltonian of the defect field strongly coupled to the host crystal by linear and quadratic terms. The exact eigenvectors and eigenvalues of the system are obtained in closed form. In order to be definite I will discuss the special case of interstitial impurities, nevertheless, the generalization to other point defects is straightforward. The very restrictive Born-Oppenheimer and Condon approximations are dropped. An important lateral result of the new transformation is the precise definition of vibrational excitations of an imperfect crystal. These excitations (which I call "phonions", since are defined in terms of phonon and defect operators) are independent of the number, spatial distribution and internal state of the defects. When expressed in terms of "phonion" operators the tightbinding state vectors of the system take a very simple form. The application of them to calculate other effects, like phonon-assisted diffusion, light absorption or luminescence, is particularly simple and straightforward. A detailed study of these applications is left, nevertheless, to a separate publication and I discuss here just 51
52
Vol. 53, No. 1
QUANTUM THEORY OF INTERSTITIAL IMPURITIES
direct implications, as deformation, self-trapping and lattice-mediated interaction between defects. The Hamiltonian of a crystal containing impurities can be written as H = no + n , ,
(1)
where {~tx } are quantum numb ers giving the'vibrational state and {rh} are the occupation numbers of impurity states. For this aim define the new annihilation and creation operators, which I will call "phonion" operators to distinguish them from the phonon excitations, bo = E (Aoxax + Boxa~)
where
X
Ho =
y.
"t hooxaxax + E
X
1
e,c*zc,+ Egx,cTc,(ax-a* x) Xl
E
--
i=1 l l . . . l
+ ~, ~.hxKtc¢lcl(ax--a¢x)(aK--a'~,), l
(2)
Y (Aox-Box) rg),,.. ,
. l i e l
*l c l I
.
.
? (5)
.. • ¢liCli,
)~X'
b~"t =
and
.
i h
Y (Bpxax + Aoxa~.) h
H1
= E ,.,c;c,, + E E gxH'¢>,' (~x - 4 , ) 1~I'
l~l'
+ ~.
X
Y
i=1 l l . . . l
+EE
14: l'XX'
hxx'u'e,*~,' (~x -- ~ ) (~x' -- ~ )
(3)
Here cox denotes the frequency of the vibrational mode (/~, q) - X of the perfect crystal, being/~ the branch and q the wavevector of the mode, ax anda*x are the corresponding phonon operators. The symbol X means X - (/a, -- q). The field operator for the impurities was expanded in the Wannier base associated to an impurity in the rigid periodic lattice• The indexes l = (c~, l) characterize tight-binding Wannier states of the impurity, a standing for the excitation level and I denotes an interstitial site. The coefficients el = ec~ are the energy levels of the impurity in a site of the rigid undeformed lattice. The third term in the right hand side of equation (2) is the well-known polaron term [5] and the fourth one collects the quadratic terms which have been discussed in a previous paragraph. They give the coupling of the impurities with the vibrations of the deformed lattice. The coefficients gxt, gxn', hxx'l and hxx' u' represent the strength of the coupling between the impurities and the phonon gas. The terms in tu' follow from the fact that the Wannier functions are not exact wavefunctions for an impurity in the periodic lattice. For tightly bound (well localized) impurities one has el, gin, hxx'l >> tu'gxn', hxx'u' for l' 4= l and H1 is a very small term. The term H0 is diagonal in the site labels I and it determines the dynamics of the crystal with we!llocalized impurities. The small term H~ collects the offdiagonal terms (l --/=l') and gives the transition probabilities between localized states of the impurities• These transitions may involve a jump to a neighbouring site or excitation with no change in the site label. The goal is now to solve H0 I{Ux }{r/l }) = E{uM{n0 I{/ax }{rh })
(4)
E(AoX--B.x)Tig,,. . . . i
*
liCl ~ e l I . . .
h
¢
• .. clicti ,
(6)
where the coefficients Aox and Box are operators • depending only on the number O perators e t1 el and having the properties
A~X = Apx, ¢ E (ApxAo'x-h
B~X = Bo*X, BpxBox')
=
6pp,,
~. ( A o x B v K - - B o x A v x ) = 0,
(7)
(8a) (9a)
X
(AoxA~x, --BoxB~x, ) = 5xx, ,
(8b)
(AoxB~x, -- BpxA~x, ) = 0.
(9b)
P
O
They are determined by the linear equations }-'~(Aox' + Box') VxX' = h(cox + ~2o)Box X'
= h(coa -- g2p)Apx,
(10)
where
vxx, =- ~. (hxx'z+ hx,~)c*~cl
(11)
t
and ~ p is a solution of the equation d e t [ 2 Vxx'wx'cox' ~2+ h(c°x--~2)6x'vJ = 0 "
The factors Tg~)l,... ti are given by
(12)
QUANTUM THEORY OF INTERSTITIAL IMPURITIES
Vol. 53, No. 1
Tx( ~ ).= g~l hw~.' 1
•"~. Xi ~ hxxi-, ~i " ' "
"" " hx~x~l~
(13)
h~'-~X~'
where
hxx'l : 2
hxx't + hx'xt h @ - ~ x~ ' "
(14)
In spite of its involved appearance this transformation will prove to be manageable and these new excitations will result in a powerful tool in the study of imperfect crystals. When expressed in terms of the b- and c-operators the Hamiltonian Ho takes the separated form H o = Hvib + / / i m p '
(15)
where the vibrational term H~b is
Hvib
= Z mx(cttcl)(btbx + ~-)--~ ½h~x,
(16)
X
and the term Him p :
y . elCtlCl @ l
.v(i-~) t t + ~ E E gxl1~h;I2...IiCz~Cl~'''C'iC'i, i=2 l I . . . l i k (17)
determines the dynamics of the degrees of freedom associated to the interstitial impurities. Since [bx, C*lCl] = [bI,ctlcl]
-0
and [Hvib,Himp] = [nvib,no]
=
[Hi,p, Ho ] = 0 both terms can be studied separately. The derivation of equations (15), (16) and (17) involve considerable algebraic manipulations and the details of them will be given in a subsequent publication [9]. The energy spectrum follows directly from equations (15), (16) and (17). One has the explicit expression
53
where the sum £' runs over all terms having at least one index l different from the rest. The terms T(xi;~. . . . zi are related to the coefficients gxt and hhh, t, appearing in the original form of Ho, by the explicit equation (13). The coefficients Apx(r/t) and Bpx(rh) do not appear in the expression for the energy equation (18). This is fortunate since the equations (10) are not easy to solve. The only term which is not closed form in equation (18) is the term in fZ~.(rh). Nevertheless one has that f2~.(rh) = w x + O(Nt~t/N) and then, for low atomic concentrations ~ x ~ t°n. The first term in the right hand side of equation (18) is, of course, the vibrational energy of the deformed crystal. The second term is the energy shift of the vibrational ground state caused by the addition of impurities. The third term collects the energies of the polarons (the impurities and their phonon clouds) as they were isolated from the rest of the impurities. The fourth term is the lattice-mediated energy of interaction between impurities. This term contains all the dependence of the energy on the relative positions of the defects. The index i = 2, 3 , . . . in the fourth term characterizes the two body, three body . . . . terms of the lattice-mediated interactions. Since the deformation field caused by a foreign atom is long-ranged the manybody terms of the lattice-mediated interaction may be important. The eigenvectors of H0 are determined by the equations
c*zc[ [{P~.~{~h}) = rh {{/ax}{rh}),
(19)
b*xbxl{m,}{nl}> = U~,l{Ux}{m}>.
(20)
The general solution of this problem is a little bit complicated because [c~, b'x] has not a simple expression. Nevertheless, with some work one can show that the many-impurity eigenstates of Ho are given by I{,x}{rh}) = l-I (bxt)ux "----"~,, a I-I(c*l ) n l a - 1 1 0 0 ) , l h v#x'
(21)
where I0 0) is the ground state of the crystal with no impurities and
["1
Q = exp } ~ b x b x
,
h
+ ~
+
et +
L g x z * h ; l . . . l rh
1
i=2
t
~'
i=2
Q-1 = exp - ½ ~ b x b x
k
gx,, Txti;z=l.)...l, r/,, -. .rh,,
Iz • • • l i
Px = 0 , 1 , 2 . . . . .
•
h
rh = 0,1,
(18)
The transformation to phonion operators can be inverted using the quadratic equations (8b) and (9b). Subtracting equation (6) from equation (5), multiplying by A~X, + B~X', summing over p and replacing equations
QUANTUM THEORY OF INTERSTITIAL IMPURITIES
54 (8b) and (9b) one obtains
t aT, --a× = Y~ (Aa× + B~x)(bp --b~) ,o
•
*
t
+2 ~ E T
the vibrational modes of the perfect crystal and the force constants. Simple closed-form expressions for'St can be obtained assuming a Debye model for the various crystal symnretries, but I will omit them here because of the lack of space. The main results shown in this letter are, nevertheless, the spectrum (18) and the tightbinding eigenvectors (21). Applicaticms of them will be published elsewhere.
A similar procedure can be used to obtain ax + ax m terms of b x + bE and then the inverse transformation. I will not write it here since the point is now to obtain a general expression for the deformation field. The ionic displacements from the sites of the perfect lattice are given by
2.
ul = Z
3.
X
2,,~7~,_ Oxein'l( ax-a~,)' ,
(24)
/.lvlivoox
REFERENCES 1.
4.
where I denotes a site of the undeformed ionic lattice and characterizes the ion attached to it. Replacing equation (23) into equation (24) one obtains two terms. One of them, containing the operators b o -- b~, does not contribute to (u l) and represents oscillations respect to the new equilibrium positions. The remaining term is the deformation operator
5. 6.
7.
8. 11 . . . I i
k
0xe i ' a ' ' rx(gq.. ,
9.
K
*
*
• liCl I Cl I • . . CliCl
i.
(25)
This operator gives the deformation field for an arbitrary impurity distribution. To evaluate it one must know just
Vol. 53, No. 1
A.M. Stoneham, Theory of Defects in Solids, Clarendon Press, Oxford (1975). C.P. Flynn, Point Defects and Diffusion, Clarendon Press, Oxford (1972). Hydrogen in Metals I and H, (Edited by G. Alefeld & J. V61k), Springer, Berlin (1978). W. Fowler, Physics of Color Centers, Academic, New York (1968). B. Jackson & R. Silbey, J. Chem. Phys. 78, 4193 (1983) (and references therein). Calculations relying on the Born-Oppenheimer approximation for defect dynamics are not discussed here since they apply just to very light impurities (F-centers and ninon diffusion). A.A. Maradudin, E.W. Montroll & G.H. Weiss, Solid State Phys. Suppl. 3 (1963). D.L. Tonks & B.G. Dick,Phys. Rev. B19, 1136 (1979); D.L. Tonks, Phys. Rev. B22, 6420 (1980). Equations (15), (16) and (17) can be easily proved in the case of linear coupling, where hxx, l = 0. To show the general case invert the transformation by the procedure described in the last paragraph, leading to ax +- a~, in terms of b~, and b*x. Then replace in equation (2) and cancel terms.
0038-1098/85 $3.00 + .00 Pergamon Press Ltd.
QUANTUM THEORY OF INTERSTITIAL IMPURITIES: ENERGY SPECTRUM, STATIONARY STATES AND DEFORMATION FIELD Miguel Lagos Facultad de Ffsca, Universidad Cat61ica de Chile, Casilla 114-D, Santiago, Chile
(Received 15 March 1984 by A.A. Maradudin) The Hamiltonian of an impurity field, tightly bound to the host crystal, is diagonalized by a new transformation. The impurities couple strongly with the phonon gas by terms which are linear and quadratic in the ionic displacements of the host. Closed-form expressions for the exact stationary states and energy spectrum follow. The analytical calculation of a number of effects like phonon-assisted tunneling and optical absorption, valid for massive impurities and high concentrations, follow as direct applications.
A NUMBER OF IMPORTANT phenomena in solid state physics are originated by point defects in crystal lattices. The dynamical study of crystals strongly coupled to a defect field is therefore of general interest in solid state theory [ 1 - 4 ] . Point defects in crystals give rise to a variety of effects having dissimilar appearance: lattice deformation, relaxation energy and self-trapping, lattice-mediated interaction between defects, frequency shifts of vibrational modes, optical and localized modes, phononassisted diffusion, absorption of light, luminescence, non-radiative decay of excited defects, etc . . . . . All these phenomena have been extensively investigated and theoretical methods and approximations, which are specific to the study of each effect, have been developed. A unified treatment is at present lacking [1, 2]. In studying any phenomenon caused by lattice imperfections one has to deal with a field of localized defects strongly interacting with the crystal and, eventually, small interaction terms causing transitions between lattice and defect states. Therefore, the first step for a general dynamic theory of crystal defects, eventually coupled with electromagnetic radiation, is to find the exact tight-binding eigenvectors and eigenvalues of a defect field strongly coupled with the host crystal. To solve the problem of tightly bound defects is the main difficulty since terms causing transitions between localized states of the defects (diffusion, optical excitations, luminescence, e t c . , . . . ) can be handled perturbatively. The strong coupling between imperfections and crystal vibrations is the cornerstone of any theory of defects. The small polaron transformation gives the exact eigenvectors of a crystal with tightly bound defects of arbitrary mass [5]. Nevertheless, this transformation, which is the main competitor of the method described here [6], assumes a potential energy
of interaction between the defects and the host crystal which is linear in the ionic displacements. Since defects couple strongly with the crystal they in general cause severe lattice deformations. Therefore, the quadratic terms in the expansion of the defect-host crystal interaction in powers of the ionic displacements cannot be dropped in the general case. On the other hand, important effects like the shift of vibrational frequencies and many-body terms of the lattice-mediated interaction between defects, are due to the quadratic coupling terms [1, 7]. The attempt to include quadratic coupling in the context of the theory of the small polaron has resulted in a very complicated formalism [8]. In this paper I propose a new transformation which diagonalizes the tight-binding Hamiltonian of the defect field strongly coupled to the host crystal by linear and quadratic terms. The exact eigenvectors and eigenvalues of the system are obtained in closed form. In order to be definite I will discuss the special case of interstitial impurities, nevertheless, the generalization to other point defects is straightforward. The very restrictive Born-Oppenheimer and Condon approximations are dropped. An important lateral result of the new transformation is the precise definition of vibrational excitations of an imperfect crystal. These excitations (which I call "phonions", since are defined in terms of phonon and defect operators) are independent of the number, spatial distribution and internal state of the defects. When expressed in terms of "phonion" operators the tightbinding state vectors of the system take a very simple form. The application of them to calculate other effects, like phonon-assisted diffusion, light absorption or luminescence, is particularly simple and straightforward. A detailed study of these applications is left, nevertheless, to a separate publication and I discuss here just 51
52
Vol. 53, No. 1
QUANTUM THEORY OF INTERSTITIAL IMPURITIES
direct implications, as deformation, self-trapping and lattice-mediated interaction between defects. The Hamiltonian of a crystal containing impurities can be written as H = no + n , ,
(1)
where {~tx } are quantum numb ers giving the'vibrational state and {rh} are the occupation numbers of impurity states. For this aim define the new annihilation and creation operators, which I will call "phonion" operators to distinguish them from the phonon excitations, bo = E (Aoxax + Boxa~)
where
X
Ho =
y.
"t hooxaxax + E
X
1
e,c*zc,+ Egx,cTc,(ax-a* x) Xl
E
--
i=1 l l . . . l
+ ~, ~.hxKtc¢lcl(ax--a¢x)(aK--a'~,), l
(2)
Y (Aox-Box) rg),,.. ,
. l i e l
*l c l I
.
.
? (5)
.. • ¢liCli,
)~X'
b~"t =
and
.
i h
Y (Bpxax + Aoxa~.) h
H1
= E ,.,c;c,, + E E gxH'¢>,' (~x - 4 , ) 1~I'
l~l'
+ ~.
X
Y
i=1 l l . . . l
+EE
14: l'XX'
hxx'u'e,*~,' (~x -- ~ ) (~x' -- ~ )
(3)
Here cox denotes the frequency of the vibrational mode (/~, q) - X of the perfect crystal, being/~ the branch and q the wavevector of the mode, ax anda*x are the corresponding phonon operators. The symbol X means X - (/a, -- q). The field operator for the impurities was expanded in the Wannier base associated to an impurity in the rigid periodic lattice• The indexes l = (c~, l) characterize tight-binding Wannier states of the impurity, a standing for the excitation level and I denotes an interstitial site. The coefficients el = ec~ are the energy levels of the impurity in a site of the rigid undeformed lattice. The third term in the right hand side of equation (2) is the well-known polaron term [5] and the fourth one collects the quadratic terms which have been discussed in a previous paragraph. They give the coupling of the impurities with the vibrations of the deformed lattice. The coefficients gxt, gxn', hxx'l and hxx' u' represent the strength of the coupling between the impurities and the phonon gas. The terms in tu' follow from the fact that the Wannier functions are not exact wavefunctions for an impurity in the periodic lattice. For tightly bound (well localized) impurities one has el, gin, hxx'l >> tu'gxn', hxx'u' for l' 4= l and H1 is a very small term. The term H0 is diagonal in the site labels I and it determines the dynamics of the crystal with we!llocalized impurities. The small term H~ collects the offdiagonal terms (l --/=l') and gives the transition probabilities between localized states of the impurities• These transitions may involve a jump to a neighbouring site or excitation with no change in the site label. The goal is now to solve H0 I{Ux }{r/l }) = E{uM{n0 I{/ax }{rh })
(4)
E(AoX--B.x)Tig,,. . . . i
*
liCl ~ e l I . . .
h
¢
• .. clicti ,
(6)
where the coefficients Aox and Box are operators • depending only on the number O perators e t1 el and having the properties
A~X = Apx, ¢ E (ApxAo'x-h
B~X = Bo*X, BpxBox')
=
6pp,,
~. ( A o x B v K - - B o x A v x ) = 0,
(7)
(8a) (9a)
X
(AoxA~x, --BoxB~x, ) = 5xx, ,
(8b)
(AoxB~x, -- BpxA~x, ) = 0.
(9b)
P
O
They are determined by the linear equations }-'~(Aox' + Box') VxX' = h(cox + ~2o)Box X'
= h(coa -- g2p)Apx,
(10)
where
vxx, =- ~. (hxx'z+ hx,~)c*~cl
(11)
t
and ~ p is a solution of the equation d e t [ 2 Vxx'wx'cox' ~2+ h(c°x--~2)6x'vJ = 0 "
The factors Tg~)l,... ti are given by
(12)
QUANTUM THEORY OF INTERSTITIAL IMPURITIES
Vol. 53, No. 1
Tx( ~ ).= g~l hw~.' 1
•"~. Xi ~ hxxi-, ~i " ' "
"" " hx~x~l~
(13)
h~'-~X~'
where
hxx'l : 2
hxx't + hx'xt h @ - ~ x~ ' "
(14)
In spite of its involved appearance this transformation will prove to be manageable and these new excitations will result in a powerful tool in the study of imperfect crystals. When expressed in terms of the b- and c-operators the Hamiltonian Ho takes the separated form H o = Hvib + / / i m p '
(15)
where the vibrational term H~b is
Hvib
= Z mx(cttcl)(btbx + ~-)--~ ½h~x,
(16)
X
and the term Him p :
y . elCtlCl @ l
.v(i-~) t t + ~ E E gxl1~h;I2...IiCz~Cl~'''C'iC'i, i=2 l I . . . l i k (17)
determines the dynamics of the degrees of freedom associated to the interstitial impurities. Since [bx, C*lCl] = [bI,ctlcl]
-0
and [Hvib,Himp] = [nvib,no]
=
[Hi,p, Ho ] = 0 both terms can be studied separately. The derivation of equations (15), (16) and (17) involve considerable algebraic manipulations and the details of them will be given in a subsequent publication [9]. The energy spectrum follows directly from equations (15), (16) and (17). One has the explicit expression
53
where the sum £' runs over all terms having at least one index l different from the rest. The terms T(xi;~. . . . zi are related to the coefficients gxt and hhh, t, appearing in the original form of Ho, by the explicit equation (13). The coefficients Apx(r/t) and Bpx(rh) do not appear in the expression for the energy equation (18). This is fortunate since the equations (10) are not easy to solve. The only term which is not closed form in equation (18) is the term in fZ~.(rh). Nevertheless one has that f2~.(rh) = w x + O(Nt~t/N) and then, for low atomic concentrations ~ x ~ t°n. The first term in the right hand side of equation (18) is, of course, the vibrational energy of the deformed crystal. The second term is the energy shift of the vibrational ground state caused by the addition of impurities. The third term collects the energies of the polarons (the impurities and their phonon clouds) as they were isolated from the rest of the impurities. The fourth term is the lattice-mediated energy of interaction between impurities. This term contains all the dependence of the energy on the relative positions of the defects. The index i = 2, 3 , . . . in the fourth term characterizes the two body, three body . . . . terms of the lattice-mediated interactions. Since the deformation field caused by a foreign atom is long-ranged the manybody terms of the lattice-mediated interaction may be important. The eigenvectors of H0 are determined by the equations
c*zc[ [{P~.~{~h}) = rh {{/ax}{rh}),
(19)
b*xbxl{m,}{nl}> = U~,l{Ux}{m}>.
(20)
The general solution of this problem is a little bit complicated because [c~, b'x] has not a simple expression. Nevertheless, with some work one can show that the many-impurity eigenstates of Ho are given by I{,x}{rh}) = l-I (bxt)ux "----"~,, a I-I(c*l ) n l a - 1 1 0 0 ) , l h v#x'
(21)
where I0 0) is the ground state of the crystal with no impurities and
["1
Q = exp } ~ b x b x
,
h
+ ~
+
et +
L g x z * h ; l . . . l rh
1
i=2
t
~'
i=2
Q-1 = exp - ½ ~ b x b x
k
gx,, Txti;z=l.)...l, r/,, -. .rh,,
Iz • • • l i
Px = 0 , 1 , 2 . . . . .
•
h
rh = 0,1,
(18)
The transformation to phonion operators can be inverted using the quadratic equations (8b) and (9b). Subtracting equation (6) from equation (5), multiplying by A~X, + B~X', summing over p and replacing equations
QUANTUM THEORY OF INTERSTITIAL IMPURITIES
54 (8b) and (9b) one obtains
t aT, --a× = Y~ (Aa× + B~x)(bp --b~) ,o
•
*
t
+2 ~ E T
the vibrational modes of the perfect crystal and the force constants. Simple closed-form expressions for'St can be obtained assuming a Debye model for the various crystal symnretries, but I will omit them here because of the lack of space. The main results shown in this letter are, nevertheless, the spectrum (18) and the tightbinding eigenvectors (21). Applicaticms of them will be published elsewhere.
A similar procedure can be used to obtain ax + ax m terms of b x + bE and then the inverse transformation. I will not write it here since the point is now to obtain a general expression for the deformation field. The ionic displacements from the sites of the perfect lattice are given by
2.
ul = Z
3.
X
2,,~7~,_ Oxein'l( ax-a~,)' ,
(24)
/.lvlivoox
REFERENCES 1.
4.
where I denotes a site of the undeformed ionic lattice and characterizes the ion attached to it. Replacing equation (23) into equation (24) one obtains two terms. One of them, containing the operators b o -- b~, does not contribute to (u l) and represents oscillations respect to the new equilibrium positions. The remaining term is the deformation operator
5. 6.
7.
8. 11 . . . I i
k
0xe i ' a ' ' rx(gq.. ,
9.
K
*
*
• liCl I Cl I • . . CliCl
i.
(25)
This operator gives the deformation field for an arbitrary impurity distribution. To evaluate it one must know just
Vol. 53, No. 1
A.M. Stoneham, Theory of Defects in Solids, Clarendon Press, Oxford (1975). C.P. Flynn, Point Defects and Diffusion, Clarendon Press, Oxford (1972). Hydrogen in Metals I and H, (Edited by G. Alefeld & J. V61k), Springer, Berlin (1978). W. Fowler, Physics of Color Centers, Academic, New York (1968). B. Jackson & R. Silbey, J. Chem. Phys. 78, 4193 (1983) (and references therein). Calculations relying on the Born-Oppenheimer approximation for defect dynamics are not discussed here since they apply just to very light impurities (F-centers and ninon diffusion). A.A. Maradudin, E.W. Montroll & G.H. Weiss, Solid State Phys. Suppl. 3 (1963). D.L. Tonks & B.G. Dick,Phys. Rev. B19, 1136 (1979); D.L. Tonks, Phys. Rev. B22, 6420 (1980). Equations (15), (16) and (17) can be easily proved in the case of linear coupling, where hxx, l = 0. To show the general case invert the transformation by the procedure described in the last paragraph, leading to ax +- a~, in terms of b~, and b*x. Then replace in equation (2) and cancel terms.