Electrical Resistivity of Transition Metals. I

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1 317 Progress of Theoretical Physics, Vol. 51, No. 2, February 1974 Electrical Resistivity of Transition Metals. Jiro YAMASHTA and Setsuro ASANO nstitute 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. ntroduction 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. n 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. n 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. n Nb, there is an inner region of holes at T, which makes one Fermi surface.

2 318 J. Yamashita and S. Asano 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/. n 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. n Table 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, the Fermi velocity of Mo is lower than this Table. 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. k. s p d f out (1)r (2)r (3)r ; (4)r (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

3 Electrical Resistivity of Transition Metals. 319 Table. Data of 14 points of Nb. No. k. s p d f out (1)r (2)H JE8 O.OC (3)H " (4)H (5)H (6)H (7)H (8)N (9)N (10)N (11)N (12)N (13)N (14)N figure, but not much lower on most part of the Fermi surfaces. The corresponding data concerning Nb are also shown in Table. 3. Electron-phonon interaction n 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. n 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). t 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

4 320 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)(x 2 +y 2 ), xz, x 2 -y 2, xyz, x 8 -(3/5)x, y 8 -(3/5)y, z 8 - (3/5)z, x(y 2 - ~), y (z 2 -x 2 ), z(x 2 - 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)a 1(k') -a 1(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 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. f we write Mq,e (k, k') as Mq<(k, k') =ex(q, f)mx+e 11 (q, f)m 11 +e,(q, f)m,, 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)} +Aa 1 {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)}, 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)}, 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))}, (6) where u=1/.j3, v=1/.j5 and w=1/.j7. Here the coupling constants of the electron-phonon interaction are given by where W 1 (r,, EF) is defined by A!,!+l = (tan "/j!- tan "/jt+l) w! (rio EF) w!+l (r,, EF)' w! (rio EF) =Rt(r,, EF) (j! (!Cr,)- tan "/j!n! (!Cr )) and JC= (EF- V 0)11 2 The values of the phase shifts and A1,1+1 of both Mo and (2) (3) (4) (5) (7) (8)

5 Electrical Resistivity of Transition Metals. 321 Table. 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 -0.C818 O.C Nb are shown in Table. For comparison, the corresponding values of copper are also shown in Table. 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 4 l and those of Nb have been given by Nakagawa and W oods. 6 l 4. Method of calculation n 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, (Mo) n=l 3 =:; gnsn=sr+sh+6sn. (Nb) n=l 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. Here, the number i runs from 1 to Nn. Then, the approximate procedure for a surface integral is as follows: and also 2.4 (9) (10) Here, p specifies one of 48 sections on the Fermi surfaces, and Bn CP) ( j, i) section p is connected with Bn Cll ( j, i) by a proper transformation. The electrical conductivity will be given by at any

6 322 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 where 1/r(k) = J{(v(k') -v(k)y/2v(k)v(k')}q(k, k')dsk,jhv(k'), (12) 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. 6 l He introduced a vector s mean free path A (k), which is to be determined by the coupled integral equations (13) Q(k, k') {A(k) -A(k')}dSk,fhv(k') =v(k). (14) Once A(k) S determined, then the relaxation time is given by r(k) = (v(k) A(k))/v(kY. (15) n 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. (A) Mo 5. Results 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 V. 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 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

7 Electrical Resistivity of Transition Metals. 323 Table V. Mean free path, relaxation time and enhancement factor in mass. Mo Nb. No. l r A No. l r A (1) (1) (2) (2) (3) (3) (4) (4) (5) (5) (6) (6) (7) (7) (8) (8) (9) (9) (10) (10) (11) (11) (12) (12) (13) (13) (14) (14) (15) (16) (17) The unit of the mean free path l is the (18) Bohr radius a0 and the unit of the relaxation (19) time is sec. (20) (21) (22) 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

8 324 J. Yamashita and S. Asano is open to question how it is modified in an improved approximation. Each surface of Mo supplies the carriers of the current, but at the same time it works as a scatterer. t is interesting to see which part is more important on the J surface whose density-of-states is high. We calculate the resistivity of a hypothetical 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 thermoelectric power at 290 K. The observed value at 290 K is about 6'"'"'7,a V degree. t is much larger than the corresponding value of Cu (2,a V degree). As mentioned previously, the hole current is dominant in Mo, so that the positive sign of the thermoelectric power is quite reasonable. The formula of the thermoelectric power is n order to calculate (JrJ(E)tJE, 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 r, H, J and N surfaces are as follows: Sr=l.278, SH=l.325, S.=2.928l6 and SN= in the unit of (2nl ay. The area of the electron surfaces becomes times larger, but that of the hole surfaces becomes times smaller than the respective Fermi surfaces. The average value of the Fermi velocity is reduced from to 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-components 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 r surface changes very little by the shift of the energy surface, because the increase of the area is just counterbalanced 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 thermoelectric power of Mo at 290 K becomes 7.4,a V degree. n fact, the thermoelectric power of Mo is not proportional to the absolute temperature through all observed temperatures. 7 l t is approximately linear in (16) T until about 600 K, but it seems to obey approximately the law:. q=at-bt 2 at high temperature. A refined theory of the thermoelectric power was proposed by Aisaka and Shimizu,Sl and Colquitt et al. also proposed the electron-electron interaction as the mechanism to introduce a T 2 term. 9 l

Electrical Resistivity of Transition Metals. 325 The mass enhancement factor due to the electron-phonon interaction S also calculated by the formula given by Hasegawa and Kasuya.

9 Electrical Resistivity of Transition Metals. 325 The mass enhancement factor due to the electron-phonon interaction S also calculated by the formula given by Hasegawa and Kasuya. 10 > Our calculated value is 0.37, while McMillan 11 > has estimated it from the superconducting transition temperature as 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. n 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 m/ A sec, while the observed value is 18 X 0-11 m/ A sec. 12 > (B) Nb The calculated values of the resistivity at 290 K and 50 K are 19.5 and 1.46/LQ em, while the corresponding observed values are 14.5 and 0.97 jlq em. 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) = The next factor may be the Fermi velocity VF (Nb) /vf (Mo) = Thus, it becomes (StotVF)Nb/ (StotVF)Mo = 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 *> 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. n our calculation the Fermi energy is uncertain about Ryd in the worst case. n 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.

10 326 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. t is rather close to the resistivity of Mo at 290 K. (f 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. t will be quite interesting to observe various transport phenomena near the melting point. References 1) N. F. Mott, Proc. Roy. Soc. A153 (1936), ) J. Yamashita and S. Asano, Prog. Theor. Phys. 50 (1973), ) W. Kohn and N. Rostoker, Phys. Rev. 94 (1954), ) A. D. B. Wood and S. H. Chen, Solid State Commun. 2 (1964), ) Y. Nakagawa and A. D. B. Wood, Phys. Rev. Letters 11 (1963), ) P. L. Taylor, Proc. Roy. Soc. A275 (1963), ) M. V. Vedernikov, Adv. Phys. 18 (1969), ) T. Aisaka and M. Shimizu, J. Phys. Soc. Japan 28 (1970), ) L. Colquitt, Jr., H. R. Fankhauser and F. J. Blatt, Phys. Rev. B4 (1971), ) A. Hasegawa and T. Kasuya, J. Phys. Soc. Japan 25 (1968), ) W. C. McMillan, Phys. Rev. 167 (1968), ) V. Frank, Appl. Sci. Res. B7 (1958), 41.

Electronic Structure of CsCl-Type Transition Metal Alloys

2119 Progress of Theoretical Physics, Vol. 48, No. 6B, December 1972 Electronic Structure of CsCl-Type Transition Metal Alloys Jiro YAMASHTA and Setsuro ASANO nstitute for Solid State Physics, University