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Influence of matrix elements on susceptibilities
Journal of Magnetism and Magnetic Materials 11.5(1992) 163-167 North-Holland
Influence of matrix elements on susceptibilities
*
W. Yeung Department of Physics, Queen Mary and Westfield College, Mile End Road, London El 4NS, UK
We consider the effect of neglecting the k dependence of the matrix elements in the calculation of susceptibilities x(q,w). We derive explicit results for the unenhanced x for a simple cubic tight-binding band and show the dramatic influence of the matrix element or form factor on the susceptibility. As a function of band filling, x(q, 0) changes from being symmetric about half filling to being asymmetric when the k dependence of the form factors are included.
With the increasing availability of ever more powerful supercomputers the calculation of response functions of susceptibilities for complex systems using first-principles band structures is now practicable. Such calculations are still formidable undertakings and sometimes one neglects the k dependence of the matrix elements in the Lindhard sum. Recently Winter et al. [l] pointed out that the matrix elements can influence the susceptibility considerably especially in the region of large q. For example large values of ,y(q, q, 0) predicting phase transitions at q may not be that large when the matrix elements or form factors are taken into account. In this communication we consider the problem of form factors and their effect on the calculation of x. Since the calculation of x for actual band structures involves the calculation and processing of masses of intermediate results it would be very useful if one can have a simple model with which to compared one’s results. For the simple text book model of free electrons one can calculate the unenhanced susceptibility analytically. Unfortunately the form factor is unity in this case and therefore one cannot learn anything about the effects of the matrix elements. Here we consider a simple cubic tight-binding band and show that for the case of q along one of the lattice directions we can derive almost analytic results within a tight-binding approximation. We use this simple one-band model to analyse the influence of the form factors on the unenhanced susceptibility. The unenhanced susceptibility may be written as
x(q + G,
4,
4 = / de’ de Jq+~,,+‘, 6)
f(E’> --f(E) .5’-•--_-iq
(1)
in terms of the joint density of states (JDOS) [21, J defined by
(2) where G is a reciprocal lattice vector, n,n’ are and indices and we have used a convention in which numerically subscripted variables are summed or integrated over their respective domains. The so-called matrix elements M are given by
Correspondence
to: W. Yeung, Dept. of Physics, Queen Mary and Westfield College, Mile End Road, London El 4NS, UK. * Presented at ICM ‘91, 2-6 September 1991, Edinburgh, Scotland.
0304-8853/92/$05.00
0 1992 - Elsevier Science Publishers B.V. All rights reserved
164
W. Yeung / Influence of matrix elements on susceptibilities
In a tight-binding LMTO [3] formalism with basis states pL,L = (Im) and two centre integrals [4] FLtL defined by FLtL(p, R) =/qLf(r)
e-iP.r(p~(r--R)
dr,
(4)
the above form factors are given by
$d~)
=&.&J,
R,) e-i“‘R%,KlbL*,r,
(5)
where b,, relates the band states 4, to the LMTO orbitals (pL. The composite index K denotes the band y1and the wave vector k. For simple cases with explicit forms for the band energies one can sometimes obtain analytic expressions for JDOS. For instance, for free electrons where ek = h2k2/2m and M is unity we have simply 7
J&‘,
1
m ‘2 E) = (2n)3 i 12 4
for
I&-&,1
<@<&4+&,
(6)
= 0 otherwise where ‘SY is the volume of the solid. Substituting expression for static susceptibility:
into eq. (1) we obtain for instance the standard
where lq = h2q2/2m. The logarithmic singularity 2,,& free electrons. We now consider the simple cubic band defined by Ed = -t(cos
= eq is just the Kohn anomaly at q = 2k,
for
ak, + cos ak, + cos ak,),
where t and a are the hopping parameter and the lattice spacing. For simplicity we choose a q in one of the lattice directions, the z-direction. The curve C where the surfaces E = Ed and E’ = E~+~ intersect lies in the X-Y plane with k, determined by q, E and E’. If we define the dimensionless quantities E’+E E’-•E t=T,n=T,5=sinT,
cos aq c=2’
(8) C=/a,2A=77+cC,
a straight forward calculation gives 7
M(k 4)
J&E’, E) = (2~)~ / c 4( ~zt)~sC (sin*ak,/2
dkx - A)(cos2ak,/2
+ A)
*
(9)
In contrast to the free-electron model the matrix elements M are not constant with respect to k. For a one-band model the b,‘s in eq. (5) are constants and are absorbed in the orbitals (pr. The only k dependence occurs in the sum over the exponent eik’R. For finite q the two-centre integral F(q, R)
W Yeung / Injluence of matrix elements on susceptibilities
165
decays rather quickly. In the spirit of the tight-binding approximation we restrict the sum over R to the origin and to the nearest neighbours R,. For simplicity we assume an s-band wave function of the form (rr~$‘/~ exp (-r/a,> for cp. In this case F(q, R) depends only on the magnitudes of q, R and the angle between them. For q along the z-direction we have two types of nearest neighbours: those with R along an perpendicular to q. Because of the delta functions in eq. (2) we note that M = I F I 2 is only required for k’s on curve C along which k, is constant and k, and k, satisfy cos ak, + cos ak, = -2A. Therefore M is constant along C and depends only on q, 77 and 5. Indeed Y k+q,k=F(q,
0) +F+(s)(cC+5)
+F-(q)(-sC+c&/s)
-2F%+.
(10)
The joint density of states may be written as
(11) where
K(A)
= l:n/:fi
/(sin28 -A;Fcos2,
+A)
(12)
’
Note that A in eq. (11) depends only on q, TJ and 5 and that K(A)
has a logarithmic singularity at
A=O.
Compared to the free-electron case the JDOS of our simple one-band model possesses a remarkable richness in structure. The free-electron JDOS is a constant with respect to 77 and 5 and drops discontinuously to zero outside a certain region whereas our one-band model has a logarithmic singularity via K(A). Furthermore it also has an inverse square-root singularity at 5 = sin aq/2. For a finite q JDOS is nonvanishing only for energy difference 5 such that I E’ - E I I 2t sin(aq/2). This leads to a restriction in the absorption spectrum which is a consequence of the periodic band structure. At absolute zero the unenhanced susceptibility is given by
X(&
47
0) = t/J&L S)
c--v-_5)-~(cL-77+5)
drl d5,
5
(13)
where fI is the step function and p = eF/t. Substituting for J, we have
24(7715) K(A)
7”
x(4, 430) = - ~
(2Tra)3
//
tsc
-
(14)
dq d[.
5
The region of integration is such that E and E’ are on opposite sides of the Fermi surface and restricted by the condition I< (
4Y5i7 Xap,(47 430) =
t(2Ta)3
’ K( ll+pl/2)
/ -1
In
[
ItI
1
dLc*
(15)
We observe that the above expression is an even function of the Fermi energy p. This integral is finite and has been evaluated numerically. A plot of xapp as a function of the Fermi energy is given in fig. 1. We see that the maximum value of x occurs at half filling.
W. Yeung / Influence of matrix elements on susceptibilities
166
I
-1
-0.5
1
0.5
Fig. 1. (a) x and (b) x with A4 replaced
by an average
as a function
of cF/t
Let us now include the k dependence of M. To do this we consider the various terms in eq. (10). For simplicity we consider a typical case where the size a. of the orbital rp is between one and two lattice spacings a. Closer scrutiny of the coefficients F * in this case shows that they are much smaller than the coefficient F(q, 0). For F’ we have cancellation because of the oscillation of a codqz) factor and for Ftwo almost equal exponentials cancel. The coefficient F ’ is of the same order of magnitude as F(q, 0). For the chosen value of q we also have c = cos aq/2 = 0 and therefore A = 77/2. Inserting the resulting expression for M = @ and integrating over 5 we have 47
X(4>49 0) =
t(2+3
/i
-1
K(I~+cLI/~)(F~~-F~(~.+~))’ ln
(16)
We now observe that the above expression for x is not symmetric about I_L= 0. The symmetry reflects particle-hole symmetry in the energies. The plot of x with the correct expression for the matrix element is also shown in fig. 1. Besides being quantitively different the curves show very different behaviours. We see now that the largest x comes from small fillings. The matrix elements are really the integrated effects of the usual selection rules. As such the character of the wave function and in particular the Fourier transform of the overlap of wave functions play an important role. In the theory of phase transitions one can show 121that there is a SDW antiferromagnetic instability if 1x(Q, 0) > 1. Thus a large value of x(Q, 0) at a particular filling will signal the onset of an antiferromagnetic state. It is seen from the figure that we can get different predictions depending on whether one includes or ignores the dependence of the matrix elements on k. For completeness we give the imaginary part of x namely Im x(4,
4, w) = -
2nV t(2%7)3
1 J1-yz
Iv K( Ii+~1/2)(F,~-FL(i.+~))2 -lJ
45,
(17)
where v = hw/2t. Again there is much structure compared to the free-electron case. In summary we have developed a simple cubic one-band model and give explicit and almost analytic expressions for the joint density of states, x(q, q, 0) and the imaginary part of x. We show that this model provides a rich structure from which one can study lattice effects and those due to matrix elements in the Lindhard expression. We have shown explicitly the influence of the matrix elements or the form factors on the susceptibility and show how they can dramatically change the behaviour of x. For
W. Yeung / Influence of matrix elements on susceptibilities
167
example we change from a behaviour which is symmetric about half filling to one which is dominated by small fillings. This can lead to very different predictions about the behaviour of the system.
Acknowledgement I wish to thank Derek Crockford for useful comments and discussions.
References [l] [2] [3] [4]
H. Winter, Z. Szotek and W.M. Temmerman, 2. Phys. B 79 (1990) 241. M.W. Long and W. Yeung, J. Phys. F 16 (1986) 769. O.K. Andersen et al., Phys. Rev. B 70 (1986) 5253. 0. Gunnarson et al., Phys. Rev. B 40 (1989) 12140.
Influence of matrix elements on susceptibilities
*
W. Yeung Department of Physics, Queen Mary and Westfield College, Mile End Road, London El 4NS, UK
We consider the effect of neglecting the k dependence of the matrix elements in the calculation of susceptibilities x(q,w). We derive explicit results for the unenhanced x for a simple cubic tight-binding band and show the dramatic influence of the matrix element or form factor on the susceptibility. As a function of band filling, x(q, 0) changes from being symmetric about half filling to being asymmetric when the k dependence of the form factors are included.
With the increasing availability of ever more powerful supercomputers the calculation of response functions of susceptibilities for complex systems using first-principles band structures is now practicable. Such calculations are still formidable undertakings and sometimes one neglects the k dependence of the matrix elements in the Lindhard sum. Recently Winter et al. [l] pointed out that the matrix elements can influence the susceptibility considerably especially in the region of large q. For example large values of ,y(q, q, 0) predicting phase transitions at q may not be that large when the matrix elements or form factors are taken into account. In this communication we consider the problem of form factors and their effect on the calculation of x. Since the calculation of x for actual band structures involves the calculation and processing of masses of intermediate results it would be very useful if one can have a simple model with which to compared one’s results. For the simple text book model of free electrons one can calculate the unenhanced susceptibility analytically. Unfortunately the form factor is unity in this case and therefore one cannot learn anything about the effects of the matrix elements. Here we consider a simple cubic tight-binding band and show that for the case of q along one of the lattice directions we can derive almost analytic results within a tight-binding approximation. We use this simple one-band model to analyse the influence of the form factors on the unenhanced susceptibility. The unenhanced susceptibility may be written as
x(q + G,
4,
4 = / de’ de Jq+~,,+‘, 6)
f(E’> --f(E) .5’-•--_-iq
(1)
in terms of the joint density of states (JDOS) [21, J defined by
(2) where G is a reciprocal lattice vector, n,n’ are and indices and we have used a convention in which numerically subscripted variables are summed or integrated over their respective domains. The so-called matrix elements M are given by
Correspondence
to: W. Yeung, Dept. of Physics, Queen Mary and Westfield College, Mile End Road, London El 4NS, UK. * Presented at ICM ‘91, 2-6 September 1991, Edinburgh, Scotland.
0304-8853/92/$05.00
0 1992 - Elsevier Science Publishers B.V. All rights reserved
164
W. Yeung / Influence of matrix elements on susceptibilities
In a tight-binding LMTO [3] formalism with basis states pL,L = (Im) and two centre integrals [4] FLtL defined by FLtL(p, R) =/qLf(r)
e-iP.r(p~(r--R)
dr,
(4)
the above form factors are given by
$d~)
=&.&J,
R,) e-i“‘R%,KlbL*,r,
(5)
where b,, relates the band states 4, to the LMTO orbitals (pL. The composite index K denotes the band y1and the wave vector k. For simple cases with explicit forms for the band energies one can sometimes obtain analytic expressions for JDOS. For instance, for free electrons where ek = h2k2/2m and M is unity we have simply 7
J&‘,
1
m ‘2 E) = (2n)3 i 12 4
for
I&-&,1
<@<&4+&,
(6)
= 0 otherwise where ‘SY is the volume of the solid. Substituting expression for static susceptibility:
into eq. (1) we obtain for instance the standard
where lq = h2q2/2m. The logarithmic singularity 2,,& free electrons. We now consider the simple cubic band defined by Ed = -t(cos
= eq is just the Kohn anomaly at q = 2k,
for
ak, + cos ak, + cos ak,),
where t and a are the hopping parameter and the lattice spacing. For simplicity we choose a q in one of the lattice directions, the z-direction. The curve C where the surfaces E = Ed and E’ = E~+~ intersect lies in the X-Y plane with k, determined by q, E and E’. If we define the dimensionless quantities E’+E E’-•E t=T,n=T,5=sinT,
cos aq c=2’
(8) C=/a,2A=77+cC,
a straight forward calculation gives 7
M(k 4)
J&E’, E) = (2~)~ / c 4( ~zt)~sC (sin*ak,/2
dkx - A)(cos2ak,/2
+ A)
*
(9)
In contrast to the free-electron model the matrix elements M are not constant with respect to k. For a one-band model the b,‘s in eq. (5) are constants and are absorbed in the orbitals (pr. The only k dependence occurs in the sum over the exponent eik’R. For finite q the two-centre integral F(q, R)
W Yeung / Injluence of matrix elements on susceptibilities
165
decays rather quickly. In the spirit of the tight-binding approximation we restrict the sum over R to the origin and to the nearest neighbours R,. For simplicity we assume an s-band wave function of the form (rr~$‘/~ exp (-r/a,> for cp. In this case F(q, R) depends only on the magnitudes of q, R and the angle between them. For q along the z-direction we have two types of nearest neighbours: those with R along an perpendicular to q. Because of the delta functions in eq. (2) we note that M = I F I 2 is only required for k’s on curve C along which k, is constant and k, and k, satisfy cos ak, + cos ak, = -2A. Therefore M is constant along C and depends only on q, 77 and 5. Indeed Y k+q,k=F(q,
0) +F+(s)(cC+5)
+F-(q)(-sC+c&/s)
-2F%+.
(10)
The joint density of states may be written as
(11) where
K(A)
= l:n/:fi
/(sin28 -A;Fcos2,
+A)
(12)
’
Note that A in eq. (11) depends only on q, TJ and 5 and that K(A)
has a logarithmic singularity at
A=O.
Compared to the free-electron case the JDOS of our simple one-band model possesses a remarkable richness in structure. The free-electron JDOS is a constant with respect to 77 and 5 and drops discontinuously to zero outside a certain region whereas our one-band model has a logarithmic singularity via K(A). Furthermore it also has an inverse square-root singularity at 5 = sin aq/2. For a finite q JDOS is nonvanishing only for energy difference 5 such that I E’ - E I I 2t sin(aq/2). This leads to a restriction in the absorption spectrum which is a consequence of the periodic band structure. At absolute zero the unenhanced susceptibility is given by
X(&
47
0) = t/J&L S)
c--v-_5)-~(cL-77+5)
drl d5,
5
(13)
where fI is the step function and p = eF/t. Substituting for J, we have
24(7715) K(A)
7”
x(4, 430) = - ~
(2Tra)3
//
tsc
-
(14)
dq d[.
5
The region of integration is such that E and E’ are on opposite sides of the Fermi surface and restricted by the condition I< (
4Y5i7 Xap,(47 430) =
t(2Ta)3
’ K( ll+pl/2)
/ -1
In
[
ItI
1
dLc*
(15)
We observe that the above expression is an even function of the Fermi energy p. This integral is finite and has been evaluated numerically. A plot of xapp as a function of the Fermi energy is given in fig. 1. We see that the maximum value of x occurs at half filling.
W. Yeung / Influence of matrix elements on susceptibilities
166
I
-1
-0.5
1
0.5
Fig. 1. (a) x and (b) x with A4 replaced
by an average
as a function
of cF/t
Let us now include the k dependence of M. To do this we consider the various terms in eq. (10). For simplicity we consider a typical case where the size a. of the orbital rp is between one and two lattice spacings a. Closer scrutiny of the coefficients F * in this case shows that they are much smaller than the coefficient F(q, 0). For F’ we have cancellation because of the oscillation of a codqz) factor and for Ftwo almost equal exponentials cancel. The coefficient F ’ is of the same order of magnitude as F(q, 0). For the chosen value of q we also have c = cos aq/2 = 0 and therefore A = 77/2. Inserting the resulting expression for M = @ and integrating over 5 we have 47
X(4>49 0) =
t(2+3
/i
-1
K(I~+cLI/~)(F~~-F~(~.+~))’ ln
(16)
We now observe that the above expression for x is not symmetric about I_L= 0. The symmetry reflects particle-hole symmetry in the energies. The plot of x with the correct expression for the matrix element is also shown in fig. 1. Besides being quantitively different the curves show very different behaviours. We see now that the largest x comes from small fillings. The matrix elements are really the integrated effects of the usual selection rules. As such the character of the wave function and in particular the Fourier transform of the overlap of wave functions play an important role. In the theory of phase transitions one can show 121that there is a SDW antiferromagnetic instability if 1x(Q, 0) > 1. Thus a large value of x(Q, 0) at a particular filling will signal the onset of an antiferromagnetic state. It is seen from the figure that we can get different predictions depending on whether one includes or ignores the dependence of the matrix elements on k. For completeness we give the imaginary part of x namely Im x(4,
4, w) = -
2nV t(2%7)3
1 J1-yz
Iv K( Ii+~1/2)(F,~-FL(i.+~))2 -lJ
45,
(17)
where v = hw/2t. Again there is much structure compared to the free-electron case. In summary we have developed a simple cubic one-band model and give explicit and almost analytic expressions for the joint density of states, x(q, q, 0) and the imaginary part of x. We show that this model provides a rich structure from which one can study lattice effects and those due to matrix elements in the Lindhard expression. We have shown explicitly the influence of the matrix elements or the form factors on the susceptibility and show how they can dramatically change the behaviour of x. For
W. Yeung / Influence of matrix elements on susceptibilities
167
example we change from a behaviour which is symmetric about half filling to one which is dominated by small fillings. This can lead to very different predictions about the behaviour of the system.
Acknowledgement I wish to thank Derek Crockford for useful comments and discussions.
References [l] [2] [3] [4]
H. Winter, Z. Szotek and W.M. Temmerman, 2. Phys. B 79 (1990) 241. M.W. Long and W. Yeung, J. Phys. F 16 (1986) 769. O.K. Andersen et al., Phys. Rev. B 70 (1986) 5253. 0. Gunnarson et al., Phys. Rev. B 40 (1989) 12140.