317
Progress of Theoretical Physics, Vol. 51, No. 2, February 1974
Electrical Resistivity of Transition Metals. I
Jiro YAMASHITA and Setsuro ASANO
Institute for Solid State Physics, University of Tokyo
Roppongi, Minatoku, Tokyo 106
(Received June 23, 1973)
The electrical resistivity of. pure transition metals Mo and Nb is calculated by a
realistic model. The Fermi vectors, Fermi velocities and wave functions are determined by
the KKR- and APW-methods. The phonon spectrum is calculated by the Born-von Karman
atomic-force-constant model. The matrix elements of the electron-phonon interaction are
given by the single site approximation. The results of the calculation are in reasonable
agreement with experiment.
§ 1.
Introduction
Transition metals have higher resistivity than simple metals. As early as
1936 Mote> proposed the so-called two-band model. Needless to say, this model
is a prototype of various models by which transport properties of transition
metals have been studied. As far as the authors know, however, the transport
properties of non-magnetic transition metals have not been elucidated on the basis
?f the real band structures which are quite familiar to any band-theorists. In
the present paper, we shall calculate the mean free path of typical non-magnetic
transition metals, Mo and Nb, on the basis of a realistic model.
§ 2. Electronic structure of Mo and Nh
For quantitative calculations of the mean free path, it is required to have
a detailed knowledge of the electronic states on the Fermi surfaces of Mo and
Nb, and of the perturbation potential by which the electrons are scattered. The
shape and the area of the Fermi surfaces of Mo and Nb are evaluated by KKR
method with the self-consistent Xa-potential.
In Mo, a close region of electrons around T makes one Fermi surface.
Another Fermi surface encloses a large closed region of holes around H. A set
of small electron pocket, or lenses, located at a position on TH, makes six equivalent Fermi surfaces. There is a small closed region of holes around N, so that
there are six equivalent Fermi surfaces. They will be referred to as the T-, H-,
Ll- and N-surface, respectively. Their area will be denoted by Sr, SH, SJ and
SN and their numerical values are: Sr=l.162, SH=l.413, SJ=2.307/6 and
SN=2.001/6 in the unit of (2n/aY, where a is the lattice constant.
In Nb, there is an inner region of holes at T, which makes one Fermi surface.
J. Yamashita and S. Asano
318
There is a multiply-connected set of hole tubes along H directions. There is
an ellipsoidal pocket of hole around N, so that there are six equivalent Fermi
surfaces. They will be referred to as the T-, H- and N-surface, respectively.
Their area will be denoted by Sr, SH and SN, and their numerical values are
Sr=l.003, SH=6.538 and SN=5.420/6 in the unit of (2n/a/.
In the actual calculation of resistivity, the 22 k-points are selected on 1/48
of the Fermi surfaces of Mo and the Fermi velocity is determined by the KKR
method, and th~ wave functions are calculated by APW method. On the other
hand, the 14 k-points are selected on 1/48 of the Fermi surface of Nb.
In Table I we show the 22 Fermi vectors, the Fermi velocities and the
amount of s-, P-, d- and !-components of the charge in each inscdbed sphere.
The wave functions on the T-, H- and ..::1-surface are almost d-like ~~dare mostly
localized within the inscribed sphere. The N-surfaces have some amount of the
P-components and higher velocity. The Fermi velocity of a monovalent freeelectron-like metal whose lattice constant is equal to Mo, is about 1.25 in the
atomic unit. As seen from Table I, the Fermi velocity of Mo is lower than this
Table I. Data of 22 points of Mo. Three components of the Fermi vector, the weight, the
Fermi velocity; The s-, p., d- and /·components of the charge in the inscribed sphere and
the amount of charge outside the inscribed sphere "out" are normalized to one. The
unit of the Fermi vector is 2n/a and the unit of the velocity is the atomic unit.
No.
k.
(1)r
(2)r
(3)r
(4)r
0.13651
0.16972
0.27955
0.35000
0.23424
0.13651
0.16972
0.06989
0.03500
0.13385
0.85178
0.14822
0.18662
(5)r
(6)H
(7)H
(8)H
(9)H
(10)H
(11)J
(12)J
(13)J
(14)J
(15)J
(16)J
(17)N
(18)N
(19)N
(20)N
(21)N
(22)N
0.81338
0.69559
0.61970
0.73775
0.23105
0.58559
0.40832
0.28951
0.54324
0.40832
0.36687
0.57531
0.50000
0.48478
0.41001
0.56258
k.
0.07610
0.03803
0.14985
0.00000
0.00000
0.15533
0.11881
0.13492
0.13000
0.36687
0.42469
0.50000
0.36517
0.41001
0.43742
0.13651
0.07890
0.06989
0.03500
8
24
24
24
0.853
0.820
0.842
0.720
0.03346
0.14822
48
0.772
1.330
1.284
0.07465
0.07610
0.03803
0.03746
0.00000
0.00000
0.00000
0.00000
0.00000
0.13000
0.00000
0.00000
0.19485
0.00000
0.12726
0.11801
8
24
24
24
1.140
0.862
1.176
0.665
48
6
6
24
24
24
24
12
12
12
24
24
0.568
0.624
0.630
0.314
0.528
0.862
1.177
0.811
1.130
1.005
24
1.050
s
p
d
0.0049
0.0041
0.0022
0.0027
0.0011
0.0229
0.0216
0.0163
0.0070
0.0148
0.8935
0.8944
0;8982
0.9089
0.9060
0.0138
0.0156
0.0189
0.0231
0.0173
0.0328
0.0265
0.0427
0.0424
0.0446
0.7615
0.7734
0.7888
0.0360
0.0348
0.0328
0.0521
0.0434
0.0000
0.0591
0.0468
0.0048
0.0499
0.0114
0.2919
0.2174
0.8012
0.7966
0.8192
0.7964
0.8050
0.8475
0.7984
0.8599
0.1996
0.4930
0.3670
0.4382
0.2770
0.0298
0.0323
0.0311
0.0286
0.0160
0.0033
0.0129
0.0000
0.0000
0.0050
0.0003
0.0032
0.0000
0.1165
0.0000
0.0000
0.0383
0.0647
0.0012
0.3119
0.2149
0.3008
0.2466
0.4537
I f
0.0222
0.0245
0.0284
0.0254
0.0199
0.0162
0.0092
0.0194
0.0158
0.0140
I out
0.0647
0.0641
0.0641
0.0606
0.0605
0.1270
0.1227
0.1176
0.1135
0.1146
0.1496
0.1158
0.1208
0.1227
0.1202
0.1032
0.3719
0.2731
0.3118
0.2891
0.3416
0.2845
Electrical Resistivity of Transition Metals. I
319
Table II. Data of 14 points of Nb.
No.
k.
p
s
d
f
I
out
(1)r
0.24000
0.10964
0.04541
0.320
0.0043
0.0239
0.8172
0.0204
0.1341
(2)H
0.18500
0.07645
0.03167
0.326
0.0037
0.8JE8
O.OC83
0.0907
0.0236
0.8346
0.0179
0.1195
0.0337
0.8124
0.0211
0.1316
(3)H
0.35000
0.09139
0.03785
0.453
0.0000
"0.0042
(4)H
0.51000
0.15478
0.03079
0.698
0.0010
(5)H
0.51000
0.15542
0.10385
0.414
0.0063
0.0764
0.7292
0.0187
0.1693
(6)H
0.66000
0.24505
0.04.874
0.839
0.0065
0.0304
0.8022
0.0242
0.1366
(7)H
0.66000
0.18241
0.12188
0.819
0.0231
0.0349
0.7559
0.0291
0.1569
(8)N
0.27370
0.27370
0.00000
0.264
0.0€85
0.1394
0.5030
0.0243
0.2646
(9)N
0.63700
0.36311
0.0000
0.1930
0.4902
0.0170
0.2997
0.50000
0.50000
0.00000
0.32551
1.037
(10)N
0.604
0.0000
0.2518
0.4582
0.0040
0.2859
(11)N
0.47658
0.26578
0.00000
0.867
0.0580
0.1776
0.4285
0.0231
0.3126
(12)N
0.61829
0.38171
0.20000
0.801
0.0000
0.2192
0.4703
0.0112
0.2992
0.938
0.546
0.0855
0.2273
0.3583
0.0140
0.314.8
0.0398
0.1983
0.4564
0.0143
0.2910
(13)N
0.36701
0.36701
0.18807
(14)N
0.50000
0.28522
0.15117
figure, but not much lower on most part of the Fermi surfaces. The corresponding
data concerning Nb are also shown in Table II.
§ 3. Electron-phonon interaction
In the present paper, the resistivity due to the electron-phonon interaction
will be considered. The evaluation of the matrix elements of the electron-phonon
interaction in transition metals is still a hard task, if sufficient accuracy is required.
Here, we shall take the "single site approximation", although it is clearly an
over-simplified model. We must rely on a model, but we do not want to introduce any empirical parameters. In a previous paper/> the electrical resistivity
and the thermoelectric power of copper were evaluated by this approximation.
The results were found to be fairly reasonable. As will be mentioned later,
the results obtained for Mo and Nb by this approximation are fairly good, so
that it seems to have a meaning as the first approximation.
First, let us summarize various notations appearing in this section. The
radius of the inscribed sphere is denoted by rt, the Fermi energy as EF, k is
the Fermi vector at the initial state and k' is that of the final state. The reciprocal lattice vectors are denoted as G.,. and the wave vector of phonon is given
by q = k'- k + G,. in the first Brillouin zone. The mode of the lattice vibration
is specified by ~ = 1, 2, 3 and e (q, f) is the polarization vector of a phonon whose
circular frequency is denoted by w(q, f). The band energy is denoted by E(k)
and the Bloch function is by C/Jk (r). It has the form in the inscribed sphere,
C/Jk(r)
= ~ a,.(k)Rt(r, EF)C,.((}, rp).
"'
(1)
Here, R 1 (r, EF) is the radial wave function of order l at EF and it 1s normalized
320
J. Yamashita and S. Asano
in the inscribed sphere. Further, Cn(O, q;) is the normalized cubic harmonics.
The index n specifys the· angular part of the wave function in the following
order: (1, y, z, x, xy, yz, z 2 -(1/2)(x2 +y 2) , xz, x 2 -y2, xyz, x 8 -(3/5)x, y 8 -(3/5)y,
z 8 - (3/5)z, x(y2 - ~), y (z2 -x2) , z(x2 - y 2) and so on). The coefficients an(k) are
normalized so as to give the ratio of the charge in the inscribed sphere. Further,
we introduce the abridged notation: a(i,j) =a,(k)a1 (k') -a1 (k)a 1 (k'). The
potential V(r) is the self-consistent muffin-tin potential which is used to calculate
the band energy and V 0 is the constant potential outside the inscribed sphere.
The phase shift for the muffin-tin potential at EF is denoted by "'J.1 (EF) which is
connected with the logarithmic derivative L 1 (rio EF) by the well-known formulae. 3>
The matrix element of the electron-phonon interaction may be written in the
single site approximation as
(2)
The integration within the inscribed sphere is transformed into the surface integral
over the inscribed sphere. The integration outside the inscribed sphere is simply
disregarded in the single site approximation. If we write Mq,e (k, k') as
Mq<(k, k') =ex(q, f)Mx+e 11 (q, f)M11 +e,(q, f)M,,
(3)
then Mx, M 11 and M, are expressed in the following way.
Mx=A,Pua(1, 4) +Apa{v(a(2, 5) +a(3, 8) +a(4, 9)) -uva(4, 7)}
+Aa1 {w(a(6, 10) +a(8, 16) -a(5, 15)) +1.5va(9, 11) -1.5vwa(7,11)
-1.5uwa (7, 14) - 0.5wa (9, 14) - 3uvwa (5, 12) - 3uvwa (8, 13)},
(4)
M:v=A,pua(1, 2) +Apa{v( -a(2, 9) +a(3, 6) +a(4, 5)) -uva(2, 7)}
+A 111 {w(a(6, 16) +a(8, 16) +a(5, 14)) -1.5va(9, 12)-1.5vwa(7, 12)
+1.5uwa(7, 15) -0.5wa(9, 15) -3uvwa(5, 11) -3uvwa(6, 13)},
(5)
M, = A,pua (1, 3)
+ Apa {v (a (2, 6) +a ( 4, 8)) + 2uva (3, 7)}
+A 111 {w(a(5, 10) -a(8, 14} +a(6, 15) +a(9, 16)) +3vwa(7, 13)
-3uvw(a(6, 12) +a(8, 11))},
where u=1/.J3, v=1/.J5 and w=1/.J7.
electron-phonon interaction are given by
A!,!+l
(6)
Here the coupling constants of the
= (tan "/j!- tan "/jt+l) w! (rio EF) w!+l (r,, EF)'
(7)
where W 1 (r,, EF) is defined by
w! (rio EF) =Rt(r,, EF) I (j! (!Cr,)- tan "/j!n! (!Cr£))
and
JC=
(EF- V 0)112 •
The values of the phase shifts and
A1,1+1
(8)
of both Mo and
Electrical Resistivity of Transition Metals. I
321
Table III. Phase shifts, Coupling constants of the electron-phonon interaction of Mo, Nb and
Cu. The coefficients A are calculated in the rydberg unit by (7) and (8).
tan 7Jo
tan 711
tan 712
tan 7Js
Ao1
A12
Azs
Mo
Nb
Cu
-1.422
-0.393
-3.480
-0.002
0.956
-0.546
-3.05
-1.346
-0.382
2.139
-0.001
0.821
-0.564
-2.52
-0.C818
O.C854
-0.132
-0.0003
0.14
-0.30
2.4
Nb are shown in Table III. For comparison, the corresponding values of copper
are also shown in Table III.
The phonon spectrum is calculated using the Born-von Karman atomic forceconstant model. The force constants of Mo have been given by Woods and Chen 4l
and those of Nb have been given by Nakagawa and W oods. 6l
§ 4.
Method of calculation
In this section, it becomes necessary to integrate a quantity A (k) over the
whole Fermi surfaces. We shall always replace such an integral by a sum over
a number of points on the Fermi surfaces. Let us write the total area of the
Fermi surfaces, Stat> as a sum of the area of the separated surfaces:
4
Stot=:E gnSn=Sr+SH+6Sd+6SN,
n=l
(Mo)
3
=I:; gnSn=Sr+SH+6SN.
n=l
(Nb)
Here, each surface is specified by n and the number of the points selected on
1/48 of the n-th Fermi surface is denoted by Nn. At Mo, we take N1 = 5, Nz = 5,
N 8 =6 and N 4 =6, and at Nb, we take N1=1, Na=6 and Na=7. Each point
bears the weight wn(i) whose value is given in Table I. Here, the number i
runs from 1 to Nn. Then, the approximate procedure for a surface integral is
as follows:
(9)
and also
(10)
Here, p specifies one of 48 sections on the Fermi surfaces, and Bn CP) ( j, i) at any
section p is connected with Bn Cll ( j, i) by a proper transformation.
The electrical conductivity will be given by
322
J. Yamashita and S. Asano
(11)
where v (k) and r (k) are the velocity and the relaxation time of each Bloch
state on the Fermi surface. The Boltzmann equation is solved in two ways. The
first is the usual relaxation time approximation, in which r(k) is given by
1/r(k)
= J{(v(k') -v(k)Y/2v(k)v(k')}Q(k, k')dSk,jhv(k'),
(12)
where
(13)
Here, M is the ion mass, N is the number of ions in the unit volume, kB is the
Boltzmann constant, T is the absolute temperature and nqE is the number of
phonon.
The second method is the one proposed by Taylor. 6l He introduced a vector
mean free path A(k), which is to be determined by the coupled integral equations
s
Q(k, k') {A(k) -A(k')}dSk,fhv(k') =v(k).
Once A(k)
IS
(14)
determined, then the relaxation time is given by
r(k)
= (v(k) ·A(k))/v(kY.
(15)
In the present paper, we reduce the coupled integral equations to a set of linear
equations of 3 X 22 variables at Mo, and 3 x14 variables at Nb. For the conductivity, the value obtained by the relaxation time approximation differs at most
a few percent from one obtained by the second method. The value of the mean
free path at each point deviates about 25% in the worst case from one obtained
by the second method.
§ 5. Results
(A)
Mo
The electrical resistivity is calculated at 290°K and 50°K. The values
obtained from (11), (14) and (15) are 5.4 and 0.14pQ em, while the corresponding experimental values are 5.3 and 0.11pQ em. The values of the mean free
path and the relaxation time at 290°K are shown in Table IV. As seen from
the Table, the relaxation time is fairly isotropic, so that the mean free path is
roughly proportional to the Fermi velocity v (k). The mean free path at every
point is sufficiently long compared with the lattice constant at room temperature.
The rates of contribution to the conductivity from four kinds of the Fermi surfaces
mentioned before are evaluated as 1, 1.57, 1.35 and 2.46. Roughly speaking
every kind of the surfaces contributes rather equally to the conductivity, but the
current carried by holes is about twice larger than that by electrons. The
323
Electrical Resistivity of Transition Metals. I
Table IV. Mean free path, relaxation time and enhancement factor in mass.
Mo
No.
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(18)
(19)
(20)
(21)
(22)
I
l
204
196
183
144
177
263
257
228
176
238
119
110
151
143
71
124
209
297
217
271
239
268
I
Nb.
r
1.16
1.16
1.05
0.968
1.10
0.955
0.965
0.965
0.988
0.969
0.866
0.936
1.17
1.10
1.09
1.13
1.17
1.22
1.29
1.16
1.15
1.24
I
A
No.
0.332
0.334
0.368
0.408
0.348
0.417
0.410
0.412
0.403
0.401
0.518
0.401
0.380
0.431
0.393
0.389
0.346
0.336
0.299
0.353
0.332
0.338
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
I
l
11.5
18.6
23.8
37.7
22.2
46.0
39.8
13.8
56.0
37.9
40.9
40.2
44.7
30.7
I
r
0.174
0.276
0.254
0.261
0.259
0.265
0.235
0.253
0.261
0.304
0.228
0.243
0.230
0.272
I
A
1.71
1.53
1.64
1.59
1.70
1.50
1.58
1.86
1.64
1.66
1.89
1.68
1.70
1.74
The unit of the mean free path l is the
Bohr radius a0 and the unit of the relaxation
time is 10-14 sec.
resistivity is mainly determined from the P-d scattering. The s-P scattering
contributes quite a little, because the s-component in wave functions is very
small on the Fermi surfaces of Mo. The d.j scattering contributes to the
resistivity, but the effect is considerably smaller than that by the P-d scattering,
because the amount of the /-components is small. The d-d scattering is forbidden in the single site approximation. The holes on the N surfaces have a
large probability of scattering to other surfaces, through the P-d scattering,
because the P-components are large on this surface and the d-components are
large on other surfaces. On the other hand, the electrons on the other surfaces
have much larger d-components than the holes on the N-surfaces, but they are
scattered mainly through the d-P scattering, because the d-d scattering is forbidden. As the P-components are large only on the N-surfaces, they are mainly
scattered to the N-surfaces by the d-P scattering, that is, by the reverse processes
of the scattering from the N-surfaces to the other surfaces. Therefore, the
holes on the N-surfaces and the electrons on the other surfaces have roughly
the same probability of scattering. We must note, however, that the result
mentioned before was obtained by the single site approximation. At present, it
324
J. Yamashita and S. Asano
is open to question how it is modified in an improved approximati on.
Each surface of Mo supplies the carriers of the current, but at the same time
it works as a scatterer. It is interesting to see which part is more important
on the J surface whose density-of-s tates is high. We calculate the resistivity
of a hypothetica l Mo by excluding 6 points on the J surface from 22 points
used before. As a result, the value of resistivity is reduced from 5.4,aQ em to
3.9,a.!Jcm.
Next, let us consider the thermoelect ric power at 290°K. The observed value
at 290°K is about 6'"'"'7 ,a VI degree. It is much larger than the correspondi ng
value of Cu (2,a VI degree). As mentioned previously, the hole current is dominant
in Mo, so that the positive sign of the thermoelect ric power is quite reasonable.
The formula of the thermoelect ric power is
(16)
In order to calculate (JrJ(E)ItJE, the equi-energy surfaces of EF+0.01 Ryd are·
considered. Again, 22 points are selected on these surfaces and the resistivity
is calculated in the same way as before. The area of
H, J and N surfaces
are as follows: Sr=l.278, SH=l.325, S.=2.928l6 and SN=1.751I6 in the unit
of (2nl aY. The area of the electron surfaces becomes 1.225 times larger, but
that of the hole surfaces becomes 0.901 times smaller than the respective Fermi
surfaces. The average value of the Fermi velocity is reduced from 0.8446 to
0.8035 in the atomic unit. This reduction of the velocity works to increase the
resistivity. The increase of the area of the J surface also works to increase
the resistivity, because the J surface is an effective scatterer. The amount of
the P-componen ts at the new N-surfaces becomes larger than the Fermi surfaces.
As a result of calculation, we find that the current carried by the new
surface changes very little by the shift of the energy surface, because the increase
of the area is just counterbala nced by the decrease of the mean free path. The
situation is nearly the same on the new J surface. Here, the mean free path
decreases by more than 20%, but the area increases by 27%. On the other hand,
the current carried by the new H surface decreases by about 25% by the shift
of the energy surface, because both the mean free path and the area are decreased.
The situation is almost the same on the new N surfaces. About 60% of the
current is carried by the positive holes, so that the total current decreases by
about 15% when the energy surfaces shift up by 0.01 Ryd from the Fermi surfaces. Then, the thermoelect ric power of Mo at 290°K becomes 7.4,a VI degree.
In fact, the thermoelect ric power of Mo is not proportiona l to the absolute
temperature through all observed temperature s. 7l It is approximate ly linear in
T until about 600°K, but it seems to obey approximate ly the law:. q=aT-bT 2
at high temperature . A refined theory of the thermoelect ric power was proposed
by Aisaka and Shimizu,Sl and Colquitt et al. also proposed the electron-ele ctron
interaction as the mechanism to introduce a T 2 term. 9l
r,
r
Electrical Resistivity of Transition Metals. I
325
The mass enhancement factor due to the electron-phonon interaction IS also
calculated by the formula given by Hasegawa and Kasuya. 10> Our calculated value
is 0.37, while McMillan11> has estimated it from the superconducting transition
temperature as 0.41. On the other hand, if the coefficient of the electronic specific
heat is derived from the calculated density-of-states at the Fermi surface, then
the observed value is by 1.32 times larger than the calculated one.
The evaluation of the Hall constant is more elaborate, because the knowledge
of the curvature of the Fermi surfaces is required. In order to estimate its order
of magnitude, we replace each Fermi surface by a spherical surface of the same
area and use the average value of the velocity on each surface. Then, the contribution from the hole surfaces is three times as large as that from the electron
surfaces. The calculated value of the Hall constant becomes 8 X 10-11 m/ A sec,
while the observed value is 18 X I0- 11 m/ A sec. 12>
(B)
Nb
The calculated values of the resistivity at 290°K and 50°K are 19.5 and
The
theoretical values are by 40% "'50% larger than the observed ones. Such an extent
of disagreement may be within a limit of accuracy of the single site approximation.
We must admit that accuracy of the present band calculation is also limited.*>
The mass enhancement factor due to the electron-phonon interaction is determined as 0.82 by McMillan. Comparison of the observed value of the coefficient of the electronic specific heat and that derived from the band calculation
gives the value of 0.8 for the same factor. The same factor is directly calculated through the matrix element of the electron-phonon interaction as done in
Mo. The theoretical value is as large as 1.7. Even if we take into account
the correction factor 1.5, which is the ratio of the calculated value of the
resistivity and the observed one, the value of the factor is still by about 1.4
times larger than McMillan's value.
Now, let us examine the reason why the resistivity of Nb is about three
times larger than that of Mo. The first factor may be the total area of the
Fermi surfaces Stot(Nb)/Stot(Mo) =1.715. The next factor may be the Fermi
velocity VF (Nb) /vF (Mo) = 0.737. Thus, it becomes (StotVF)Nb/ (StotVF)Mo = 1.26.
Thus, the difference in resistivity must be in the relaxation time. As the phonon
spectrum will be the most important factor for determining the relaxation time,
the force constants of Mo are used to calculate the resistivity of Nb at 290°K
1.46/LQ em, while the corresponding observed values are 14.5 and 0.97 jLQ em.
*> The resistivity is quite sensitive to the conditions on the Fermi surface, so that the very
accurate determination of the Fermi energy is required to have a good quantitative result of the
resistivity. In our calculation the Fermi energy is uncertain about 0.005 Ryd in the worst case.
In fact, the approach by the muffin-tin potential does not seem to be adequate to treat the
electron-phonon interaction, because the good knowledge of the wave functions in the outer region
is necessary for accurate determination of the matrix elements of the electron-phonon interaction.
However, such refinement has no meaning within the single site approximation.
326
J. Yamashita and S. Asano
to eliminate the difference in the spectrum. Then, the value of the resistivity
is reduced to 7.0p.Q em. It is rather close to the resistivity of Mo at 290°K.
(If the correction factor 1.5 is introduced, then the theoretical value of the
resistivity of Nb is further reduced to 4.7 p.Q em.) We see that the difference
in the conductivity of Mo and Nb comes mainly from the difference in the Debye
temperature.
As the mean free paths at some points are considerably short even at room
temperature, they may be close to the interatomic distance near the melting
point. It will be quite interesting to observe various transport phenomena near
the melting point.
References
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