Physica B 172 (1991) 383-391 North-Holland

Temperature variation of the dynamical orbital susceptibility o: metals

J. Singh Department of Physics, Punjab Agricultural University, Ludhiana-141004 Punjab, India

Received 9 December 1988 Revised 12 July 1989

A general expression for the temperature dependent dynamical orbital magnetic susceptibility x,,,( q, w, T) is derived in the nonlocal model potential theory. Here q, o and Tare wave vector, frequency and temperature, respectively. Using the Shaw and Harrison model wave function transformation xOrb(q, w, T) is separated into two parts: the first is the free electron contribution and the second is the depletion hole contribution associated with the ion-core. The latter includes the Bloch character of the electrons. The formalism is applied to study the q, w and T dependence of x,,,( q, w, T) in the case of Al and Au where the depletion hole contribution is expected to be small. xorb( q, o, T) is compared to the temperature variation of the dynamical spin susceptibility. The temperature variation of the static orbital susceptibility is studied in Al and Au. The calculated values exhibit reasonably good agreement with the experimental values of the static orbital susceptibility in case of Au.

1. Introduction

The theoretical study of the dynamical mag- netic susceptibility x( q, w, T) is of fundamental importance as it can be directly related to the momentum and energy dependence of the neu- tron inelastic scattering cross-section which de- scribes the dynamics of the system [l, 21. In the one-electron approximation x(q, w, T) can be written as a sum of spin susceptibility x,( q, o, 7’)

and orbital susceptibility x,,,( q, w, T) neglecting the spin-orbit interaction. Hebborn and March [2], using a density matrix method, gave the general theory of x,( q, 0, T) and x&q, w, T) for Bloch electrons. The spin susceptibility, both temperature independent ,ys(q, w, 0) and tem- perature dependent x,(q, w, T), for the free electron metals and d-band metals, is extensively studied and reasonably well understood theoreti- cally [2-81. On the other hand xorb( q, w, T) has

significantly lower symmetry and consequently needs more labour to estimate it. Lowde and Windsor [l] and Hebborn and March [2] calcu- lated the temperature independent orbital sus- ceptibility ,yor,,(q, w, 0) for the free electron metals and found it to be of the same order of magnitude as x,( q, w, 0). Singh et al. [5] calcu- lated x,,,( q, w, 0) for the Bloch electrons using a model potential approach. The temperature vari- ation of xorb( q, w, T) in metals has not been investigated still and a good deal of attention is required in this direction. In the present text we give a formalism of x,,,( q, w, T) for metals with Bloch electrons which is applicable to all types of metallic systems. In section 2 we outline the theoretical formalism for xorb(q, w, 7’) and in section 3 the temperature variation of x,,,( q, w,

7’) is studied as applied to the simple metals Al and Au. In the last section the conclusions are drawn.

0921-4526/91/$03.50 0 1991- Elsevier Science Publishers B.V. (North-Holland)

384 3. Singh i Dynamical orbital susceptibility of metals

2. Theory

The general expression for Xor,,(q, w, T) is given as [Z].

where

M k.k+g = ( +k(,(r)i em’q’r &I tbk+q(r)) ( +k+q(r)i e’q.’ & 1 (cl,cr)> (2)

Here F~, g and m are the Bohr magneton, Lande’s splitting factor and the effective electron mass. E is a positive infinitesimal parameter corresponding to the adiabatic switching on of the perturbing field. E, ir the energy eigenvalue of the Bloch wave t,bk(r). The Fermi-Dirac distribution function f(Ek, T) i> defined as

(3)

where E,(T) is the temperature dependent chemical potential and k, is the Boltzmann constant. One should note that in defining eq. (1) the magnetic field is assumed to be in the z-direction and the wave vector q along the y-direction [2].

From now on we use the units k, = h = m, = 1 for convenience with m, as free electron mass. The first term of eq. (1) can be solved analytically and one can write

xori3(4, w, n = l,(q) + I?( q, w, T) , (4)

where

Z,(q)=-9 (2m)Cf(E,,T)=~:(liA’). k

-2g$.&; .

I,(q, 0, r> = 2

llm c [f(Ek, T, -f(Ek+,, T)l M

4 s-o k E k+q - E, - o + ie k.k+q 3

(5)

(6)

(7)

and

A = ql2k,. (8)

xl is the Landau diamagnetic susceptibility for the free electrons. Using the properties of a BIoch wave in the Wigner-Seitz approximation it can be shown that

M k.k+q = /i+k+q(dt & i+k+q(r))i2. (‘j’)

J. Singh I Dynamical orbital susceptibility of metals 385

The overlap matrix elements can be evaluated easily in the nonlocal model potential approach. Shaw and Harrison.[9] gave a model wave function transformation defined as,

I+&@)) = (l- 2) IA(r)) , (10)

where VM(r, E) is the nonlocal model potential and c#+( ) r is the model wave function. The energy derivatives of VM(r, E) have the properties of a projection operator, i.e.

av, * av, (4 aE =aE. (11)

The transformation (10) is sufficiently general to include the Heine-Abarenkov model potential [ 10, 1 l] and the Harrison transition metal pseudopotential [12]. In the d-band metals eq. (10) holds provided the pole in V, at E = E, (d-band energy) is handled appropriately by replacing (E - Ed)-’ by (E-E,+iiW,)-* h w ere W, is the d-bandwidth. For simplicity one can take &k(r) to be a single plane wave defined as

]~$~(r)) = Jk) = [0]-“” exp(ik . r) ,

where 0 is the crystal volume. Substitute eq. (10) in eq. (9) and retain

get

M k,k+q = -k:P + P(k + 41,

where

P(k+q)=(k+q[-2$k+q).

(12)

terms only up to first order to

(13)

(14)

dV,laE is assumed to be a first order quantity. p(k + q) is called depletion hole and is analogous to the orthogonalization hole in the orthogonalized plane wave theory. Substitute eq. (13) in eq. (6) and then in eq. (4) to get

where

Q+; z&L w, 0 = 2 lim c [f(‘% T) -f(4+q, Ulk:

9 *--to & E k+q - E, - w + ie ’

z;yq, 0, T) = 2gZCLk lim c t.f(E,, 0 -f(E,+,, W:

E k+q - E, - o + ie P(k + 4) - 9 r-0 k

(16)

(17)

ZE (q, o, T) and I:“( q, co, T) are the free electron and depletion hole contributions to Z2( q, o, T) and hence to xorb( q, o, T). Shaw and Harrison [9] defined the averaged depletion hole by summing over all the occupied electronic states as

P(q, T) = ; F f(% T)P(k + q) , (18)

386 J. Singh I Dynamical orbital susceptibility of metals

where Z is the atomic valency. Singh [13] has shown that the exact depletion hole P(k + q) and the averaged depletion hole p( q, T) yield the same contribution towards the magnetic spin susceptibility over all the q, o and T values. Changing the summation in eq. (18) into an integration and using eq. (14) we get [14],

(19)

where

/?,(k+q)=$(2Z+l)(d&) j[jl(]k+q\r)]‘idr. (20)

R, is the model potential radius and jl( (k] r) is the spherical Bessel function of order 1. dA,ldE are the energy derivatives of the potential well depth A,(E) in the Heine-Abarenkov model potential. dA,ldE for I = 0 and 1 can be calculated in the same way as by Singh and Prakash [14] and for 1 = 2 it is given as

[141,

dA* UEF)(EF - -%I dE = (E, - E)* + (1 W,)’ ’ (21)

where the singularity at E = E, is handled by using a t-matrix. Notice that dA,ldE has resonant behaviour at E = E, and is analogous to the s-d hybridization in the d-band metals [ll, 121. When /I( q, T) is used in place of /3(k + q) in eq. (17), the orbital magnetic susceptibility, given by eq. (15), becomes

xordq, ~3 T) = Z,(q) + [l + P(q, 731 Z:(q, w, T) . (22)

Note that ZI( q) is real and temperature independent, therefore the temperature dependence of xorb( q, co, T) comes from p( q, T) and I:( q, w, T). Further Z: (q, w, T) is a complex quantity and its real and imaginary parts can be separated by using the identity,

1 x -+ i.5

= P i 7 i7F 6(x) , 0 (23)

where 6(x) is the Dirac delta function. When eq. (23) is used the imaginary part of Xorb(q, w, T) from eqs. (16) and (22) becomes

Im x,,dq, 0, T) = [l + P(q, T)l Im I:(q, 0, T)

= -9 [I+ P(q, VI T %W,, T) -f(%+g, W(E,+, - 4 - w>. (24)

Changing the summation over k into integration and substituting k, = k sin0 cos 4, one gets

Im xorb(qy w T) = -TT ;:j [1+P(q,T)ljdkjd8jd9k4(f(E,,T)-f(E,,,,T)1 0 0 0

x WE,+, - E, - w) sin38 cos2+ . (25)

J. Singh I Dynamical orbital susceptibility of metals 31

The angular part of the integral in eq. (25) can be solved analytically to get,

Im xorb(4~ my T) = 32;;;A3 X:. [I+ P(4, T)1[2ml,(C w, T) - (7 - $)*,(q. o, T)] , (26

where

m

I,(% 0, T) = I dE, EJf(E,o U - f(E, + ~3 VI , (27 _!&32

rx

&(!7, 0, T) = I WME,~ T) - f(E, + 0, VI l&_$'

=Tln e (

w/T +ee(w,TvT

1 + e~(17,w,T)lT > ’

and

l (q,w,T)= -L mo +; I 1 2

2m 9 -J%(T). (291

The real part of I:( q, w, T) can be evaluated from Im Z: (q, w, 7') by using the Kramers-Kronig relation as,

Re Zi(q, W, T) = 2 -IT

Hence the real part of the orbital susceptibility is given as

Rex,,dq~~~ T)=4(d+[l+P(q, T)lRez:(q,w, T). (31)

The eqs. (26) and (31) for the orbital susceptibility are general and valid at all the temperatures, provided one knows E,(T).

One can study the limiting cases of x,,,b(q, w, T) and compare them with the existing results. At absolute zero eqs. (26) and (31) reduce to,

Re x0&, w, 0) =

(32)

and

388 J. Singh I Dynamical orbital susceptibility of metals

Im x0&, 64 0) = - $ xfL[l + P(4, (VI

where

W = ol2k?, (34)

Equations (32) and (33) are essentially the same as obtained by Hebborn and March [2] except that the depletion hole contribution /3( q, 0) is also present which includes the Bloch character of electrons. The static orbital susceptibility has only a real part given as.

XOd4? 03 0) = 8A 32 X:~[l+c((q,O)][l+A*- (l~/\h’)‘lnl+I], (35)

which is the usual expression for the free electron metals [2] if p( q, 0) = 0. In the limit q-,0, eq. (35) reduces to an effective Landau diamagnetism x,_ given as,

XL = [1+ PC% ONX:. . (36)

@(O, 0) is the depletion hole contribution in the limit q = T = 0 and can be positive or negative depending upon the electronic structure of the metal.

3. Calculations and results

The magnitude of /3( q, T) depends upon the nature of the metal under consideration. In free electron metals, p(O, 0) is of the order of 10% and can safely be neglected [14]. But in metals with a partially filled d-band (the so-called transi- tion metals) p(0, 0) is quite large and the major contribution comes from the d-band [13, 141. In the present text we apply the formalism to the metals Al and Au. In Au the d-band is complete- ly filled and lies well below the Fermi energy E, which allows us to treat it as a nearly free electron metal like Al. Therefore, in the present calculation of ,yor,,(q, w, T) for Al and Au we neglect the depletion hole contribution i.e. p( q, T) = 0. One should note that the formalism in section 2 is applicable at all temperatures pro- vided one knows the variation of E,(T) with T. The behaviour of E,(T) is well established at low and high T values, therefore we calculate x,,,( q, o, T) in these limits. The physical pa- rameters needed for Al and Au are given in

table 1. At low temperatures E,(T) is approxi- mately given as,

(37)

where T, is the Fermi temperature. Equation (37) is not valid at high temperatures, therefore we obtain E,(T) at high temperatures, by satis- fying the condition of conservation of electrons (7, 81, i.e.

2Cf(E,, T)=NZ. (38) k

Here N is the total number of atoms with val-

Table 1 Fermi momentum k, and atomic volume fl, in atomic units

and Fermi temperature TF in kelvin for Al and Au.

Metal k, fi,, 7-P

Al 0.927 111.0 1.357 x 10’

Au 0.638 114.0 6.428 x 10”

J. Singh I Dynamical orbital susceptibility of metals 389

ency Z and the factor of two is due to the spin degeneracy.

The temperature variation becomes more ap- parent if we plot the reduced orbital suscep- tibility defined as iorb( q, w, T) = ,yorb( q, o, T) / j&,, (0, 0, 0). Figures 1 and 2 show Im iorb(q, o, T) as a function of w for Al at low and high temperatures, respectively. At low temperatures, Im ,&,( q, w, T) remains unaffected at small w values but decreases at large w values. Further, the spread of Im ,f,,,,(q, o, T) in w is not affected at low temperatures. At high tempera- tures Im j&( q, w, T) decreases at all values of w and the spread of ,&,( q, co, T) in w increases with the increase of temperature. The decrease in the magnitude of Im iorb(q, w, T) and the increase of spread in w is due to the thermal

x103 I

0.12 g=o.2a.u. 2

3

4 -a 5 0.08 5

v= -0

tG O.OL

E

0.00 0.0 2.0 L-O 6.0

q=o.La.u. 0.3

12 3

0.2 IAl L

0.1

t/

o(oV1 o(eVJ

Fig. 1. Im j,,,,( q, w, T) as a function of w for Al at 141 = 0.2 and 0.4a.u. The curves labelled 1, 2, 3, 4 and 5 are for T = 50, 130, 210, 290 and 370 K, respectively.

2.0 w(eV)

Fig. 2. Im iorb(q, w, T) as a function of o for Al at 141 = 0.05 a.u. The curves labelled 1, 2, 3, 4 and 5 are for T/T,=O.O, 0.2, 0.4, 0.8 and 1.2, respectively.

excitation of the electrons around the Fermi level or the smearing of the Fermi distribution around the Fermi level with the increase of T. We have also calculated the imaginary part of the reduced spin susceptibility f,( q, w, T) = x,( q, w, T)l,y,(O, 0, 0) for Al at high tempera- tures following Jullien et al. [7], see fig. 3. Comparing figs. 2 and 3 it is evident that the temperature variation of Im &,,(q, w, T) is much weaker than that of Im X,(q, w, T) as is expected. It is due to the fact that in Im &,,( q, w, T) the overlap part M,,,,, = -kz (see eq. (24) in the free electron limit) is much less than the overlap part in &(q, w, T) which is unity. Further the two terms in eq. (26) cancel each other.

The usually measured quantity is the static orbital susceptibility x,,,( q, 0, T). The tempera- ture variation of xorb(q, 0, T) for Al at low T values is shown in fig. 4. It exhibits the decrease in the absolute value of x,,,,(q, 0, T) with the increase in T and approaches a constant value at large T values. Again the temperature variation becomes weaker with the increase of q. Recent- ly, Plumier et al. [15] measured experimentally the diamagnetic susceptibility in Au at different magnetic fields and calculated the average sus- ceptibility. Therefore, it is worthwhile to calcu-

w (0V)

Fig. 3. Im ,f,(q, w, T) as a function of w for Al at 1q( = 0.05a.u. The description is the same as that of fig. 2.

390 J. Singh I Dynamical orbital susceptibility of metals

X10”

- 0.30 - Al

q.0.L

0.0 I / I 0 100 200 300 LOO

T("K)

Fig. 4. xrb(q, 0, T) as a function of Tfor Al at )q) = 0.1,0.2 and 0.4 a.u.

late xorb(q, 0, T) for Au. The calculated values of xorb( q, 0, T) for Au as a function of tempera- ture are shown in fig. 5. As the experimental values are measured at finite magnetic field so we choose a finite field wave vector q such that the value of xorb( q, 0, T) matches with the experimental value at 0 K. From fig. 5 it is evident that the calculated results are in reason- able agreement with the experimental values at low T values while these are smaller at higher T values. The maximum decrease in (xorb( q, 0, T)I is 5% in the temperature range O-300 K. The

q=O.787 i

0.001 0 100 300

T('='K)

Fig. 5. xorb(q, 0, T) as a function of T for Au.

faster decrease of the calculated diamagnetic sus- ceptibility as compared to the experimental val- ues at higher T values may be due to the follow- ing reasons: first, eq. (37) overestimates the decrease of E,(T) with increasing T which in turn makes Ix,,,( q, 0, T)j to decrease faster with increasing T. Second, the orbital and spin paramagnetism [16, 171 in Au also contribute towards the susceptibility but these are not in- cluded in the present calculation. The faster decrease of orbital paramagnetic susceptibility and of spin susceptibility with temperature, see fig. 3, weakens the decrease of the total diamag- netic susceptibility of Au. Third, the effect of thermal expansion on the susceptibility should also be taken into account because the tempera- ture variation of xorh(q, 0, T) in Au is very small. Further, we want to point out that with the explicit inclusion of the depletion hole contri- bution the temperature variation of xorb( q, 0, T) remains approximately the same except that its magnitude will change by about 10%.

The electron-electron interaction can also be included phenomenologically in the total nonin- teracting magnetic response function [2], ob- tained by adding spin and orbital contributions, and not to the orbital response function alone. The inclusion of electron-electron interaction will enhance the value of the magnetic response function.

4. Conclusions

A formalism for the temperature variation of the orbital dynamical susceptibility xorb( q, o, T) is developed in the model potential approach and it consists of two contributions: the free electron contribution and the depletion hole con- tribution. The latter takes care of the Bloch character of the electrons in the solid. The for- malism is applied to study the temperature vari- ation of xorb( q, o, T) for Al and Au in the free electron approximation. Reasonable agreement between theory and experiment in case of Au is obtained and shows that the major contribution to the susceptibility is from the orbital diamag- netism. In the future we plan to calculate ,y,,,( q,

J. Singh I Dynamical orbital susceptibility of metals 391

o, T) for a transition metal where the depletion [31 hole contribution is significant [5, 13, 141. [41

Finally, we conclude that this is the first at- tempt, to the best of our knowledge, to study the temperature variation of the orbital susceptibility in metallic systems.

151

[61 171

Acknowledgements

I am thankful to Professor M. Shimizu and Professor K.N. Pathak for fruitful discussions. The financial support from CSIR under scheme No. 656/88/EMR-II is gratefully acknowledged.

References

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[2] J.E. Hebborn and N.H. March, Adv. Phys. 19 (1970) 175.

PI

[91

1101 WI t121 1131 [I41 1151

[I61

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