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An analytical model for the density of states
PHYSICS LETTERS
Volume 48A, number 1
AN ANALYTICAL
20 May 1974
MODEL FOR THE DENSITY
OF STATES
T. OGAWA and T. OGAWA Department
of physics, Faculty of Science, Kyoto University, Kyoto, Japan
Received 3 March 1974 Analytical expressions are proposed for the density of states which approximate those of only-nearest-neighbouroverlapping bands (ONNOB). They have the correct type of singularities and the correct values of some lowest order moments.
For a given dispersion relation E(k) expressing the energy E as an analytical function of the momentum k, the density of states, defined by p(E) = JI (E - E(k)) dk/l
dk (The integration should be performed over a single Brillouin-zone .) ,
generally includes some elliptic integrals. Therefore if one wishes to perform some calculations using such a density of states one has been forced to carry out these calculations numerically or to use some unrealistically simplified models: constant, parabolic, elliptic etc. In this paper a model for the density of states (MDS) is proposed which enables us to make analytical calculations with results satisfactory both qualitatively and quantitatively for practical purposes. The fundamental principle in its derivation is to give the correct type of singularities and the correct values of some lowest order moments. The density of states has so-called Van Hove singularities (VHS) [ 1,2] at the energies E = E, = E(k,), where k, satisfies the equation [grad @‘(k)lk=k,
=0 ,
(1)
and by introducing a proper rectangular coordinate system (.!j,n, c), the Taylor expansion of E(k) at k = k, can be written as E=E,
taq;
tbq;
+C+0(q3),
(2)
where 4 E k - k,. Then the type of VSH are classified according to the signs of a, b and c as shown in table 1. The procedure to determine MDS for a given E(k) is as follows: 1) Evaluation of some lowest order moments
Table 1 Classification of van Hove singularities (VHS). Dimensionality
Classifification
Signs of a,b,c
Type of singularity
1
type ICI type I I
+
I/G
type
++
e(A)
+_-
-lnlAl f%- A)
i-t+
fi
2
110
type II I
type 112 type type type type
3
III0 III 1 I& III3
-
+++-- - -
-1/A
-x/xi >$
In the last column, A = E - E,. In the case where any of a, b and c is zero, higher order terms in the Taylor expansion govern the type of the singularities.
M, = j
Enp(E)dE=
J[E(k)]“dk/Jdk,
(3)
-00 by momentum integration or by counting the number of ways of return at n th step of nearest neighbour walk on the lattice (in the case of ONNOB discussed later). 2) Investigation of the type of VHS appearing in the genuine density of states. 3) Assumption of the functional form of MDS, P,+#‘), having the correct type of VHS investigated in 2). 4) Determination of the parameters appearing in pi&E), though the condition that MDS has the correct values of some lowest order moments,
51
Volume 48A, number 1 pn =
~“,YI~(E)dE
PHYSICS LETTERS
= M, .
(4)
In the following, three examples of MDS are shown: ONNOB of square-, simple cubic- and body centred cubic lattices. Square: E(k,, kY) = cos k, + cos kY. l)h!f,= 1,h’f,=1,bf,=9/4,(M,,+, =o). 2) type II, at E(n, n) = - 2, type III at E(n, 0) = 0, type II, at E(O,O) = 2. 3) P,~(E) =Aln(2/lEl) +B t CE2 for 1El =G2. 4) A = 0.098877, B = 0.146851, C = 0.003204. Simple cubic: E(k, , kv , k,) = cos k, t cos ky + cos kZ. 1) MO = 1, M2 = 312, M4 = 45/8, (M2n+l = 0). 2) type III, at E(n, rr, n) = -3, type III, at E(rr ,n, 0) = - 1, type III2 at E(n, 0,O) = 1, type III, at E(O,O,O) = 3. 3) p,u(E)=A~+B{JZ-8(IEI-l)JiEI_I} t-C(E2 -9) for 1El < 3. 4)A=0.030491, B=0.213231, C=O.O12118. Body centered cubic: E(k,, ky , k,) = cos k, cos ky cos k, . 1) MO = 1, M;! = l/8, M4 = 27/512, (M2n+l = 0). 2) type III, at E(n, 0,O) = - 1, type II, at E(n/2,n/2,0)=o*‘, type 1113 at E(O,O,O) = 1, type of (lnA)2 at E(n/2, n/2, n/2) = 0, where E = qxqyqz * 2. 3) P,~(E)=AJI - E2 - BlnlEl+ (ElGl.
C(lnlE1)2
for
*l This cIroice of the type of singularity contradicts the statement in the caption of table 1. According to [ 31, this singularity is of order Aa(lnA)2. The present choice, however, makes the results better. We think that it is because higher order singularities make it effectively of a logarithmic type.
*2 This singularity was discussed in two papers quoted in [ 3 1.
52
20 May 1974
4) A = 0.171489, B = 0.206523, C= 0.079395. In order to test the validity of MDS, some higher order moments are estimated &MS = 1.00027, tices, ps/M6 = 1.00188, p6/M6 = 0.99616,
pg/M8 = 1.00057 for square latps/M8 = 1.00457 for S.C.lattice, p8/M8 = 0.99130 forb.c.c.lattice.
We do not insert the figure showing direct comparison of MDS with that obtained numerically [4], because agreement of them is so good that it is impossible to draw two distinguishable curves. These facts show that MDS is applicable for many problems with excellent reliability. It will be especially powerful for Fermi-type problems where the density of states for the whole energy region is required in the investigation of the particle density dependence. Finally, we refer to Jelitto’s paper [5] in which similar expressions have been derived as experimental formulae from the results of numerical calculations. It should be noticed that the present formulae are simple and the derivation is non-numerical. The authors wish to thank Professor Tohru Morita for valuable discussions.
References [l] L. Van Hove, Phys. Rev. 89 (1953) 1189. [2] J.G. Phillips, Phys. Rev. 104 (1956) 1263. [3] S. Katsura and T. Horiguchi, .I. Math. Phys. 12 (1971) 230; T. Morita andT. Horiguchi, J. Math. Phys. 12 (1971) 986. [4] T. Morita and T. Horiguchi, Table of lattice Green’s function for the cubic lattices (Values at the ‘origin), 197 1, Appl. Math. Res. Group, Dept. Appl. Sci., Faculty of Engineering, Tohoku Univ., Sendai, Japan. [5] R.J. Jelitto, J. Phys. Chem. Solids 30 (1969) 609.
Volume 48A, number 1
AN ANALYTICAL
20 May 1974
MODEL FOR THE DENSITY
OF STATES
T. OGAWA and T. OGAWA Department
of physics, Faculty of Science, Kyoto University, Kyoto, Japan
Received 3 March 1974 Analytical expressions are proposed for the density of states which approximate those of only-nearest-neighbouroverlapping bands (ONNOB). They have the correct type of singularities and the correct values of some lowest order moments.
For a given dispersion relation E(k) expressing the energy E as an analytical function of the momentum k, the density of states, defined by p(E) = JI (E - E(k)) dk/l
dk (The integration should be performed over a single Brillouin-zone .) ,
generally includes some elliptic integrals. Therefore if one wishes to perform some calculations using such a density of states one has been forced to carry out these calculations numerically or to use some unrealistically simplified models: constant, parabolic, elliptic etc. In this paper a model for the density of states (MDS) is proposed which enables us to make analytical calculations with results satisfactory both qualitatively and quantitatively for practical purposes. The fundamental principle in its derivation is to give the correct type of singularities and the correct values of some lowest order moments. The density of states has so-called Van Hove singularities (VHS) [ 1,2] at the energies E = E, = E(k,), where k, satisfies the equation [grad @‘(k)lk=k,
=0 ,
(1)
and by introducing a proper rectangular coordinate system (.!j,n, c), the Taylor expansion of E(k) at k = k, can be written as E=E,
taq;
tbq;
+C+0(q3),
(2)
where 4 E k - k,. Then the type of VSH are classified according to the signs of a, b and c as shown in table 1. The procedure to determine MDS for a given E(k) is as follows: 1) Evaluation of some lowest order moments
Table 1 Classification of van Hove singularities (VHS). Dimensionality
Classifification
Signs of a,b,c
Type of singularity
1
type ICI type I I
+
I/G
type
++
e(A)
+_-
-lnlAl f%- A)
i-t+
fi
2
110
type II I
type 112 type type type type
3
III0 III 1 I& III3
-
+++-- - -
-1/A
-x/xi >$
In the last column, A = E - E,. In the case where any of a, b and c is zero, higher order terms in the Taylor expansion govern the type of the singularities.
M, = j
Enp(E)dE=
J[E(k)]“dk/Jdk,
(3)
-00 by momentum integration or by counting the number of ways of return at n th step of nearest neighbour walk on the lattice (in the case of ONNOB discussed later). 2) Investigation of the type of VHS appearing in the genuine density of states. 3) Assumption of the functional form of MDS, P,+#‘), having the correct type of VHS investigated in 2). 4) Determination of the parameters appearing in pi&E), though the condition that MDS has the correct values of some lowest order moments,
51
Volume 48A, number 1 pn =
~“,YI~(E)dE
PHYSICS LETTERS
= M, .
(4)
In the following, three examples of MDS are shown: ONNOB of square-, simple cubic- and body centred cubic lattices. Square: E(k,, kY) = cos k, + cos kY. l)h!f,= 1,h’f,=1,bf,=9/4,(M,,+, =o). 2) type II, at E(n, n) = - 2, type III at E(n, 0) = 0, type II, at E(O,O) = 2. 3) P,~(E) =Aln(2/lEl) +B t CE2 for 1El =G2. 4) A = 0.098877, B = 0.146851, C = 0.003204. Simple cubic: E(k, , kv , k,) = cos k, t cos ky + cos kZ. 1) MO = 1, M2 = 312, M4 = 45/8, (M2n+l = 0). 2) type III, at E(n, rr, n) = -3, type III, at E(rr ,n, 0) = - 1, type III2 at E(n, 0,O) = 1, type III, at E(O,O,O) = 3. 3) p,u(E)=A~+B{JZ-8(IEI-l)JiEI_I} t-C(E2 -9) for 1El < 3. 4)A=0.030491, B=0.213231, C=O.O12118. Body centered cubic: E(k,, ky , k,) = cos k, cos ky cos k, . 1) MO = 1, M;! = l/8, M4 = 27/512, (M2n+l = 0). 2) type III, at E(n, 0,O) = - 1, type II, at E(n/2,n/2,0)=o*‘, type 1113 at E(O,O,O) = 1, type of (lnA)2 at E(n/2, n/2, n/2) = 0, where E = qxqyqz * 2. 3) P,~(E)=AJI - E2 - BlnlEl+ (ElGl.
C(lnlE1)2
for
*l This cIroice of the type of singularity contradicts the statement in the caption of table 1. According to [ 31, this singularity is of order Aa(lnA)2. The present choice, however, makes the results better. We think that it is because higher order singularities make it effectively of a logarithmic type.
*2 This singularity was discussed in two papers quoted in [ 3 1.
52
20 May 1974
4) A = 0.171489, B = 0.206523, C= 0.079395. In order to test the validity of MDS, some higher order moments are estimated &MS = 1.00027, tices, ps/M6 = 1.00188, p6/M6 = 0.99616,
pg/M8 = 1.00057 for square latps/M8 = 1.00457 for S.C.lattice, p8/M8 = 0.99130 forb.c.c.lattice.
We do not insert the figure showing direct comparison of MDS with that obtained numerically [4], because agreement of them is so good that it is impossible to draw two distinguishable curves. These facts show that MDS is applicable for many problems with excellent reliability. It will be especially powerful for Fermi-type problems where the density of states for the whole energy region is required in the investigation of the particle density dependence. Finally, we refer to Jelitto’s paper [5] in which similar expressions have been derived as experimental formulae from the results of numerical calculations. It should be noticed that the present formulae are simple and the derivation is non-numerical. The authors wish to thank Professor Tohru Morita for valuable discussions.
References [l] L. Van Hove, Phys. Rev. 89 (1953) 1189. [2] J.G. Phillips, Phys. Rev. 104 (1956) 1263. [3] S. Katsura and T. Horiguchi, .I. Math. Phys. 12 (1971) 230; T. Morita andT. Horiguchi, J. Math. Phys. 12 (1971) 986. [4] T. Morita and T. Horiguchi, Table of lattice Green’s function for the cubic lattices (Values at the ‘origin), 197 1, Appl. Math. Res. Group, Dept. Appl. Sci., Faculty of Engineering, Tohoku Univ., Sendai, Japan. [5] R.J. Jelitto, J. Phys. Chem. Solids 30 (1969) 609.