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High-field magnetoresistance of metals by Kubo-Mott formula
ANNALS OF PHYSICS:70, 150-170 (1972)
High-Field Magnetoresistance of Metals by Kubo-Mott Formula* A. 0. E. ANIMALU+ Department of Physics, University of Missouri, Rolla, Missouri 65401 Received September 24, 1970
A completely quantum-mechanical calculation of the galvanomagnetic tensor olLy (including magnetic breakdown effects) for nearly free-electron (NFE) metals in strong dc magnetic fields (~~7 > 1) is presented. The method employed is the Kubo formula for olLVin a particularly convenient form introduced recently by Mott: this involves the matrix elements, <&* I j I &), of the current operator, j = (e/m)(-itiV + eA/c), where curl A defines the applied field H, and & is the one-electron wave function which we determine to first order in a weak pseudopotential, V, by NFE perturbation of Landau (harmonic oscillator) eigenstates for a free electron in a magnetic field. It is shown that the contributions to such matrix elements of the current can be represented by Feynman graphs for a second-order scattering process which involves an electron-coulomb vertex (given by V) and the emission or absorption of a fictitious excitation (characterized by a vertex given by a current (efi/m)(A,/r)l’z, A, being the area of the quantized cyclotron orbit in k-space, and an energy transfer of amount Aw, in the plane normal to H). For a pseudopotential, V, having any number of Fourier components, the method enables us to include phase coherence exactly in the Pippard network of electron orbits coupled by magnetic breakdown, and to exhibit three types of oscillatory effects due to phase coherence, electron density of states (Shubnikov-de Haas effect) and the so-called “zone” oscillations. Analytical results are given for the linear chain and the hexagonal network and compared with experiment.
1. INTRODUCTION In a number of previous articles [l-3], we have developed a practical method for applying the Kubo formalism [47] to quantitative calculation of the frequencydependent conductivity tensor of simple, nearly free-electron (NFE) metals in zero magnetic field. The purpose of this article and the next [8] is to extend our technique [l-3] to the calculation of the dc gulvanomagnetic tensor, uu, , of NFE * Work supported by grant NSF GU-2587 to Physic Department, University of Missouri, Rolla, Missouri. + Present address: Department of Physics, Drexel University, Philadelphia, Pennsylvania 19104. 0 1972 by Academic Press, Inc.
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metals in magnetic fields that are sufficiently strong to induce magnetic breakdown [9]. The essence of our method is to begin with the spectral form of the Kubo formula [4, 51 in the particularly convenient form introduced recently by Mott [6], and apply NFE perturbation theory in the spirit of Pippard [lo] to the determination of the electron wave functions required to evaluate the appropriate matrix elements of the current operator. The periodic lattice potential, V, whose Fourier components determine the topology of the Fermi surface and hence the structure of the Pippard network of electron orbits coupled by magnetic breakdown [IO], will be treated as a weak but otherwise arbitrary pseudopotential. This procedure will provide, for the first time, a completely quantum-mechanical treatment of the problem and will be applicable in practice to any arbitrary Pippard network. We note, however, that various methods of calculating aUYexist in current literature [ll-141 and that the oscillatory effects associated with the occurence of magnetic breakdown are reasonably well understood, as the review by Stark and Falicov [9] indicates. But the common shortcoming of all previous calculations stems from the fact that they rely on semiclassical methods [li, 121. As a consequence, no previous method has succeeded in providing an exact treatment of phase colrerence in an arbitrary Pippard network. The most recent effort by Chambers [13] to improve on the situation by employing a form of the Kubo formula and diagrammatic (Green’s function) method appears to have led to some slight success in the interpretation of experimental results [13, 141, but the method is also basically semiclassical. In this article, therefore, our primary objective is to provide a complete and simple quantum-mechanical picture which will transcend all previous quantitative calculations and reveal the origins of the three types of oscillatory galvanomagnetic effects that arise in the theory, viz., density of states, phase coherence, and the so-called “zone” oscillations. At the same time, our analysis will provide a simple interpretation of the usual behaviors of the galvanomagnetic tensor, such as the quadratic rise and saturation (in high fields) [15] and anomalous behaviors like linearity [16]. The outline of the paper is as follows. In Section 2, we shall develop a general theory of the galvanomagnetic tensor (including magnetic breakdown effects) from the Kubo formula; explicit expressions for the wave functions and matrix elements appearing in the Kubo formula will be given in terms of a weak but otherwise arbitrary local pseudopotential; and the results will be simulated in a number of relaxation times which will bring out the connection of our approach to the effective-path methods [I 1, 121. In Section 3, the theory will be applied to specific Pippard networks of special interest, namely the linear chain and the hexagonal network and compared with experiment. The paper will be concluded in Section 4.
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ANIMALU
2. THEORY OF THE GALVANOMAGNETIC
2.1. The Kubo-Mott
TENSOR
Formula
According to the general theory of irreversible processesdeveloped by Kubo [4], the current density induced by an oscillating electric field of frequency w in a metal under the influence of a dc magnetic field H = curl A, is given by the galvanomagnetic tensor,
%“(W) =s J mdt
’ dh eciwt(j,,(0) jv(t + ix)),
0
0
(CL,v = x, Y, 4.
(2.1)
In this formula, /3 = ~/KT, K being the Boltzmann constant and T the absolute temperature; ju(t) represents the p-component of the quantum-mechanical current operator in the Heisenberg representation, so that in a medium characterized by a Hamiltonian, X, jw(t) = eiflt/fij,(O) e-i*t/fi, j,(O) = (eQz)(-iV
+ eA/fic), = j, .
c2.2)
Also,
= c P?c k
defines the thermal average of an operator C, where pk = e-OEk e--BEs > lx s
(2.3)
Ek and #k being the eigenvalue and eigenfunction of 2. It is well known that the above form of the Kubo formula is not suitable for practical computation of o,, in a real metal. We shall therefore consider its spectral form, due to Greenwood [5], which involves only the stationary states of 8:
The two forms in (2.1) and (2.4) have been shown to be equivalent in Ref. [3]: the second form is, however, more useful for calculation. In this paper, we shall be primarily concerned with the limit of zero frequency
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(w -t 0) and low temperatures, in particular, T = 0°K. In this limit, a particularly convenient form of (2.4) has recently been derived by Mott [6]. One observes from the first equation in (2.4) that as the frequency tends to zero, the factor (pk, - pk.)/ (.Ek, - E,) approaches ap/~?E, and (2.4) takes the form (also due to Greenwood [5]):
Mott [b] noted, further, that at low temperatures, 3p/aE, approaches a S-function centered at the Fermi level E, . Thus, (2.5) reduces to what we shall call the Kubo-Mott formula:
This form is suitable for calculating the rlc conductivity under any condition, subject only to the uncertainty principle, dt < h/AE. When a magnetic field is present, we may interpret this relation with At = l/wC and AE = /T/T, so that it takes the form w,.~ > 1, W, being the cyclotron frequency of an electron tied to a closed orbit by the magnetic field, and T the relaxation time due to electronimpurity scattering (or electron-phonon scattering at finite temperatures). In addition to the low-temperature approximation. we shall make two additional approximations in the application of the Kubo-Mott formula (2.6) as follows: (i) Independent-particle approximation, consistent field method, so that the stationary by solving the one-electron wave equation, X$1< -ti (l/2111)(--i/7V
based on the HartreeeFock selfstates $I< and E, will be determined
(- eA/c)2 $I,,,-+ V$J,; = E&J,, ,
(2.7)
P’ being an effective lattice potential; (ii) Nearly free-electron approximation, based on the pseudopotential method [17], so that the lattice potential V can be treated as a weak perturbation of the Landau eigenstates. given by the eigenfunctions and eigenvalues of the unperturbed Hamiltonian, Z.
L
( 1/2w)(
--~/IV
-$
eA,/c)“.
(2.8)
In reality, of course, the very presenceof the magnetic field vitiates the concept of a pseudopotential, and one should more correctly speak of a “magnetic” pseudopotential (cf., “optical” pseudopotential [l, 21): this idea is taken up in AILl 1 AI1
is Ref.
[8].
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ANIMALU
It turns out, however, that a “magnetic” rather good approximation by an ordinary assume this fact in what follows.
pseudopotential can be replaced to a (zero magnetic field) one and we shall
Finally, we shall introduce an element of simplification, by considering forms of V which conserve the k,-component of the wave vector, since the quantization of the electron motion in the magnetic field occurs in the plane normal to the field direction (which will be taken to be the z-axis). Thus, we may, in the gauge defined in connection with (2.18) below, set
(A I .L I h~)
(2.9)
where summation over reciprocal lattice vectors such that k,, - k, = g, is to be understood for an arbitrary form of V. To eliminate one energy a-function from (2.6) we need the following relations between summation and integration over the quantum state (nk,k,) in a magnetic field (see, Eq. (1.21) of Ref. [7]): C --?, & 7%
j+” dk, = $ -cc
1 7$q
2
(E = iPkz2/2m); (2.10)
-& where X = 2/(/ic/eH). y% , are of the form:
s dk,, Also,
the unperturbed
energies, i.e., the eigenvalues
c(n, k,) = (n + +) &J, + fi2kz2/2m,
of
(2.11)
OJ, = (eH/mc) being the cyclotron frequency. Now, k,-conservation requires us to set k,, = k, in the integrand of (2.6) (from the summation over k,,) and integrate the result over k, to find
(2.13) In (2.13) we have used the fact that summation over IZ’ reduces to setting n’ = n and introducing the factor (l/fiiwJ from the S-function, and k, in Juy is to be given the value determined by the integration over k, in the step leading to (2.12). The essential point to grasp in the reduction of (2.6) to (2.13) is that k,-conservation implies ~1’ = 12but one should resist the temptation to conclude on the
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basis of this that (2.6) would then involve the meaningless factor [a(~, - ~(n, k,)]” which is clearly not the case. It is readily checked that (2.13) is equivalent to the final result:
The new feature of the Kubo-Mott formula can be appreciated directly from (2.6) and (2.14). If in (2.6) the matrix elements (c/J~~,I.j, I #,J are comtatzt, then the formula reduces simply to
(J,”=
(2.15)
where N(E,) is the density of states at the Fermi level and C is a constant. In this case, the conductivity is proportional to the square of the density of states as pointed out by Mott [6]. On the other hand, if in (2.14), JUL,depends only on g, , then the expression will reduce to the form
where N(+)
=
c ~(EF (n.vzL,)
+I, k,))
(2.16b)
is the free-electron density of states at the Fermi level (in a magnetic field). Accordingly, we are able to exhibit the Shubnikov-de Haas Effect, resulting from the periodicity of N(cF) in l/H, without much effort. Any other oscilIatory effects must occur through the function J,,,, which depends on geometry: we shall show that phase coherence effects in Pippard network models of electron orbits coupled by magnetic breakdown arise in this way. On the basis of (2.14) the calculation of the galvanomagnetic tensor for any form of a lattice potential that conserves the kz-component of the electron wave-vector k reduces to evaluation of the matrix elements of the current operator. This will be considered next. 2.2. Calculation
of ($k, 1j 1 $,> by NFE Perturbation
Method
Let us suppose that the orbits of an electron in a nearly free-electron (NFE) metal under a magnetic field are completely determined in band-theoretic sense from a knowledge of some weak but otherwise arbitrary lattice potential, V. In order to calculate the galvanomagnetic tensor for the transport of electrons in such orbits under any condition, including the possibility of magnetic breakdown,
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ANIMALU
we require two basic steps: namely, the calculation of (#,j lj, ] &J by a NFE perturbation method, and the reduction of the Kubo-Mott formula with such matrix elements. These will now be considered begining with the calculation of the matrix elements of the current operator. According to NFE perturbation theory, we find (2.17) Here, I V) stands for I k, , k, , n), where in the r-representation,
(r I k, , k, , n> = (LW1
exp(ik,x + ik,z) u,(y - yo)
(2.18)
(J+, = Pk,), is an unperturbed Landau function, i.e., an eigenfunction of (2.8) in the gauge [18], A = (--Hy, 0, 0), with H parallel to the z-axis; u, is a normalized simple harmonic oscillator eigenfunction of parity n, given by u,(y) H, being a Hermite
= (2% ! hn-)1/2 exp(--y2/2h2)
polynomial
H,(v),
(2.19)
of degree n;
h = (hc/eH)l12 = (~?/rno,)~/~ is the radius of the lowest (ground
state) Landau cyclotron
(2.20) orbit, and
E, = ~(n, k,) = (n + 4) fiw, + A2ka2/2m.
(2.21)
It is pertinent to make two general comments here. The first is that, strictly speaking, we should take the full magnetic translation symmetry of the Bloch Hamiltonian in (2.7) when constructing the unperturbed Landau functions in (2.18). Usually this would involve the use of the so-called symmetry adaptedfunctions [19], I xKn), which is given in our gauge by (I’ / XA
= (&Jz)F1
2 exp (isk,N’a, .9=--m
+ ik,z + i [k” - F]
x) u,(y - yJ, (2.22)
where yS = h2(k, - 2nsG/a,). Here ai (i = 1, 2, 3) are primitive translation vectors of the lattice, and a3 defines the direction of the z-axis; and for the Bloch states to be well defined with a good k-vector, it has to be assumed [19] that CD= H . a, 8 and N’ being relatively
x
a2 = (t/N’)(2dic/e),
(2.23)
prime integers, i.e., the flux CDacross a unit cell must be
HIGH-FELD
MAGNETORESISTANCE
OF
1.57
METALS
rational. It turns out, however, that the matrix elements of the current operator (2.3) are the same for both (2.18) and (2.22) and so we choose the simpler representation (2.18). But the domain of k, in (2.22) is limited only to (--/N’a, , + n/N’s,) and that of k, to (-255-t/a,, + 2x//a,), in any selection rule involving, say (xrPfz / exp(iq . t-)1x~,,) for the Fourier components of the lattice potential V: this will be discussedmore fully in Al1 where such detail is pertinent. A second comment is that a necessarycondition for the perturbation expansion (2.17) to be useful, i.e., convergent, is 1
(2.24)
which, as shown by Fischbeck [20] (see, also Section 4 of AH), is equivalent to Blount’s criterion for magnetic breakdown, H > Ho z ttlcA”/h ; e j E, ,
(2.25)
rl being the band gap at the pertinent Brillouin zone face where the breakdown occurs, and EF the Fermi energy. Consequently, the NFE perturbation method is, by fortunate accident, applicable to situations involving magnetic breakdown. We observe, however, that the Kubo-Mott formula is valid under the condition W,T > 1, and so the NFE perturbation method applies quite generally to high-field galvanomagnetic effects, particularly those associated with magnetic breakdown. Now, the calculation of the matrix elements of the current operator in the Schrodinger representation (2.3) is quite straightforward, but sufficiently intriguing to warrant a careful exposition of the salient steps. Like all problems involving magnetic fields, it is quite messy,but the physical content will be quite transparent. From the properties of harmonic oscillator eigenfunctions given by Schiff [21], we obtain, in our gauge, ‘,v’ 1jx / v,, =:
+(eM-,Jtn)
S(tz’,
17)
S(k’, k) -
(eh/td’) X,($8(k’,
k),
,j-
j L’,,
==
+(e/kl/tn)
S(n’,
77)
(2.26)
Qk’, k).
where h’
=-:
(/7/2t77w,.)‘~“;
/ifL+/
=
(I7
-7
I)l/”
S(t7’,
I? +
1)
&
tzW(t7’,
and we have introduced the abbreviated Kronecker a-function, S(n’,
t7)
:=
s*,,,,i
;
S(k’, k) ~-1 8~,~,k,8k,~,7~, .
II
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ANIMALU
The results in (2.26) are to be applied repeatedly to the complete expression for the matrix elements of the current between the states (2.17), i.e.,
+ 1 (v" 1j, 1 v) '",,! Y
"_I z"'*I
.
(2.27)
The first term in this expression can be dropped out as of no consequence in magnetoresistance. The remaining terms, by virtue of (2.26) reduce to the following expressions:
= ‘Elley; {k;(v’ 1v / v) - k,(v 1v 1v’)*} (n’)ljZ (v’ -
- (e~~m~~1 -
(etl/mhl)
1
1 I V 1 V) +
(n’ + 1)1/Z (V’ + 1 --1 I/ I v) E” - E,‘+1
E”- E”,-l
(n + 1)1/Z (v + 1 1 VI v’}” E”l - Ev+1
(n’)l/2 (v’ -
1 I V 1 v) _
E, _ E,,_l
+
@(v
E,'
-
1 1 V / v’)* 3 E,-1 I
(n’ + 1)1/Z (v’ + 1 I V / V) % - E,'+1
1 V + (iefi/mhl) 1 (n + 1)rj2 (V + 1 1u’)* E”’ - %+1
_
n112(u
-
E"'
1
1 -
I VI v’)* E,-1
1
19 (2.29)
(h
I A I kc>
= \-ey Y{k,‘(v’j v /v) - k,(v1v I v’>*},
(2.30)
Since these expressions look quite complicated, it is helpful to think of the various contributions in terms of Feynman graphs (Fig. 1) representing fundamental processes, as follows. Of the six terms in (2.28), the first, second, third, and fifth are represented by Figs. la-d, respectively. To see this, let us evaluate the contributions from these graphs.
HIGH-FIELD
MAGNETORESISTANCE
OF
159
METALS
FIG. 1. Contributions to the matrix elements of the current [Eq. (2.28)]; the graphs are evaluated in the text: (a) in Eq. (2.3la), (b) in Eq. (2.31b), (c)in Eq. (2,31c), and (d) in Eq. (2.31d).
From Fig. la. we find (cf. the rules set out in Ref. [3]): 1 jD(Er) b
S&
k’) a(~~ -
E,,,) GO(~, , l i)Ci; j 7~’i ~1, = jn(k,‘)
G&E, , l ,,)~:v 1 1~’~ v.’
(2.3Ja) and, from Fig. 1b. c .jJk,)
&ii, k) @es -
E,,) G,,(E,, , ci)(ti
1 V’ I v’> * = j,(k,,)
G,,(E,, , E~)(v / I’ 1 ~‘)a*,
(2.31b) where G”(E,, . Ebb) = (cc,- E~,)-I is the unperturbed one-electron Green’s function for an electron in the magnetic field, and ,j,(k,) = efik,.m
(2.32a)
is the current associated with a Drude (free) electron, which may be said to be equivalent to the coupling between an electron (shown as solid line) and a zerofrequency photon (shown as dashed line). The other vertex represents electroncoulomb scattering (with vertex V shown as a curly line). The two graphs give rise to a Drude-like conductivity of the type ascribed by Pippard to “quasiparticles”: their sum gives the first two terms of (2.28). The two terms represent such a quasiparticle moving along the k,-direction in k-space: the quasiparticle does not, in
160
ANIMALU
fact, move in a “straight line” because of the simple-harmonic y component of motion superimposed on the free &-motion; it is as if the particle “trembles” along its path. From Fig. lc, we find
(2.31~) and, from Fig. Id,
where j,(n)
= (e/W2/nzX’)
= (efi/m)(An/7r)1/2,
(2.32b)
A, = (2rreHn/Ac) being the area of the n-th quantized orbit in k-space, by virtue of Onsager’s theorem [22]. The current j,(n) represents the contribution at the vertex (shown as a wavy line) in Fig. lc and in Fig. Id, where the quasiparticle appears to absorb or emit an energy quantum liw, associated with the quantized orbits of a charged particle in a magnetic field. The absorption or emission of this quantum causes the quasiparticle to make an interlevel Landau transition from one quantized orbit to a neighboring one in the plane normal to the magnetic field. The remaining (fourth and sixth) terms in (2.28) are given by another pair of graphs similar to those shown in Figs. lc and Id, but with the arrows reversed, i.e., conjugate graphs. It is useful to note that the sum of Figs. la to Id represent the value of the quantity (Q!J~,ij, + ijU / I&). Alternatively, the sum of Figs. la and lb and the graphs conjugate to Figs. lc and Id represent the value of (4,~ Ij, - ij, I #,). Thus, the various terms have definite physical significance, and should be retained in the analysis, on equal footing. It is also worth pointing out here that the interpretation of the Feynman graphs in terms of “quasiparticles” is purely formal; in particular, we see no fundamental reason for abandoning the concept as Stark [15] and Falicov et al. [12] seem to suggest: instead, we shall see that the present estimate of the Drude part of aUVis in error. Finally, we sunzmarize the results of the analysis by specialising them to &-conserving lattice potentials, V, of any form. This requires us to set k,’ = k, and n’ = n, which together imply E,’ = E, , in the energy denominators. To prevent the contributions from Figs. la and lb, which we shall denote by ($,J ljUD 1 $,>,
HIGH-FIELD
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161
METALS
from being kjinite, we introduce an impurity relaxation time T, characterizing (Drude) intraferel transition, by setting, (2.33)
E,,’- E, = ihjr
in both terms: we find from the corresponding terms in (2.28) and (2.30) (multiplying and dividing by h’ for later convenience). (2.34)
u’,,,, j,O 1z/J,~:= (ie-rjt~d’) D,,(v’, v) l,j,=,i; ‘1 r- x or I D,,(v’. 1’) =: h’(k,\‘\V’ 1 v ~1’ -- k,,\:v / v 11” ,*j.
(2.35)
The J’-component is zero. The contribution from Figs. lc and Id, and two similar graphs, corresponding to the remaining terms in (2.28) and (2.29), which we shall denote by J,!J,,, ~,jUc 14,;. , are.fjnite and characterize itlterlevel transitions from one closed orbit to another: on setting E,,’--: E,.in these terms, we find (2.36) where 1')
C'i'(v'. f
{(/(
-~
{(/J -I-
.+ l)l:L'\'>"
l)"",'v
+ $
1 1 V'
1 1 v
~ ,'
v' -
*
-
,Jl:"':','
(f2')1;','1,' -
]
y
1 vJ‘
1 1 C' *j,
I'
;
(2.37)
The I component is zero. These are the results we are after. And we now proceed to usethem to complete the reduction of the galvanomagnetic tensor. 2.3 The :2/agnetoconc~~~ctiL~it~
in Relasatiorl
Time Approximation
Becauseof the analytical complexity of the preceding analysis. it is convenient to display the end results in terms of certain relaxation times which will simulate the dependence of o,, on the matrix elements of the lattice potential contained in the functions D, and C(-) defined above. The results apply to k-,-conserving lattice potentials V which are at present arbitrary, and can in principle generate any geometry of electron orbits in a magnetic field. By substituting the appropriate products, ;.lclk I .i, I $iL,,~~:$iT,I.i, I & ‘, into the formula (2.14) we find. for the diagonal components,
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ANIMALU
where the ratios, N/T, etc., of the carrier number density to the relaxation T, etc., are given by virtue of (2.34) and (2.36) as N/T, = (477/h) c I DAY’, 41” &EF N/r,*
(a = x or z),
4
= (47r/lz) 1 1 C(*)(Y’, v)I” S(+ -
times,
(2.39)
EJ,
N/T,, = (45+5) C 1D,(v, v’) C(+)(v’, v) - h.c. Ia 8(+ - EJ In these results, the D, and C(*) are to be evaluated at n’ = n, and h.c. stands for the hermitian conjugate of the preceding term. These results have standard forms but for the possibility that the cross-term (characterized by N/T,,) in gZe may not vanish, whenever the product D&J+) is pure imaginary. We observe, however, that 7, will, in general, be different from 7, so that the Drude part of the conductivity is different from the usual (zero-field) form Ne2r/m, to which it reduces if the magnetic field is turned off. If the relaxation times are constant (independent of wc), then the results are in accord with Onsager’s relations: u&“(H) = (T,~(-H), except for the cross-term in ore which may be important for the linear magnetoresistive behavior of certain metals [16]. The forms of the off-diagonal components of u,, can be expressed similarly in terms of certain relaxation times, but only tentative signs can be assigned without any further information about the D, and C I*). Proceeding as above, we find from (2.14) (2.34), and (2.36): uYx = Ne2r/mw,r,’ u1/z = Ne2r/mucrz’
+ Ne2/mwc2rcC
= --uxg , (2.40)
= -oz2/,
u m - Ne2T2/mTDD f Ne2T/mw,T,,
= -(sz5 ,
where N/T,’ = (4+5)
N/T,, = N/T,,
c C’-‘(v,
(47/B) c [iC’-‘(v, Y
v’) DJv’,
v) S(Q -
v’)] c(+yv’,
v) S(EF - E”),
= (457/A) c Dz(v, v’) Dz(y’, v> a(+ -
N/T,, = (4n/fi) c C”‘(v, ”
v’)[iDz(v’,
E,),
(2.41)
E,),
v)] 8(cF -
E,,).
Again, in these results, the D, and C (f) are to be evaluated at n’ = n, and we have absorbed factors of i in appropriate places where expressions are manifestly pure imaginary or zero by virtue of k, conservation or definition. The essence of these results is that if cross terms and the field dependence of
HIGH-FELD
MAGNETORESISTANCE
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163
the relaxation times are neglected, then our present method yields well-known semi-classicalforms of the magnetoresistivity tensor. Tn particular, for k,-conserving lattice potentials, we shall see(Eq. (3.8) below] that D; will be an oddfunction of k, so that the integrals (over k,) in the three relaxation times T:‘, T,., , and 7DD of (2.41) will vanish, giving ullz = u*: == 0. Then, the components of the transrerse magnetoresistivity tensor become: p,u z g,,/l CJI: p.ry = -pyc = %.,/I fJ :, py” = OJ,,,/ fJ I.
(2.42)
where I 0 1=: ‘s,,uva + &, . Observe that, in general, pJz # put, . The main novel feature of the present method consists, therefore, of the procedure it provides for practical computation of the relaxation times defined in this section.
3. APPLICATIONS
AND COMPARISON
WITH EXPERIMENT
3.1. Arbitrary 2-Ditmnsional Pippard Network; “Zone Oscillations” In this section, we shall apply the preceding theory to specific models of the Pippard network of electron orbits coupled by magnetic breakdown and compare the results (where possible) with experiments. Consider an arbitrary AZ-conserving periodic lattice potential of the form: I’(r) = 1 S(g) V, exp[ig,s -I- ig,y], zz
(3.1)
where S(g) is the geometrical structure factor for a given lattice structure and I’, is the Fourier component of the periodic potential associatedwith the reciprocal lattice vector g = (g, , g, , 0). Observe that the potential is constant in the z-direction, and consequently the model may be said to generate an arbitrary 2-dimensional Pippard network of coupled orbits in k-space. For the hexagonal network (Fig. 2) which is of practical interest, g is restricted (in the basal plane of the hexagonal lattice) to the six reciprocal lattice vectors of equal length: 1127T/LzMO, 2/d3,0),
* (27r/u)(l, -l/43,
O), j, (?Tf/a)(l, lj\J?, 01,
(3.2)
with corresponding structure factors (referred to the origin midway between the two atoms within a primitive cell of hexagonal close-packed metals): S(ig)
= -1.
-t1,
+1,
(3.3)
respectively. There is only one Fourier component V, of the lattice potential in this case. Observe that the first pair of g’s in (3.2) generate a one-dimensional linear chain of coupled orbits, if considered independently, while the last two pairs
164
ANIMALU
FIG. 2. Hexagonal network of coupled orbits in k-space; the large hole orbit “a” and the needle orbit “b” are the only ones that exist in low magnetic fields; the circular orbit “c” is the only one that exists if complete breakdown occurs.
(taken together) generate a rectangular network of coupled orbits. This should be born in mind since, as we shall see,each Fourier component of the lattice potential contributes independently to the matrix elements that determine G,,,. The calculation of the matrix elements that appear in the functions, D, and Cc*), defined by (2.35) and (2.37) is straightforward, but again quite complicated. For this reason, we must once more provide the intermediate results for the general case before returning presently to specific simple examples. Consider a general Fourier component of the form: V = V&q)
exp(iq,x + iq,y)
(3.4)
observing that, if S(q) = S,,,, , then S*(q) = a,,-, . Since kc is conserved, we need only matrix elements between Landau eigenstateswith k,’ = k, for II’ = n. Thus, for C(A) we need
= S(q)V,exp(+iX2k,q,)f,+,,,(-qq, , d ~A,~--L,.+~, 1 = S(q)v, exp(+ih2k,q,)f,-,,,(-q,, 4J hs,--k,,+g, t *= S*(q)v, exp(-jiX2k,‘q,)f,+I,,(q,, --4J &d,-kz,-gL T (3.5) * = S*(q) V, exp(--ih2k,‘q,),f,-,,,(q,,
-%,I &,,--P,,--o,,
HIGH-FIELD
MAGNETORESISTANCE
OF
165
METALS
where .L&l.?A4S . 41,) = 1‘ IK c/y exp(+iq,y) -cc
u,dy
(3.6)
+ x’%J u,(y).
From these expressions, we see by substitution in (2.37) that the contribution to 0’) from one Fourier component, q = +g is given by: c’(l) = S(g) 1’; exp(ih%,g,)((n
+ I)l;‘,f,-~,,,(-g,~,
g!,) ~ /VjJ,-r,,(-g,.
g,):
I’ [I + exp(i?7,,.g,,)lbk,,,,-,,d,,,J, . (3.7) C”
= S(g) I-,, exp(ih”k,g,)[(Ir
+ l)1”.f;,.8 I.li(-g.,, , g,,) + +?fn-,,,,(-gl
, g,,)]
:,~[I - vN~~Ys~g,,)l&,.f--1,..9, . Similarly,
the D, defined by (2.35) reduce to
We see, as previously noted, that D, is an o&function of kl , so that the integrals (over k,) in the relaxation times of (2.41) which involve D, will vanish, giving a,,3 = u,,, = 0. and a transverse magnetoresistivity tensor of the form (2.42). Typically, we find (3.9a) U’VY= Ne2/rmuc2rc-, where NjTc
= (4ni’h)
c
vg2
[(n
$- l)l,‘Zf,Z.n?l (6&
. g,) $- ~ll?fL*,n(-,G
, g?,Ill”
(nl,,h;g)
(3.9b)
‘.’ 2(1 ~ cos x”&.gI,) 6(t, - E(l?,I?:))
and similarly for the other components. From (3.9b) we observe the dependence of the galvanomagnetic tensor on oscillatory functions, namely, (1 - cos XZg2gI/) which may be called “zone” oscillation, S(E~- <(II, k,)) which gives rise to the Shubnikov-de Haas effect. and thef’s which introduce phasecoherenceoscillations as we shall seein Section 3.2. It is of interest to note that, by virtue of (2.23). h2g,g, = 2nN’lf
= 2&,/(2rhc/e)
= e@,/hc,
(3.9c)
where @,,is the flux of the magnetic field across the first Brillouin zone. Thus, the
166
ANIMALU
“zone” oscillations provide a basis for experimental quantization in a periodic lattice [23].
study of the nature of flux
3.2. The Linear Chain; Phase Coherence Oscillations The above results take a particularly simple analytic form for a coupled orbits and exhibit oscillatory effects due to phase coherence chain through the overlap integral defined by (3.6). To generate a k-space, we consider only the first pair of reciprocal lattice vectors the lattice potential has the simple one-dimensional form:
linear chain of in such a linear linear chain in in (3.2) so that
(3.10)
v = +3~gP(dew(iu) + S*(g) exp(--iuK
where g = (2n/a)(2/d3). By setting g, = 0 and g, = g in (3.5) to (3.8), we see that D, = 0 and C(-) = 0, while [for the first Fourier component in (3.10)] we find
C’+’ = 2~,%9 ewG~2kddKn+ l)1/2fn,n+1(0, s> - nl%n,n-l(O,811&,*.k, = 2ivAg) exp(ih2k,g)(h’g)f,,,(g, 0) &,*,k, , where, in the last step we have used the well-known integrals given by Rodriguez and Quinn [24]:
properties
of the overlap
fn*.n(O,d = (-Z-Y-” fn,,n(g, O), (n’ - n - 5>.kAg,
0) = F”[(n
(3.11)
(3.12)
+ 1)1/2fn~.n+l(g, 0) +
nl?fn,,n-l(g, O)],
E = (h’g)2 = hg2/2mw, evaluated for n’ = n. Explicitly,
.fnn(g,0) = j---m4~ + h2g)4Ld & = exp(- $3 L,(t) N (-1)”
(not sin 24)-lj2
sin[n(2$
-
(3.13)
sin 24) + r/4],
where cos2 q3 = 8/4n and the last expression in (3.13) is the asymptotic value for large n. Thus, in the linear chain, if r---f co in such a way that r/r2 = r/r,’ the transverse galvanomagnetic tensor is ~00
= Ne2r/m + Ne2/mw,2r,+;
ux?l = Ne2/mw,;
!Z 1, then
~?J?J= 0
(3.14)
HIGH-FIELD
MAGNETORESISTANCE
167
OF METALS
from (2.38) and (2.40) (apart from a cross term in uZZ), where N/7,-‘. = (4n/h) 8 V,y/?g2/2r?zW,)
1 1fn.Jg, (rkk,k,)
O)]” 6(E, -
E(Il, I?,)) (3.15)
N 16xV,” Here, &“(E~) is the free-electron density of states defined by (2.16b), and lzF CY E&J~ is the maximum value of IZ (typically, ~1~‘v 105H, H being in kilogauss in simple metals). We have used the fact that the two Fourier components in (3.10) contribute equally and independently to gZa. As we have pointed out earlier, (3.15) exhibits the Shubnikov-de Haas effect through N(E~). The oscillations due to phase coherence arise from its dependence on .fnF+ as follows. From the asymptotic formula (3.13) and the definition of nF , we find sin3[n,(2+,
lfn,,?+45, O>l” ‘v E
- sin 2+F) f
r/4]
F
(3.16) hw,
- sin”[h’A, cFn sin 24~ where co? +F = f/4??, and, following
+ n/4],
the analysis given by Reitz [25],
A, = (IZ~/~~)(~C#J~- sin 2~$~)
(3.17)
defines the area of the free-electron orbit in k-space outside the first Brillouin zone, A, being the notation used by Pippard (Fig. 3 of Ref. [lo]). The oscillatory behavior of (3.15) is now evident from (3.16). In order to bring out the connection of the oscillations with magnetic breakdown, let us introduce the breakdown field H,, as follows. We recognize that (3.16), apart from the oscillatory factor sin”[X2A, + r/4], is precisely the expression first derived by Pippard (Ref. [lo], Eq. (5)) for the magnitude of the Landau level broadening. Accordingly, the Pippard expression for the breakdown field is Ho = 4m~~;~/[(n from which it follows
sin 2~$~)l/~ h 1e 1 cF]
(3.18)
that
NIT,+ N smo, t 2:
! .N(cF)
sin2[h?A, + n/4].
This shows that l/rc+ cc w,, = eH,,/mc, so that TV+ may be interpreted relaxation time due to magnetic breakdown [9].
(3.19) as the
168
ANIMALU
Thus, the transverse
magnetoresistivity
P llv = (muc2r/Ne2) + E2
pyy = u,,/I oZy 12,is
8nw,(k2g2/2m) JIT(cF) sin2[h2A, + n/4].
(3.20)
This may be compared with equations (11.18) and (111.53)of Ref. 191,which were obtained by semiclassicalarguments. It is clear that the first term in this expression applies to the low magnetic field contribution to the magnetoresistivity from open orbits, while the second is the saturation (high field) value when magnetic breakdown of the open orbit occurs in the linear chain. Observe that the linear chain does not exhibit any “zone” oscillation. The calculation is readily generalized to a rectangular network, which may be thought of astwo linear chains at right angles: in this case there will be another contribution to (3.20) of the same form as the secondterm but having a different value of g and A, , i.e. a different amplitude and phase: this gives rise to beats in the oscillatory magnetoresistance, which is consistent with the data on tin [14]. 3.3. The Hexagonal Network; Comparism with Experiment The hexagonal network (Fig. 2) may be thought of as three linear chains inclined at an angle of 60” to each other. This implies that, by suitable choice of coordinate system, it may be possible to express the contributions from each of the three linear chains in the form of the second term in (3.20) with the sameamplitude but
2oti
Y
OO
’
I
I IO MAGNETIC
I
I
I
FIELD
STRENGTH
I 30
20 (in
I
I 40
KILOGAUSS)
FIG. 3. Experimental result showing phase coherence oscillations (after R. W. Stark, quoted as Fig. 26 in Ref. [9]) for a magnesium sample with low dislocation density at 1.1 “K.
HIGH-FIELD
MAGNETORESISTANCE
OF
METALS
169
different phases: this will give rise to phase coherence oscillations of different periods corresponding to the areas of the “needle” orbit (shown as b in Fig. I?), the “lens” orbit, and possibly the circle (shown as c in Fig. 2). On the basis of (3.9), one may also expect “zone” oscillations. These results are in accordance with the various experimental results obtained for hexagonal close-packed metals, like magnesium: the curve shown in Fig. 3 is a sample of the experimental results obtained by Stark (quoted in Ref. [9]). We have not attempted to fit our theory to the rather complicated oscillations, since we believe that very little will be learned from such an attempt beyond the qualitative understanding provided by the above analysis of the hexagonal network in terms of three linear chains. An exact calculation (in the framework of the present theory) is however possible. Since each Fourier component of the lattice potential contributes independently to the relaxation times (2.39) and (2.41). as indicated typically for oUU in (3.9), all that is required in practice is to sum expressions like (3.9b) over the six reciprocal lattice vectors g in (3.2). Since thef’s in (3.9b) are complicated, we have not taken htis step.
4.
SUMMARY
AND
CONCLUSIONS
The objective of this paper has been to extend the use of the Kubo formula for the conductivity to situations involving high magnetic tieids in a nearly free-electron metal, without resorting to semiclassical approximations. This procedure has enabled us to obtain the general form of the galvanomagnetic tensor (Section 2.3) and to exhibit special features, like the linearity of the magnetoresistancearising from certain cross terms in the matrix elements of the current, and oscillatory effects due to density of states (Shubnikov-de Haas effect), phase coherence in a network coupled by magnetic breakdown and “zone” oscillations. The main result of the paper is the explicit formula (3.20) for the transverse magnetoresistivity of a linear chain: this has provided additional insight into the sources of the various oscillatory effects mentioned above and is not completely equivalent to the usual results obtained by semiclassicalanalysis [9]. The result is only an approximation to the exact formula in (3.15) which includes phase coherence esactlv: this feature of the present method constitutes an improvement on the current situation [9]. To obtain absolute values of the magnetoresistance, the parameters 7, N and w,, may be treated as adjustable (cf. Refs. [l2, 131 to fit experiment. Finally, for most simple network models, we can obtain the galvanomagnetic tensor by decomposing the network into a number of linear chains and using symmetry considerations: this, as we saw, is due to the fact that each Fourier component of the lattice potential contributes independently to the galvanomagnetic tensor.
170
ANIMALU REFERENCES
1. A. 0. E. ANIMALU, 2. A. 0. E. ANIMALU, 3. A, 0. E. ANIMALU, 4. R. KUBO, J. Phys.
5. 6. 7. 8. 9. 10. 11. 12.
13. 14. 15. 16.
Phys. Phys. Phys. Sot. GREENWOOD, Proc. MOTT, Phil. Mug.
Rev. 163 (1967), 557. Rev. 163 (1967), 562. Rev. B 2 (1970), 282. Japan 12 (1957), 570. D. A. Phys. Sot. (London) Ser. A 71 (1958), 585. N. F. 13 (1966), 989. R. KUBO, S. J. MIYAKE, AND N. HASHITSUME, Solid State Phys. 17 (1965), 269. A. 0. E. ANIMALU, Ann. Phys. (N.Y.) 70 (1972), 171. For a recent review, see R. W. STARK AND L. M. FAUCOV, Progr. Low Temp. Phys. 5 (1968), 20. A. B. PIPPARD, Proc. Roy. Sot. (London) Ser. A 270 (1962), 1. A. B. PIPPARD, Proc. Roy. Sot. (London) Ser. A 287 (1965), 165. L. M. FALICOV AND P. R. SIEVERT, Phys. Rev. A 138 (1965), 88; L. M. FALICOV, A. B. PIPPARD, AND P. R. SIEVERT, Phys. Rev. 151 (1966), 498. W. G. CHAMBERS, Phys. Rev. 165 (1968), 799. J. 0. STROM-OSLEN, Phys. Rev. 180 (1969), 726. R. W. STARK, Phys. Rev. A 135 (1964), 1698. R. J. BALCOMBE, Proc. Roy. Sot. (London) Ser. A 275 (1963), 113; J. R. REITZ AND A. W. OVERHAUSER,Phys. Rev. 171 (1968), 749. W. A. HARRISON, “Pseudopotentials in the Theory of Metals, Benjamin, 1966.
17. 18. This gauge is most convenient for dealing with the usual choice of xyz-axes in hexagonal close-packed crystals [see Eq. (3.2)]. Our formula for (I,,~ is gauge-invariant. 19. J. ZAK, Phys. Rev. A 136 (1964), 776. 20. H. J. FISCHBECK,Phys. Status Solidi 22 (1967), 235, 649. 21. L. I. SCHIFF,“Quantum Mechanics,” pp. 64, 65, McGraw-Hill, New York, 1955. 22. L. ONSAGER, Phil. Msg. 43 (1952), 1006. 23. The assumption that the flux of the magnetic field across a unit cell of the crystal lattice is rational multiple of (ch/e) (see, e.g., Ref. [19]) has not been tested experimentally. 24. J. J. QUINN AND S. RODRIGUEZ,Phys. Rev. 128 (1962), 2487; see the Appendix of this reference. 25. J. R. REITZ, J. Phys. Chem. Solids 25 (1964), 53.
High-Field Magnetoresistance of Metals by Kubo-Mott Formula* A. 0. E. ANIMALU+ Department of Physics, University of Missouri, Rolla, Missouri 65401 Received September 24, 1970
A completely quantum-mechanical calculation of the galvanomagnetic tensor olLy (including magnetic breakdown effects) for nearly free-electron (NFE) metals in strong dc magnetic fields (~~7 > 1) is presented. The method employed is the Kubo formula for olLVin a particularly convenient form introduced recently by Mott: this involves the matrix elements, <&* I j I &), of the current operator, j = (e/m)(-itiV + eA/c), where curl A defines the applied field H, and & is the one-electron wave function which we determine to first order in a weak pseudopotential, V, by NFE perturbation of Landau (harmonic oscillator) eigenstates for a free electron in a magnetic field. It is shown that the contributions to such matrix elements of the current can be represented by Feynman graphs for a second-order scattering process which involves an electron-coulomb vertex (given by V) and the emission or absorption of a fictitious excitation (characterized by a vertex given by a current (efi/m)(A,/r)l’z, A, being the area of the quantized cyclotron orbit in k-space, and an energy transfer of amount Aw, in the plane normal to H). For a pseudopotential, V, having any number of Fourier components, the method enables us to include phase coherence exactly in the Pippard network of electron orbits coupled by magnetic breakdown, and to exhibit three types of oscillatory effects due to phase coherence, electron density of states (Shubnikov-de Haas effect) and the so-called “zone” oscillations. Analytical results are given for the linear chain and the hexagonal network and compared with experiment.
1. INTRODUCTION In a number of previous articles [l-3], we have developed a practical method for applying the Kubo formalism [47] to quantitative calculation of the frequencydependent conductivity tensor of simple, nearly free-electron (NFE) metals in zero magnetic field. The purpose of this article and the next [8] is to extend our technique [l-3] to the calculation of the dc gulvanomagnetic tensor, uu, , of NFE * Work supported by grant NSF GU-2587 to Physic Department, University of Missouri, Rolla, Missouri. + Present address: Department of Physics, Drexel University, Philadelphia, Pennsylvania 19104. 0 1972 by Academic Press, Inc.
150
HIGH-FIELD
MAGNETORESISTANCE
OF
METALS
151
metals in magnetic fields that are sufficiently strong to induce magnetic breakdown [9]. The essence of our method is to begin with the spectral form of the Kubo formula [4, 51 in the particularly convenient form introduced recently by Mott [6], and apply NFE perturbation theory in the spirit of Pippard [lo] to the determination of the electron wave functions required to evaluate the appropriate matrix elements of the current operator. The periodic lattice potential, V, whose Fourier components determine the topology of the Fermi surface and hence the structure of the Pippard network of electron orbits coupled by magnetic breakdown [IO], will be treated as a weak but otherwise arbitrary pseudopotential. This procedure will provide, for the first time, a completely quantum-mechanical treatment of the problem and will be applicable in practice to any arbitrary Pippard network. We note, however, that various methods of calculating aUYexist in current literature [ll-141 and that the oscillatory effects associated with the occurence of magnetic breakdown are reasonably well understood, as the review by Stark and Falicov [9] indicates. But the common shortcoming of all previous calculations stems from the fact that they rely on semiclassical methods [li, 121. As a consequence, no previous method has succeeded in providing an exact treatment of phase colrerence in an arbitrary Pippard network. The most recent effort by Chambers [13] to improve on the situation by employing a form of the Kubo formula and diagrammatic (Green’s function) method appears to have led to some slight success in the interpretation of experimental results [13, 141, but the method is also basically semiclassical. In this article, therefore, our primary objective is to provide a complete and simple quantum-mechanical picture which will transcend all previous quantitative calculations and reveal the origins of the three types of oscillatory galvanomagnetic effects that arise in the theory, viz., density of states, phase coherence, and the so-called “zone” oscillations. At the same time, our analysis will provide a simple interpretation of the usual behaviors of the galvanomagnetic tensor, such as the quadratic rise and saturation (in high fields) [15] and anomalous behaviors like linearity [16]. The outline of the paper is as follows. In Section 2, we shall develop a general theory of the galvanomagnetic tensor (including magnetic breakdown effects) from the Kubo formula; explicit expressions for the wave functions and matrix elements appearing in the Kubo formula will be given in terms of a weak but otherwise arbitrary local pseudopotential; and the results will be simulated in a number of relaxation times which will bring out the connection of our approach to the effective-path methods [I 1, 121. In Section 3, the theory will be applied to specific Pippard networks of special interest, namely the linear chain and the hexagonal network and compared with experiment. The paper will be concluded in Section 4.
152
ANIMALU
2. THEORY OF THE GALVANOMAGNETIC
2.1. The Kubo-Mott
TENSOR
Formula
According to the general theory of irreversible processesdeveloped by Kubo [4], the current density induced by an oscillating electric field of frequency w in a metal under the influence of a dc magnetic field H = curl A, is given by the galvanomagnetic tensor,
%“(W) =s J mdt
’ dh eciwt(j,,(0) jv(t + ix)),
0
0
(CL,v = x, Y, 4.
(2.1)
In this formula, /3 = ~/KT, K being the Boltzmann constant and T the absolute temperature; ju(t) represents the p-component of the quantum-mechanical current operator in the Heisenberg representation, so that in a medium characterized by a Hamiltonian, X, jw(t) = eiflt/fij,(O) e-i*t/fi, j,(O) = (eQz)(-iV
+ eA/fic), = j, .
c2.2)
Also,
= c P?c k
defines the thermal average of an operator C, where pk = e-OEk e--BEs > lx s
(2.3)
Ek and #k being the eigenvalue and eigenfunction of 2. It is well known that the above form of the Kubo formula is not suitable for practical computation of o,, in a real metal. We shall therefore consider its spectral form, due to Greenwood [5], which involves only the stationary states of 8:
The two forms in (2.1) and (2.4) have been shown to be equivalent in Ref. [3]: the second form is, however, more useful for calculation. In this paper, we shall be primarily concerned with the limit of zero frequency
HIGH-FIELD
MAGNETORESISTANCE
OF
METALS
153
(w -t 0) and low temperatures, in particular, T = 0°K. In this limit, a particularly convenient form of (2.4) has recently been derived by Mott [6]. One observes from the first equation in (2.4) that as the frequency tends to zero, the factor (pk, - pk.)/ (.Ek, - E,) approaches ap/~?E, and (2.4) takes the form (also due to Greenwood [5]):
Mott [b] noted, further, that at low temperatures, 3p/aE, approaches a S-function centered at the Fermi level E, . Thus, (2.5) reduces to what we shall call the Kubo-Mott formula:
This form is suitable for calculating the rlc conductivity under any condition, subject only to the uncertainty principle, dt < h/AE. When a magnetic field is present, we may interpret this relation with At = l/wC and AE = /T/T, so that it takes the form w,.~ > 1, W, being the cyclotron frequency of an electron tied to a closed orbit by the magnetic field, and T the relaxation time due to electronimpurity scattering (or electron-phonon scattering at finite temperatures). In addition to the low-temperature approximation. we shall make two additional approximations in the application of the Kubo-Mott formula (2.6) as follows: (i) Independent-particle approximation, consistent field method, so that the stationary by solving the one-electron wave equation, X$1< -ti (l/2111)(--i/7V
based on the HartreeeFock selfstates $I< and E, will be determined
(- eA/c)2 $I,,,-+ V$J,; = E&J,, ,
(2.7)
P’ being an effective lattice potential; (ii) Nearly free-electron approximation, based on the pseudopotential method [17], so that the lattice potential V can be treated as a weak perturbation of the Landau eigenstates. given by the eigenfunctions and eigenvalues of the unperturbed Hamiltonian, Z.
L
( 1/2w)(
--~/IV
-$
eA,/c)“.
(2.8)
In reality, of course, the very presenceof the magnetic field vitiates the concept of a pseudopotential, and one should more correctly speak of a “magnetic” pseudopotential (cf., “optical” pseudopotential [l, 21): this idea is taken up in AILl 1 AI1
is Ref.
[8].
154
ANIMALU
It turns out, however, that a “magnetic” rather good approximation by an ordinary assume this fact in what follows.
pseudopotential can be replaced to a (zero magnetic field) one and we shall
Finally, we shall introduce an element of simplification, by considering forms of V which conserve the k,-component of the wave vector, since the quantization of the electron motion in the magnetic field occurs in the plane normal to the field direction (which will be taken to be the z-axis). Thus, we may, in the gauge defined in connection with (2.18) below, set
(A I .L I h~)
(2.9)
where summation over reciprocal lattice vectors such that k,, - k, = g, is to be understood for an arbitrary form of V. To eliminate one energy a-function from (2.6) we need the following relations between summation and integration over the quantum state (nk,k,) in a magnetic field (see, Eq. (1.21) of Ref. [7]): C --?, & 7%
j+” dk, = $ -cc
1 7$q
2
(E = iPkz2/2m); (2.10)
-& where X = 2/(/ic/eH). y% , are of the form:
s dk,, Also,
the unperturbed
energies, i.e., the eigenvalues
c(n, k,) = (n + +) &J, + fi2kz2/2m,
of
(2.11)
OJ, = (eH/mc) being the cyclotron frequency. Now, k,-conservation requires us to set k,, = k, in the integrand of (2.6) (from the summation over k,,) and integrate the result over k, to find
(2.13) In (2.13) we have used the fact that summation over IZ’ reduces to setting n’ = n and introducing the factor (l/fiiwJ from the S-function, and k, in Juy is to be given the value determined by the integration over k, in the step leading to (2.12). The essential point to grasp in the reduction of (2.6) to (2.13) is that k,-conservation implies ~1’ = 12but one should resist the temptation to conclude on the
HIGH-FIELD
MAGNETORESISTANCE
OF
METALS
155
basis of this that (2.6) would then involve the meaningless factor [a(~, - ~(n, k,)]” which is clearly not the case. It is readily checked that (2.13) is equivalent to the final result:
The new feature of the Kubo-Mott formula can be appreciated directly from (2.6) and (2.14). If in (2.6) the matrix elements (c/J~~,I.j, I #,J are comtatzt, then the formula reduces simply to
(J,”=
(2.15)
where N(E,) is the density of states at the Fermi level and C is a constant. In this case, the conductivity is proportional to the square of the density of states as pointed out by Mott [6]. On the other hand, if in (2.14), JUL,depends only on g, , then the expression will reduce to the form
where N(+)
=
c ~(EF (n.vzL,)
+I, k,))
(2.16b)
is the free-electron density of states at the Fermi level (in a magnetic field). Accordingly, we are able to exhibit the Shubnikov-de Haas Effect, resulting from the periodicity of N(cF) in l/H, without much effort. Any other oscilIatory effects must occur through the function J,,,, which depends on geometry: we shall show that phase coherence effects in Pippard network models of electron orbits coupled by magnetic breakdown arise in this way. On the basis of (2.14) the calculation of the galvanomagnetic tensor for any form of a lattice potential that conserves the kz-component of the electron wave-vector k reduces to evaluation of the matrix elements of the current operator. This will be considered next. 2.2. Calculation
of ($k, 1j 1 $,> by NFE Perturbation
Method
Let us suppose that the orbits of an electron in a nearly free-electron (NFE) metal under a magnetic field are completely determined in band-theoretic sense from a knowledge of some weak but otherwise arbitrary lattice potential, V. In order to calculate the galvanomagnetic tensor for the transport of electrons in such orbits under any condition, including the possibility of magnetic breakdown,
156
ANIMALU
we require two basic steps: namely, the calculation of (#,j lj, ] &J by a NFE perturbation method, and the reduction of the Kubo-Mott formula with such matrix elements. These will now be considered begining with the calculation of the matrix elements of the current operator. According to NFE perturbation theory, we find (2.17) Here, I V) stands for I k, , k, , n), where in the r-representation,
(r I k, , k, , n> = (LW1
exp(ik,x + ik,z) u,(y - yo)
(2.18)
(J+, = Pk,), is an unperturbed Landau function, i.e., an eigenfunction of (2.8) in the gauge [18], A = (--Hy, 0, 0), with H parallel to the z-axis; u, is a normalized simple harmonic oscillator eigenfunction of parity n, given by u,(y) H, being a Hermite
= (2% ! hn-)1/2 exp(--y2/2h2)
polynomial
H,(v),
(2.19)
of degree n;
h = (hc/eH)l12 = (~?/rno,)~/~ is the radius of the lowest (ground
state) Landau cyclotron
(2.20) orbit, and
E, = ~(n, k,) = (n + 4) fiw, + A2ka2/2m.
(2.21)
It is pertinent to make two general comments here. The first is that, strictly speaking, we should take the full magnetic translation symmetry of the Bloch Hamiltonian in (2.7) when constructing the unperturbed Landau functions in (2.18). Usually this would involve the use of the so-called symmetry adaptedfunctions [19], I xKn), which is given in our gauge by (I’ / XA
= (&Jz)F1
2 exp (isk,N’a, .9=--m
+ ik,z + i [k” - F]
x) u,(y - yJ, (2.22)
where yS = h2(k, - 2nsG/a,). Here ai (i = 1, 2, 3) are primitive translation vectors of the lattice, and a3 defines the direction of the z-axis; and for the Bloch states to be well defined with a good k-vector, it has to be assumed [19] that CD= H . a, 8 and N’ being relatively
x
a2 = (t/N’)(2dic/e),
(2.23)
prime integers, i.e., the flux CDacross a unit cell must be
HIGH-FELD
MAGNETORESISTANCE
OF
1.57
METALS
rational. It turns out, however, that the matrix elements of the current operator (2.3) are the same for both (2.18) and (2.22) and so we choose the simpler representation (2.18). But the domain of k, in (2.22) is limited only to (--/N’a, , + n/N’s,) and that of k, to (-255-t/a,, + 2x//a,), in any selection rule involving, say (xrPfz / exp(iq . t-)1x~,,) for the Fourier components of the lattice potential V: this will be discussedmore fully in Al1 where such detail is pertinent. A second comment is that a necessarycondition for the perturbation expansion (2.17) to be useful, i.e., convergent, is 1
(2.24)
which, as shown by Fischbeck [20] (see, also Section 4 of AH), is equivalent to Blount’s criterion for magnetic breakdown, H > Ho z ttlcA”/h ; e j E, ,
(2.25)
rl being the band gap at the pertinent Brillouin zone face where the breakdown occurs, and EF the Fermi energy. Consequently, the NFE perturbation method is, by fortunate accident, applicable to situations involving magnetic breakdown. We observe, however, that the Kubo-Mott formula is valid under the condition W,T > 1, and so the NFE perturbation method applies quite generally to high-field galvanomagnetic effects, particularly those associated with magnetic breakdown. Now, the calculation of the matrix elements of the current operator in the Schrodinger representation (2.3) is quite straightforward, but sufficiently intriguing to warrant a careful exposition of the salient steps. Like all problems involving magnetic fields, it is quite messy,but the physical content will be quite transparent. From the properties of harmonic oscillator eigenfunctions given by Schiff [21], we obtain, in our gauge, ‘,v’ 1jx / v,, =:
+(eM-,Jtn)
S(tz’,
17)
S(k’, k) -
(eh/td’) X,($8(k’,
k),
,j-
j L’,,
==
+(e/kl/tn)
S(n’,
77)
(2.26)
Qk’, k).
where h’
=-:
(/7/2t77w,.)‘~“;
/ifL+/
=
(I7
-7
I)l/”
S(t7’,
I? +
1)
&
tzW(t7’,
and we have introduced the abbreviated Kronecker a-function, S(n’,
t7)
:=
s*,,,,i
;
S(k’, k) ~-1 8~,~,k,8k,~,7~, .
II
158
ANIMALU
The results in (2.26) are to be applied repeatedly to the complete expression for the matrix elements of the current between the states (2.17), i.e.,
+ 1 (v" 1j, 1 v) '",,! Y
"_I z"'*I
.
(2.27)
The first term in this expression can be dropped out as of no consequence in magnetoresistance. The remaining terms, by virtue of (2.26) reduce to the following expressions:
= ‘Elley; {k;(v’ 1v / v) - k,(v 1v 1v’)*} (n’)ljZ (v’ -
- (e~~m~~1 -
(etl/mhl)
1
1 I V 1 V) +
(n’ + 1)1/Z (V’ + 1 --1 I/ I v) E” - E,‘+1
E”- E”,-l
(n + 1)1/Z (v + 1 1 VI v’}” E”l - Ev+1
(n’)l/2 (v’ -
1 I V 1 v) _
E, _ E,,_l
+
@(v
E,'
-
1 1 V / v’)* 3 E,-1 I
(n’ + 1)1/Z (v’ + 1 I V / V) % - E,'+1
1 V + (iefi/mhl) 1 (n + 1)rj2 (V + 1 1u’)* E”’ - %+1
_
n112(u
-
E"'
1
1 -
I VI v’)* E,-1
1
19 (2.29)
(h
I A I kc>
= \-ey Y{k,‘(v’j v /v) - k,(v1v I v’>*},
(2.30)
Since these expressions look quite complicated, it is helpful to think of the various contributions in terms of Feynman graphs (Fig. 1) representing fundamental processes, as follows. Of the six terms in (2.28), the first, second, third, and fifth are represented by Figs. la-d, respectively. To see this, let us evaluate the contributions from these graphs.
HIGH-FIELD
MAGNETORESISTANCE
OF
159
METALS
FIG. 1. Contributions to the matrix elements of the current [Eq. (2.28)]; the graphs are evaluated in the text: (a) in Eq. (2.3la), (b) in Eq. (2.31b), (c)in Eq. (2,31c), and (d) in Eq. (2.31d).
From Fig. la. we find (cf. the rules set out in Ref. [3]): 1 jD(Er) b
S&
k’) a(~~ -
E,,,) GO(~, , l i)Ci; j 7~’i ~1, = jn(k,‘)
G&E, , l ,,)~:v 1 1~’~ v.’
(2.3Ja) and, from Fig. 1b. c .jJk,)
&ii, k) @es -
E,,) G,,(E,, , ci)(ti
1 V’ I v’> * = j,(k,,)
G,,(E,, , E~)(v / I’ 1 ~‘)a*,
(2.31b) where G”(E,, . Ebb) = (cc,- E~,)-I is the unperturbed one-electron Green’s function for an electron in the magnetic field, and ,j,(k,) = efik,.m
(2.32a)
is the current associated with a Drude (free) electron, which may be said to be equivalent to the coupling between an electron (shown as solid line) and a zerofrequency photon (shown as dashed line). The other vertex represents electroncoulomb scattering (with vertex V shown as a curly line). The two graphs give rise to a Drude-like conductivity of the type ascribed by Pippard to “quasiparticles”: their sum gives the first two terms of (2.28). The two terms represent such a quasiparticle moving along the k,-direction in k-space: the quasiparticle does not, in
160
ANIMALU
fact, move in a “straight line” because of the simple-harmonic y component of motion superimposed on the free &-motion; it is as if the particle “trembles” along its path. From Fig. lc, we find
(2.31~) and, from Fig. Id,
where j,(n)
= (e/W2/nzX’)
= (efi/m)(An/7r)1/2,
(2.32b)
A, = (2rreHn/Ac) being the area of the n-th quantized orbit in k-space, by virtue of Onsager’s theorem [22]. The current j,(n) represents the contribution at the vertex (shown as a wavy line) in Fig. lc and in Fig. Id, where the quasiparticle appears to absorb or emit an energy quantum liw, associated with the quantized orbits of a charged particle in a magnetic field. The absorption or emission of this quantum causes the quasiparticle to make an interlevel Landau transition from one quantized orbit to a neighboring one in the plane normal to the magnetic field. The remaining (fourth and sixth) terms in (2.28) are given by another pair of graphs similar to those shown in Figs. lc and Id, but with the arrows reversed, i.e., conjugate graphs. It is useful to note that the sum of Figs. la to Id represent the value of the quantity (Q!J~,ij, + ijU / I&). Alternatively, the sum of Figs. la and lb and the graphs conjugate to Figs. lc and Id represent the value of (4,~ Ij, - ij, I #,). Thus, the various terms have definite physical significance, and should be retained in the analysis, on equal footing. It is also worth pointing out here that the interpretation of the Feynman graphs in terms of “quasiparticles” is purely formal; in particular, we see no fundamental reason for abandoning the concept as Stark [15] and Falicov et al. [12] seem to suggest: instead, we shall see that the present estimate of the Drude part of aUVis in error. Finally, we sunzmarize the results of the analysis by specialising them to &-conserving lattice potentials, V, of any form. This requires us to set k,’ = k, and n’ = n, which together imply E,’ = E, , in the energy denominators. To prevent the contributions from Figs. la and lb, which we shall denote by ($,J ljUD 1 $,>,
HIGH-FIELD
MAGNETORESISTANCE
OF
161
METALS
from being kjinite, we introduce an impurity relaxation time T, characterizing (Drude) intraferel transition, by setting, (2.33)
E,,’- E, = ihjr
in both terms: we find from the corresponding terms in (2.28) and (2.30) (multiplying and dividing by h’ for later convenience). (2.34)
u’,,,, j,O 1z/J,~:= (ie-rjt~d’) D,,(v’, v) l,j,=,i; ‘1 r- x or I D,,(v’. 1’) =: h’(k,\‘\V’ 1 v ~1’ -- k,,\:v / v 11” ,*j.
(2.35)
The J’-component is zero. The contribution from Figs. lc and Id, and two similar graphs, corresponding to the remaining terms in (2.28) and (2.29), which we shall denote by J,!J,,, ~,jUc 14,;. , are.fjnite and characterize itlterlevel transitions from one closed orbit to another: on setting E,,’--: E,.in these terms, we find (2.36) where 1')
C'i'(v'. f
{(/(
-~
{(/J -I-
.+ l)l:L'\'>"
l)"",'v
+ $
1 1 V'
1 1 v
~ ,'
v' -
*
-
,Jl:"':','
(f2')1;','1,' -
]
y
1 vJ‘
1 1 C' *j,
I'
;
(2.37)
The I component is zero. These are the results we are after. And we now proceed to usethem to complete the reduction of the galvanomagnetic tensor. 2.3 The :2/agnetoconc~~~ctiL~it~
in Relasatiorl
Time Approximation
Becauseof the analytical complexity of the preceding analysis. it is convenient to display the end results in terms of certain relaxation times which will simulate the dependence of o,, on the matrix elements of the lattice potential contained in the functions D, and C(-) defined above. The results apply to k-,-conserving lattice potentials V which are at present arbitrary, and can in principle generate any geometry of electron orbits in a magnetic field. By substituting the appropriate products, ;.lclk I .i, I $iL,,~~:$iT,I.i, I & ‘, into the formula (2.14) we find. for the diagonal components,
162
ANIMALU
where the ratios, N/T, etc., of the carrier number density to the relaxation T, etc., are given by virtue of (2.34) and (2.36) as N/T, = (477/h) c I DAY’, 41” &EF N/r,*
(a = x or z),
4
= (47r/lz) 1 1 C(*)(Y’, v)I” S(+ -
times,
(2.39)
EJ,
N/T,, = (45+5) C 1D,(v, v’) C(+)(v’, v) - h.c. Ia 8(+ - EJ In these results, the D, and C(*) are to be evaluated at n’ = n, and h.c. stands for the hermitian conjugate of the preceding term. These results have standard forms but for the possibility that the cross-term (characterized by N/T,,) in gZe may not vanish, whenever the product D&J+) is pure imaginary. We observe, however, that 7, will, in general, be different from 7, so that the Drude part of the conductivity is different from the usual (zero-field) form Ne2r/m, to which it reduces if the magnetic field is turned off. If the relaxation times are constant (independent of wc), then the results are in accord with Onsager’s relations: u&“(H) = (T,~(-H), except for the cross-term in ore which may be important for the linear magnetoresistive behavior of certain metals [16]. The forms of the off-diagonal components of u,, can be expressed similarly in terms of certain relaxation times, but only tentative signs can be assigned without any further information about the D, and C I*). Proceeding as above, we find from (2.14) (2.34), and (2.36): uYx = Ne2r/mw,r,’ u1/z = Ne2r/mucrz’
+ Ne2/mwc2rcC
= --uxg , (2.40)
= -oz2/,
u m - Ne2T2/mTDD f Ne2T/mw,T,,
= -(sz5 ,
where N/T,’ = (4+5)
N/T,, = N/T,,
c C’-‘(v,
(47/B) c [iC’-‘(v, Y
v’) DJv’,
v) S(Q -
v’)] c(+yv’,
v) S(EF - E”),
= (457/A) c Dz(v, v’) Dz(y’, v> a(+ -
N/T,, = (4n/fi) c C”‘(v, ”
v’)[iDz(v’,
E,),
(2.41)
E,),
v)] 8(cF -
E,,).
Again, in these results, the D, and C (f) are to be evaluated at n’ = n, and we have absorbed factors of i in appropriate places where expressions are manifestly pure imaginary or zero by virtue of k, conservation or definition. The essence of these results is that if cross terms and the field dependence of
HIGH-FELD
MAGNETORESISTANCE
OF METALS
163
the relaxation times are neglected, then our present method yields well-known semi-classicalforms of the magnetoresistivity tensor. Tn particular, for k,-conserving lattice potentials, we shall see(Eq. (3.8) below] that D; will be an oddfunction of k, so that the integrals (over k,) in the three relaxation times T:‘, T,., , and 7DD of (2.41) will vanish, giving ullz = u*: == 0. Then, the components of the transrerse magnetoresistivity tensor become: p,u z g,,/l CJI: p.ry = -pyc = %.,/I fJ :, py” = OJ,,,/ fJ I.
(2.42)
where I 0 1=: ‘s,,uva + &, . Observe that, in general, pJz # put, . The main novel feature of the present method consists, therefore, of the procedure it provides for practical computation of the relaxation times defined in this section.
3. APPLICATIONS
AND COMPARISON
WITH EXPERIMENT
3.1. Arbitrary 2-Ditmnsional Pippard Network; “Zone Oscillations” In this section, we shall apply the preceding theory to specific models of the Pippard network of electron orbits coupled by magnetic breakdown and compare the results (where possible) with experiments. Consider an arbitrary AZ-conserving periodic lattice potential of the form: I’(r) = 1 S(g) V, exp[ig,s -I- ig,y], zz
(3.1)
where S(g) is the geometrical structure factor for a given lattice structure and I’, is the Fourier component of the periodic potential associatedwith the reciprocal lattice vector g = (g, , g, , 0). Observe that the potential is constant in the z-direction, and consequently the model may be said to generate an arbitrary 2-dimensional Pippard network of coupled orbits in k-space. For the hexagonal network (Fig. 2) which is of practical interest, g is restricted (in the basal plane of the hexagonal lattice) to the six reciprocal lattice vectors of equal length: 1127T/LzMO, 2/d3,0),
* (27r/u)(l, -l/43,
O), j, (?Tf/a)(l, lj\J?, 01,
(3.2)
with corresponding structure factors (referred to the origin midway between the two atoms within a primitive cell of hexagonal close-packed metals): S(ig)
= -1.
-t1,
+1,
(3.3)
respectively. There is only one Fourier component V, of the lattice potential in this case. Observe that the first pair of g’s in (3.2) generate a one-dimensional linear chain of coupled orbits, if considered independently, while the last two pairs
164
ANIMALU
FIG. 2. Hexagonal network of coupled orbits in k-space; the large hole orbit “a” and the needle orbit “b” are the only ones that exist in low magnetic fields; the circular orbit “c” is the only one that exists if complete breakdown occurs.
(taken together) generate a rectangular network of coupled orbits. This should be born in mind since, as we shall see,each Fourier component of the lattice potential contributes independently to the matrix elements that determine G,,,. The calculation of the matrix elements that appear in the functions, D, and Cc*), defined by (2.35) and (2.37) is straightforward, but again quite complicated. For this reason, we must once more provide the intermediate results for the general case before returning presently to specific simple examples. Consider a general Fourier component of the form: V = V&q)
exp(iq,x + iq,y)
(3.4)
observing that, if S(q) = S,,,, , then S*(q) = a,,-, . Since kc is conserved, we need only matrix elements between Landau eigenstateswith k,’ = k, for II’ = n. Thus, for C(A) we need
-%,I &,,--P,,--o,,
HIGH-FIELD
MAGNETORESISTANCE
OF
165
METALS
where .L&l.?A4S . 41,) = 1‘ IK c/y exp(+iq,y) -cc
u,dy
(3.6)
+ x’%J u,(y).
From these expressions, we see by substitution in (2.37) that the contribution to 0’) from one Fourier component, q = +g is given by: c’(l) = S(g) 1’; exp(ih%,g,)((n
+ I)l;‘,f,-~,,,(-g,~,
g!,) ~ /VjJ,-r,,(-g,.
g,):
I’ [I + exp(i?7,,.g,,)lbk,,,,-,,d,,,J, . (3.7) C”
= S(g) I-,, exp(ih”k,g,)[(Ir
+ l)1”.f;,.8 I.li(-g.,, , g,,) + +?fn-,,,,(-gl
, g,,)]
:,~[I - vN~~Ys~g,,)l&,.f--1,..9, . Similarly,
the D, defined by (2.35) reduce to
We see, as previously noted, that D, is an o&function of kl , so that the integrals (over k,) in the relaxation times of (2.41) which involve D, will vanish, giving a,,3 = u,,, = 0. and a transverse magnetoresistivity tensor of the form (2.42). Typically, we find (3.9a) U’VY= Ne2/rmuc2rc-, where NjTc
= (4ni’h)
c
vg2
[(n
$- l)l,‘Zf,Z.n?l (6&
. g,) $- ~ll?fL*,n(-,G
, g?,Ill”
(nl,,h;g)
(3.9b)
‘.’ 2(1 ~ cos x”&.gI,) 6(t, - E(l?,I?:))
and similarly for the other components. From (3.9b) we observe the dependence of the galvanomagnetic tensor on oscillatory functions, namely, (1 - cos XZg2gI/) which may be called “zone” oscillation, S(E~- <(II, k,)) which gives rise to the Shubnikov-de Haas effect. and thef’s which introduce phasecoherenceoscillations as we shall seein Section 3.2. It is of interest to note that, by virtue of (2.23). h2g,g, = 2nN’lf
= 2&,/(2rhc/e)
= e@,/hc,
(3.9c)
where @,,is the flux of the magnetic field across the first Brillouin zone. Thus, the
166
ANIMALU
“zone” oscillations provide a basis for experimental quantization in a periodic lattice [23].
study of the nature of flux
3.2. The Linear Chain; Phase Coherence Oscillations The above results take a particularly simple analytic form for a coupled orbits and exhibit oscillatory effects due to phase coherence chain through the overlap integral defined by (3.6). To generate a k-space, we consider only the first pair of reciprocal lattice vectors the lattice potential has the simple one-dimensional form:
linear chain of in such a linear linear chain in in (3.2) so that
(3.10)
v = +3~gP(dew(iu) + S*(g) exp(--iuK
where g = (2n/a)(2/d3). By setting g, = 0 and g, = g in (3.5) to (3.8), we see that D, = 0 and C(-) = 0, while [for the first Fourier component in (3.10)] we find
C’+’ = 2~,%9 ewG~2kddKn+ l)1/2fn,n+1(0, s> - nl%n,n-l(O,811&,*.k, = 2ivAg) exp(ih2k,g)(h’g)f,,,(g, 0) &,*,k, , where, in the last step we have used the well-known integrals given by Rodriguez and Quinn [24]:
properties
of the overlap
fn*.n(O,d = (-Z-Y-” fn,,n(g, O), (n’ - n - 5>.kAg,
0) = F”[(n
(3.11)
(3.12)
+ 1)1/2fn~.n+l(g, 0) +
nl?fn,,n-l(g, O)],
E = (h’g)2 = hg2/2mw, evaluated for n’ = n. Explicitly,
.fnn(g,0) = j---m4~ + h2g)4Ld & = exp(- $3 L,(t) N (-1)”
(not sin 24)-lj2
sin[n(2$
-
(3.13)
sin 24) + r/4],
where cos2 q3 = 8/4n and the last expression in (3.13) is the asymptotic value for large n. Thus, in the linear chain, if r---f co in such a way that r/r2 = r/r,’ the transverse galvanomagnetic tensor is ~00
= Ne2r/m + Ne2/mw,2r,+;
ux?l = Ne2/mw,;
!Z 1, then
~?J?J= 0
(3.14)
HIGH-FIELD
MAGNETORESISTANCE
167
OF METALS
from (2.38) and (2.40) (apart from a cross term in uZZ), where N/7,-‘. = (4n/h) 8 V,y/?g2/2r?zW,)
1 1fn.Jg, (rkk,k,)
O)]” 6(E, -
E(Il, I?,)) (3.15)
N 16xV,” Here, &“(E~) is the free-electron density of states defined by (2.16b), and lzF CY E&J~ is the maximum value of IZ (typically, ~1~‘v 105H, H being in kilogauss in simple metals). We have used the fact that the two Fourier components in (3.10) contribute equally and independently to gZa. As we have pointed out earlier, (3.15) exhibits the Shubnikov-de Haas effect through N(E~). The oscillations due to phase coherence arise from its dependence on .fnF+ as follows. From the asymptotic formula (3.13) and the definition of nF , we find sin3[n,(2+,
lfn,,?+45, O>l” ‘v E
- sin 2+F) f
r/4]
F
(3.16) hw,
- sin”[h’A, cFn sin 24~ where co? +F = f/4??, and, following
+ n/4],
the analysis given by Reitz [25],
A, = (IZ~/~~)(~C#J~- sin 2~$~)
(3.17)
defines the area of the free-electron orbit in k-space outside the first Brillouin zone, A, being the notation used by Pippard (Fig. 3 of Ref. [lo]). The oscillatory behavior of (3.15) is now evident from (3.16). In order to bring out the connection of the oscillations with magnetic breakdown, let us introduce the breakdown field H,, as follows. We recognize that (3.16), apart from the oscillatory factor sin”[X2A, + r/4], is precisely the expression first derived by Pippard (Ref. [lo], Eq. (5)) for the magnitude of the Landau level broadening. Accordingly, the Pippard expression for the breakdown field is Ho = 4m~~;~/[(n from which it follows
sin 2~$~)l/~ h 1e 1 cF]
(3.18)
that
NIT,+ N smo, t 2:
! .N(cF)
sin2[h?A, + n/4].
This shows that l/rc+ cc w,, = eH,,/mc, so that TV+ may be interpreted relaxation time due to magnetic breakdown [9].
(3.19) as the
168
ANIMALU
Thus, the transverse
magnetoresistivity
P llv = (muc2r/Ne2) + E2
pyy = u,,/I oZy 12,is
8nw,(k2g2/2m) JIT(cF) sin2[h2A, + n/4].
(3.20)
This may be compared with equations (11.18) and (111.53)of Ref. 191,which were obtained by semiclassicalarguments. It is clear that the first term in this expression applies to the low magnetic field contribution to the magnetoresistivity from open orbits, while the second is the saturation (high field) value when magnetic breakdown of the open orbit occurs in the linear chain. Observe that the linear chain does not exhibit any “zone” oscillation. The calculation is readily generalized to a rectangular network, which may be thought of astwo linear chains at right angles: in this case there will be another contribution to (3.20) of the same form as the secondterm but having a different value of g and A, , i.e. a different amplitude and phase: this gives rise to beats in the oscillatory magnetoresistance, which is consistent with the data on tin [14]. 3.3. The Hexagonal Network; Comparism with Experiment The hexagonal network (Fig. 2) may be thought of as three linear chains inclined at an angle of 60” to each other. This implies that, by suitable choice of coordinate system, it may be possible to express the contributions from each of the three linear chains in the form of the second term in (3.20) with the sameamplitude but
2oti
Y
OO
’
I
I IO MAGNETIC
I
I
I
FIELD
STRENGTH
I 30
20 (in
I
I 40
KILOGAUSS)
FIG. 3. Experimental result showing phase coherence oscillations (after R. W. Stark, quoted as Fig. 26 in Ref. [9]) for a magnesium sample with low dislocation density at 1.1 “K.
HIGH-FIELD
MAGNETORESISTANCE
OF
METALS
169
different phases: this will give rise to phase coherence oscillations of different periods corresponding to the areas of the “needle” orbit (shown as b in Fig. I?), the “lens” orbit, and possibly the circle (shown as c in Fig. 2). On the basis of (3.9), one may also expect “zone” oscillations. These results are in accordance with the various experimental results obtained for hexagonal close-packed metals, like magnesium: the curve shown in Fig. 3 is a sample of the experimental results obtained by Stark (quoted in Ref. [9]). We have not attempted to fit our theory to the rather complicated oscillations, since we believe that very little will be learned from such an attempt beyond the qualitative understanding provided by the above analysis of the hexagonal network in terms of three linear chains. An exact calculation (in the framework of the present theory) is however possible. Since each Fourier component of the lattice potential contributes independently to the relaxation times (2.39) and (2.41). as indicated typically for oUU in (3.9), all that is required in practice is to sum expressions like (3.9b) over the six reciprocal lattice vectors g in (3.2). Since thef’s in (3.9b) are complicated, we have not taken htis step.
4.
SUMMARY
AND
CONCLUSIONS
The objective of this paper has been to extend the use of the Kubo formula for the conductivity to situations involving high magnetic tieids in a nearly free-electron metal, without resorting to semiclassical approximations. This procedure has enabled us to obtain the general form of the galvanomagnetic tensor (Section 2.3) and to exhibit special features, like the linearity of the magnetoresistancearising from certain cross terms in the matrix elements of the current, and oscillatory effects due to density of states (Shubnikov-de Haas effect), phase coherence in a network coupled by magnetic breakdown and “zone” oscillations. The main result of the paper is the explicit formula (3.20) for the transverse magnetoresistivity of a linear chain: this has provided additional insight into the sources of the various oscillatory effects mentioned above and is not completely equivalent to the usual results obtained by semiclassicalanalysis [9]. The result is only an approximation to the exact formula in (3.15) which includes phase coherence esactlv: this feature of the present method constitutes an improvement on the current situation [9]. To obtain absolute values of the magnetoresistance, the parameters 7, N and w,, may be treated as adjustable (cf. Refs. [l2, 131 to fit experiment. Finally, for most simple network models, we can obtain the galvanomagnetic tensor by decomposing the network into a number of linear chains and using symmetry considerations: this, as we saw, is due to the fact that each Fourier component of the lattice potential contributes independently to the galvanomagnetic tensor.
170
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