222
Progress of Theoretical Physics, Vol. 16, No. 3, September 1956
The Energy Band Structure of the Metallic Copper
--The Orthogonalized Plane Wave Method-Mitsuru FUKUCHI
Institute of Science and Technology, University of Tokyo, Tokyo
{Received May 24, 1956)
The electronic band structure in the metallic copper is worked by the orthogonalized plane wave
method in which the crystal potential is constructed by using both the Hartree's self-consistent field
of Cu+ ion cores with exchange and conduaion electrons with the Slater's approximate exchange
potential.
The approximate energy versus k curve is obtained, which shows the strong resemblance to the
it
free electron behaviour along three directions in space, i.e., (1.0.0), {1.1.1) and {1.1.0). The results
of the various methods of approximations performed so far are compared with ours together with
some critical discussions.
§ 1. Introduction
As is well known, it is one of the fundamental problems of the quantum theory
of solids to obtain the accurate knowledge about the electronic band structure of the actual
crystals to which the manifold of physical properties characteristic of solid state can be
ascribed at least in principle.
Although many efforts along these lines have been made
so far and are still being continued, they do not seem to have attained any convincing
conclusions even for rather simple crystals.
As for the alkali metals, for which the most thorough investigations have been made
so far, the different methods of approximations are known to give rise to some discrepancies
in the detailed behaviours of the band s~cture, although the general behaviours are
consistent with each other, getting agreements with experiment, as shown, for sodium, by
Lage & Bethe1 > and by Howarth & Jones/> and, for lithium, by Schiff,'ll Kohn &
Rostoker/l by Parzen5l and by Parmenter.6> Even for the simplest metals, therefore, the
problem does not seem to be settled completely.
As for the polyvalent metals such as beryllium, 7' magnesium, 8l titanium,9l iron10l
and nickel, 11 > the various methods of approximations have been applied to deal with the
electronic band structure, however the results of the computation seem to be far less
reliable compared with those of alkali metals owing to the polyvalent character of the
respective crystals.
Between the alkali metals and the polyvalent metals there are monovalent noble
metals, whose band structures have been studied also by many workers. Above all, the
most detailed investigations have been performed for the metallic copper by Fuchs,12'
223
The Energy Band Structure of the Metallic Copper
Krutter/ 3> Tibbs14l and Howarth15),lBl, who have adopted the Wigner-Seitz cellular method
and the augmented plane wave method, respectively. Although their calculations provide
useful knowledge about the band structure of the metallic copper, there still remain
discrepancies, qualitative or quantitative, in the detailed behaviours of energy E versus
wave number k according to the methods of approximations mentioned above. In particular,
the final results obtained are remarkably sensitive to the potential to be used, as shown
by Howarth.
In view of such unsettled situation in metallic copper, it seems to be highly desirable
to study the band structure in copper by the use of the orthogonalized plane wave method
(O.P.W.) proposed by Herring17l and to compare the results with those obtained by
other methods, since the O.P.W. method has already been proved to be powerful for
alkali metals and even for other non·metallic crystals by Parmenter18l and Herman.19l Since
the conduction electrons in the monovalent noble metals are known to behave similarly
to free electrons on the empirical basis, we may expect that , the O.P.W. method gives
presumably a good approximation to this problem. In this article, therefore, we shall
describe the result of our application of O.P.W. to the metallic copper, which is compared
with the previous results.
§ 2.
Outline of the method of computation
It is widely recognized that the O.P.W. method has many advantages: at first this
method is based upon the secular problem in which the unperturbed wave functions satisfy
automatically the boundary conditions and, moreover, the self-consistent procedure can be
proceeded, without much labour, in this method. Secondly the potential need not be
spherically symmetric about each of the atoms in the crystal. Further, as far as the conduction electrons in metallic copper do concern, O.P.W. expansion will be expected to show
presumably good convergence in view of Parmenter's results for the metallic lithium. 20>
Following Herring's original ideas, each O.P.W. may be written as
(2 ·1)
k is the reduced· wave
X~ (k: ;) the Bloch sum for
where
h a translation vector in the reciprocal lattice,
and A~ (k+ ii) the orthogonalizing coefficient,
number vector,
core-electron
which is given by
(2·2)
+h: ;:)
The variational procedure based on the linear combination of X(k
the secular equation, whose matrix elements are given by,
leads to
(2·3)
and
(X(k+h: ;), HX.(k+h;: ;:)) = (h+k) 2ah, !;i+ U[U -hj
-2~E~(k) A~*(k+ii) A~(k+U),
(2 ·4)
M. Fuk:uchi
224
where U[K]=l/.Qo.
cu&),
expi{k,-~))ilo represents the Fourier coefficient of the crystal
potential U (;) .
The core functions
(2·5)
R,)
are constructed approximately from the atomic orbital <p~ (;of Hartree's self-consistent
field with exchange for Cu +, and further the overlaps between the neighbouring cores are
approximately neglected. Then we have E~ (k) =Enz, which represents atomic energy
belonging to the quantum numbers of n and I.
If the potential U(;) is once given, the relation between the energy E and the wave
vector
k may
be given by solving the following secular equation,
(2·6)
which is explicitly written by using (2 · 3) and (2 · 4).
As pointed out in the literature, the accuracy of the solutions of (2 · 6) will be
mainly governed by the appropriateness of the assumed crystal potential and the number
of O.P.W. functions contained in (2 · 6). The criterion for the latter is assured by
checking the good convergence of the expansion in the actual calculations. However,
there seems to be no convincing criterion for constructing an appropriate crystal potential,
except for the comparison of the final results with the experiment, in spite of a decisively
important effect of the crystal potential on the final results.
In fact, Parmenter20> has constructed in his O.P.W. calculation the mentioned crystal
potential for Li by placing the neutral Li atom with electron configuration (ls) 2 (2s)"18
(2p) ~/s at each lattice point, whereas, Hermann has assumed for each C atom in diamond
to "be in valence state of (ls)2(2s) 1 (2p) 3 • 5S. Although both authors have actually started
from the picture of free atoms as mentioned above, we shall attempt to construct the
crystal potential in copper by using the model of the free ion cores at lattice points plus
free conduction electrons except for a concerned electron. The field of Cu + ion at each
lattice point is given by the solution of Hartree-Fock equation for Cu + ion, i.e., 22>
Vcou! (r) = -2/r· Zv(r)
(2. 7)
and the contribution from the conduction electrons is approximately replaced by the uniform
charge density, following Herring's proposition/ 3> by taking account of the fact that
Wigner-Seitz method for alkali metals has given actually the nearly uniform charge
distribution of conduction electrons except for the very vicinity of the nuclei at lattice
points.
In other words, the assumption of the uniform charge density may be taken to represent the starting potential of our procedure. The exchange interaction of a conduction
electron with an ion core is taken into account approximately by using Slater's exchange
potential, on account of which the potential energy of a conduction electron due to an
ion core at each lattice point may be given by
The Energy Band Structure of the Metallic Copper
225
(2 ·8)
Then, it follows,
co
U[K]=3/r/K·[- (2/K) +) (rv(r) +2) sinKrdr],
0
(2·9)
for K=FO (r8 =radius of a s-sphere).
As shown by Herring/3} U[o], i.e., the mean value of u(;) over s-polyhedron, is
approximately equal to the mean value averaged over s-sphere of the potential caused by
both ion core located at the center and the uniform charge density of other conduction
electrons. The mentioned approximation for U[o] seems to be reasonably good in copper,
because the first term in ( 3 · 2) is essentially equal to -2/r outside the s-sphere and
the charge neutrality may be well preserved within the s-sphere.
Thus we obtain,
r,
U[O]= (3 /r,
3) )
r2 v (r) dr+ (2 · 4/r.)- 6 (9 /32 tr2 )
113 •
(1/r.),
(2 ·10)
0
the second and third terms of which denote the contribution from the conduction electrons.
As the ion-core within the crystal is approximately replaced by free Cu + ion in the
above calculation, our procedure will not be considered to give a satisfactory answer to
the problem for the reason that the effect of overlaps or distortions of the relatively large
3d-cores is not fully taken into account by such a procedure. Recently, such an effect
has actually been partly worked for the electronic band structures of the metallic iron24 >
but the method adopted there does not seem to be sufficiently satisfactory.
§ 3. Numerical results and discussions
The secular equations of ( 2 · 6) are resolved into factors by the use of group
theoretical considerations and our actual procedure has been chosen as follows.
For the state corresponding to a fixed k-vector, we take into consideration X(k+ h: ;:)
corresponding to the smaller values of lk+hl in succession, from which the linear combinations are set up to solve the corresponding secular equation. This may be justified
from the fact that the mean energy corresponding to each O.P.W., i.e.,
X(k+h:-;)
is
approximately determined by the relative magnitudes of lk+hl and the O.P.W. with
lower mean energies are considered to make dominant contributions in the linear combination.
At the edge points in the Brillouin zone, however, not only one wave number vector
k but
give the minimum magnitude of Ik+hi, as seen
- - - in the case of (2nja)(~·~·O), (2rrja)(-!·-!·1) and (277"/a)(i·-!·1) at a point Kin
Fig. 1. In such a case we consider only the states which belong to the irreducible representation in the gwup of k25> appearmg in the linear combinations and the states not
also some other vectors
k+ h may
226
M. Fukuchi
appearing in the linear combinations of the minimum I. k+hi are disregarded, for these
correspond certainly to the states of higher energy which are not usually occupied by
electrons.
Kz
Kx
Fig. 1. The first Brillouin zone of a face-centered cubic crystal. Letters
indicate points associated with wave functions of particular symmetries.
For the state corresponding to X in Fig. 1 the minimum of
k+h being taken as
(2Tr/a) (0.0.1) and (21rja) (o.o.i).
Ik+hi
becomes 2Tr/a,
Thus, from the group-theoretical
procedure, we have X 8 and X/ states, of which the eigenfunctions become
X( 0. 0.1 :-;)
+X(o.o.i: ;) and X(0.0.1: ;) -X(o.o.i: ;), respectively.
Actual computations have been carried out along each of the directions in k--space,
i.e., LI(0.0.1), A(1.1.1) and 2(1.1.0) in Fig. 1, and in particular at the edge points
X, L, K and the origin F in the figure, where the considerably rapid convergences are
2.chieved, as shown in Fig. 2. The symmetry along each of the above mentioned directions in k--space becomes lower than those of edge points, so that the reduction of the
secular equation in the former cases is incomplete compared with the latter, resulting in
the relatively poor convergence.
From the comparison with the results on other substances, the degree of the convergence in copper is estimated to be intermediate between
those of the alkali metals and the valence crystals as is expected.
In Table I, the values of E computed for a number of k-values are shown.
Since the computed energies have been obtained only for the limited number of k-values, we have attempted to represent E (k) by the rather simple polynomial of k; the
coefficients of which are determined by the least square method, as shown in Table IL
Since this polynomial-expression is expected to be valid for small k--values, the values of
The Energy Band Structure of the Metallic Copper
227
E (Rdy)
number .of O:P.W. functions.
Ls
4
5
Xl
-0.5
4
:2
3
2
p
v
.5 p
4
--~----------------~r.
4
Fig. 2. The degree of convergence m the O.P.W. expansion. The number attached
to the points on the curves are the order of the secular equation m the reduced form.
Table I.
The energy-values at various wave number k.
(1.0.0) direction
k
E
(1.1.1) direction
k
n
(Ryd.)
(1.1.0) direction
E
k
n
E
n
---
0
r.
-1.282
0.2
As
0.4
0
r.
-1.282
27
0.2
13
-1.062
9
-0.850
9
-0.541
9
0.9771
I.
I,
I,
I,
K,
-1.200
-0.260
11
0.9771
Kr/
-0.344
8
0.7978
I pI
-0.135
6
27
0
r.
-1.282
-1.211
15
0.2
A,
-1.204
12
d,
-1.089
15
0.4
A,
-1.073
11
0.6
0.6
d,
-0.882
14
0.6
-0.868
11
0.7978
0.9212
Xtl
-0.499
20
0.7978
-0.696
24
0.9212
x.
-0.108
24
0.7978
A.
Lp t
L.
27
-0.393
22
I
n is the number of the O.P.W. Junction5 contamed in the expanston.
0.4
-
228
M. Fukuchi
Table II.
The coefficients of the powers of kin the E(k)-curve.
Ek=Eo+ Et k~ + E~ k4 +5E3(kx~ky~ + ky~k.~ +k.~kx~ -1/5•k4 )
For copper
-1.282
1.249
-0.194
0.265
For lithium
-0.6832
0.723
0.039
-0.033
The values for lithium are quoted from the reference 4.
edge points, i.e., X, L and K, have not been used for its computation. Furthermore,
the above mentioned expression may be considered not to be sufficiently accurate for the
detailed behaviour near the Fermi surface in metallic copper.
The similar energy-expression by a polynomial of k has been worked for lithium, by
Kohn and RostoLer, 26> with the results of £ 2 =0.039, £ 3 = -0.033 which are much nearer
to the free electron values than ours for copper. The rela•ively large values of E2 and
E~ for the case of copper would be presumably ascribed to the rather large 3d-core of
copper ion.
In Figs. 3, 4 and 5 the energy curves along each direction in k-space are shown
E.-E,
(Ryd.)
1.0
1.0
1.0
0.5
r
0.2
0.4
0.(
0.8
X
Fig. 3. E versus k along
the (0.0.1) directton.
Fig. 4. E versus k along
the (1.1.1) direction.
Fig. S. E versus k along
the (1.1.0) direction.
Broken lines express the energy-curves of fl'ee electwns.
The Energy Band Structure of the Metallic Copper
229
together with those of the free electrons for comparison.
The lowering of the eigenvalue of the state L 1/ compared with that of L, is observed
to agree with the suggestion by Mott/7) who pointed out, from the X-ray emission
spectrum of copper, that at points near the (1.1.1)-planes inside the first Brillouin zone,
the symmetry is predominantly of P-type.
The energy bands of metallic copper have been already computed by the WignerSeitz method28l and the augmented plane wave method29l also.
In Table III, therefore,
the results by the various methods are listed for comparison.
Table III.
The results by various methods of approximations.
H-F. ion core potential
(The relative values to
-
Lp r
gap
x.
r.
state, (eV))
gap
ws
8.86
10.56
1.7
11.37
10.83
0.5
APW
OPW
7.83
9.25
1.4
9.92
11.65
1.7
6.11
5.99
12.09
7.98
4.1
15.97
10.65
5.3
13.91
12.76
free
I
8.66
I
11.54
-
-
12.99
Hartree ion core potential
ws
8.34
8.58
0.2
14.87
11.06
3.8
APW
6.88
8.80
1.9
8.68
10.87
2.2
5.59
5.74
As is seen in Table III, the energy values of the conduction electrons in the metallic
copper given by the O.P.W. method behave in a quite similar way to that of free electrons
along each of three directions in k-space, whereas the results by the augmented plane
wave method show much deviation from the free electron values along the (1.1.0)directions. In both directions of (1.0.0) and (1.1.1), however, the band widthn obtained
by the three methods are nearly equal, while there are some discrepancies in the relative
positions of E(X,) -E(XP1 ) and E(L,,) -E(Lv1 ) and in the energy gaps.
In fact, our
values for the energy gaps are larger than those of the others and are seen to rather
resemble to the results of Tibbs.:m) These discrepancies might be caused by our use of
Slater's exchange potential in the form of free electron approximation, since the contribution from this potential is rather large and then the inaccuracy involved in Slater's averaged
exchange potential is taken to have sensitive influence upon the final results.
Although out result does not give an accutate information on the Fermi energy
sutface, it is inferred from our computation that the magnitude of mjm* on the Fermi
sutface will be much smaller in the ( 1.1.1) -direction.
In view of the results of our computation together with the other data the assumed
crystal potential is observed to give rise to remarkably sensitive influence upon the final
energy values. It will, therefore, be highly needed to follow the self-consistent procedure
in the actual calculation and examine carefully the effect of crystal potential upon the
230
M. Fukuchi
final result.
The investigation of both 4s-and 3d-bands along these lines are now in
progress.
The author should like to express his sincere thanks to Prof. T. Muto and J.
Yamashita for suggesting the problem and for their kind guidance throughout the present
work.
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See reference 16.
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