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Zero sound in anisotropic metals
ANNALS
OF PHYSICS:
60,
27-45 (1970)
Zero
Sound
in Anisotropic
Metals*
Y. C. CHENG AND N. D. MERMIN+ Laboratory
of Atomic
and Solid State Physics, Ithaca, New York 14850
Cornell
University,
Received March 18, 1970
The Gor’kov-Dzyaloshinskii theory of zero sound in anisotropic metals is generalized, and applied to specific metals. The main generalizations are: (a) sufficient conditions for the existence of zero sound are derived without the assumption of weak coupling: (b) examples are discussed in which these conditions are satisfied even when the propagation vector does not lie along or close to a symmetry axis. A survey of metallic Fermi surfaces indicates that zero sound may be possible in the noble metals along directions close to the [ill] direction, and in beryllium and graphite along directions close to the c axis. It may also be possible in the semimetal bismuth (and perhaps antimony and arsenic) along certain nonsymmetry directions. In all these cases sufficient conditions on the Landau F-function are derived for zero sound modes to exist at infrared frequencies (and, in some cases, at radio frequencies.) The conditions can be satisfied in the weak coupling limit, provided Fhas the appropriate sign. Unfortunately this is beyond reliable theoretical calculation, and the final question of whether the modes exist is best settled by experiment.
I. INTRODUCTION
In 1963 Gor’kov and Dzyaloshinskii [l] pointed out that Silin’s [2] analysis of zero sound in a charged isotropic Fermi liquid [3,4] required substantial modification in suitably anisotropic metals. In particular, they found that Silin’s conclusion, that the existence of zero sound required unrealistically large values of the Landau F-function, could be replaced by conditions that could be realized, for special directions of propagation in suitable materials, by an F-function of arbitrarily small magnitude, provided only its sign satisfied the proper condition. Their paper has since received surprisingly little attention, in part, no doubt, due to their conclusion that such modes would be extremely difficult to detect. We see no reason to disagree with their pessimistic conclusion, but nevertheless feel that it is appropriate to reformulate, make rigorous, and extend their analysis, for several reasons: * Work supported by the National Science Foundation + Alfred P. Sloan Foundation Fellow.
27
through Grant GP-9402.
28
CHENG AND MERMIN
(i) Starting with the paper of Platzman and Walsh [5] in 1967 there has been a considerable body of work on the detection of Fermi liquid modes and the theory underlying their interpretation [6-131. Most of this work has concentrated on measurements in the alkali metals, considered as isotropic electron Fermi liquids, and relies on the presence of a static magnetic field for the modes on which it concentrates. However, more recently [ 13, 14],l efforts have been made to extend the analysis, still in a magnetic field, to metals with anisotropic Fermi surfaces. Such investigations might go seriously astray if some of the spectacular consequences of anisotropy discovered by Gor’kov and Dzyaloshinskii in the absence of a magnetic field are not properly taken into account. In particular, the features responsible for allowing the propagation of zero sound in weakly interacting Fermi liquids, in the absence of a magnetic field, can materially alter the conditions for propagation of the modes in a magnetic field, provided similar anisotropies are present. The detection of the field dependent modes is within the range of present experimental technique, so the extension of Gor’kov and Dzyaloshinskii theory to the presence of a magnetic field, though difficult, is not without practical interest. (ii) The analysis of Gor’kov and Dzyaloshinskii applies only in the weak coupling limit. Although this is probably the case of practical interest, we have been able to extend their analysis to the case of a general F-function of arbitrary strength, deriving sufficient conditions for the existence of zero sound, that reduce to those of Gor’kov and Dzyaloshinskii in the weak coupling limit. (iii) Knowledge of metallic Fermi surfaces has been considerably extended since the paper of Gor’kov and Dzyaloshinskii. We can now discuss the Gor’kov and Dzyaloshinskii analysis in the context of particular metals. (iv) We point out the possibility that zero sound may propagate in certain semimetals along special directions which are not, however, directions of symmetry, as in the analysis of Gor’kov and Dzyaloshinskii. In this paper we take up points (ii), (iii), and (iv). Since the analysis is substantially more difficult in a magnetic field, we shall defer discussion of that case to a subsequent paper, in which it is investigated in some simplified models. The basic point underlying the analysis of Gor’kov and Dzyaloshinskii as well as our own is quite simple: zero sound with wave vector q in a charged Fermi liquid with small F-function is impossible if q . v(k) is maximum at only a single k on the Fermi surface. (Here v(k) is the electronic velocity.) This is, of course, always the case for an isotropic Fermi liquid, where q . v(k) is maximum at the single value of k parallel to q. The reason for this limitation is that the zero sound oscillations 1 This paper [14] is concerned with spin oscillations, but the general consequences of anisotropy we wish to emphasize are equally relevant in that case.
29
ZERO SOUND
must have negligible charge and current oscillations or they will be completely dominated by macroscopic electromagnetic interactions, being negligibly influenced by many body interactions. Now if the charge and current densities in a mode are negligible, then the kinetic equation (see next section) will have the form: (w - q . v) v(k) = q . v J’ -$$+k,
k’) I&‘).
When f is very small then v can be large only when OJis very close to q * vmax . If there is only a single maximum, then Y will be appreciable only at a single point of k-space and there will necessarily be a charge density associated with the mode. If, however, q . v has more than one maximum, it is possible to find solutions with zero charge (and, if necessary, current) density. In Section II, we formulate a variational principle that makes these crude considerations rigorous for F-functions of arbitrary strength. From this we can derive sufficient conditions for the existence of zero sound by a suitable choice of trial functions. We consider only cases with q . v(k) maximum for more than one k on the Fermi surface, since f is almost certainly not so large that zero sound can exist with a single maximum. Multiple maxima can occur in some metals when q is along a symmetry direction (as pointed out by Gor’kov and Dzyaloshinskii) but also in certain others when q is along special directions which are not directions of symmetry. We will discuss these two cases in Section III and Section IV respectively.
II. THE VARIATIONAL PRINCIPLE
In this section we generalize to the charged anisotropic case a variational argument 1151 used to discuss the existence of zero sound in neutral isotropic Fermi liquids (He3). We start with the kinetic equation [2, 61:
(As is customary in investigating the existence of modes, we make the (generally unrealistic) assumption that the relaxation time can be made large enough to satisfy the condition WT > 1.) Here w and 4 are the angular frequency and wave vector of the zero sound mode, v is related to the deviation from equilibruim 6n of the quasi-particle distribution function by an
=
_
($)
yei(a+-d
30
CHENG AND MERMIN
and the angular brackets represent an average over the Fermi surface:2 dS deF)
=
1
4p3u
(4)
*
The quantity V is defined by
8k) = (1 + F>40 = 44 + 1 $$+k = v(k) + -
1
dEF)
s
k’) 40 $$$
t’(k, k’) v@‘),
(5)
where F(k, k’) = g(+) f(k, k’) and f(k, k’) is the spin-independent part of the Landau Fermi liquid interaction function. Finally, k,2 = 4rrezg(+) and 11and 1 refer to the direction of q. Equation (2) is derived in the form used here for isotropic charged Fermi liquids in [6]; the procedure in the anisotropic case is exactly the same. For zero sound to exist it is necessary that (2) have a solution for a real value of w that exceeds the largest value assumed by q * v(k) as k ranges over the Fermi surface: ” > 4uYBx@).
(6)
If this condition is not met, the mode will merge with the continuum and undergo catastrophic Landau damping. In addition, we are only interested in modes with frequencies well below the plasma frequency. This amounts to the condition:
k&‘F3 w,
(7)
where up is a typical Fermi velocity and k,vF is on the order of the plasma frequency. In deriving the variational principle we must also exploit the stability condition?
> 0,
(8)
for any disturbance v(k) whatsoever, provided only that
(9)
(v) = 0.
(10)
and Conditions
(9) and (10) mean that the disturbance v(k) creates no net charge and
%We have set fi = 1. AIL integrals are over the Fermi surface.
SSee, for example, 1161 for the isotropic case. It can be extended to the anisotropic case.
ZERO
31
SOUND
current densities and hence no macroscopic electromagnetic energy. For such disturbances the stability condition (8) is the same as in the neutral Fermi liquid. It is convenient to express the stability condition in terms of 17= (1 + F)v. Equations (81, (9), and (lb) become (G*(l - B)C) > 0
(11)
(VV) = 0
(14
((1 - B,)lQ = 0.
(13)
for any V, provided only that
and
Here B(k, k’) is defined in operator notation as
B=
1F
’
(14)
i.e.,
B(k k’) = Fe, k’) - -&
1 4$;;!t) B(k, k”) FW, k’),
(15)
B(k, k’).
(16)
and 1 BoW = g(EF) .r dS(k’) 4dv(k’)
Equation (2) is to be solved in the space of all functions #(k) with k varying over the Fermi surface. The scalar product of 4(k) and 4(k) is defined as (17)
The condition that the frequency of zero sound be much smaller than the plasma frequency requires the solution of (2) to have negligible projection on u,, , for if the solution has an appreciable projection on z1,,, the term in v I will dominate all the other terms in the equation when k,,vr > W. As discussed in [6] and [8], we can drop the term containing vII in (2) provided we project the resulting equation out of the vfl subspace: k,b P wPS - qv,,PC - wBPC -j- v . &PC>] [ q2c2 L
= 0,
32
CHENG
AND
MERMIN
where P=
1 - 16,,)(6,, 1;
6, =-,
v II (v2IIy2
(19)
Here we have also anticipated that w/q will be of order VFfor the modes of interest, and therefore have neglected w2 in comparison with q2c2. Note that it follows from Eq. (18) that since PG has no component in the v,, subspace, it also has no component in the (1 - B,) subspace, i.e., we have ((1 - B,,) PF) = 0. (This is just the assertion that zero longitudinal current implies zero charge density.) We can therefore rewrite (18) as V=
[Q (+)
Q + QBQ - -$$
Q I v,) *
(20)
where Q is the projection operator which projects out both the v, and (1 - B,) subspaces. This is convenient to do, since the explicit appearance of the projection operator Q insures that two of the restrictions (12) and (13) appearing in the stability condition (11) will be automatically satisfied. For simplicity, we only consider crystals with inversion symmetry, so that (v,,(l - B,)) = 0, and therefore 1 - B,, Q = 1 - I %X6,, I - I i&b I; B= ((1 - Bo)2)1’2 * (21) In Eq. (20), we consider w as a fixed parameter and q, a variable. The condition for zero sound is that H, have eigenvalue 1 for some real q between 0 and o/v:“. (We take w to be positive. The range of q between 0 and -cu/v~~ need not be considered, since the spectrum of (2) is symmetric about 0 for a crystal with inversion symmetry.) Since H, is Hermitian for real w and q, and since B is bounded, the maximum eigenvalue Aa of H, is a continuous function of q. Now it follows from the stability conditions (1 l), (12), and (13) that all the eigenvalues of Hg are less than 1 when q = 0: for functions with no component in the v, subspace, H,,=, = QBQ and all the eigenvalues of QBQ are less than 1 in this subspace because of (1 l), (12), and (13); for functions with components in the vI subspace, Hp.+, is dominated by the term
- +$ QI v,>*(v, I Q, the eigenvalues of which approach - co as q -+ 0. Therefore, A,,, < 1. If for some q. between 0 and o/v? we can show that h, > 1,
(22)
ZERO
33
SOUND
then by continuity Xql will equal 1 for some q1 between q. and 0, and zero sound will exist with wave vector q1 . To derive a sufficient condition for the existence of zero sound, we can therefore use the variational principle that tells us that b > (#*H&j
= (#*
[Q (+j
Q + QBQ - $$
for any trial function #(k) satisfying the normalization
Q I VA . (v, I Q] #),
(23)
condition
cldr*+> = 1.
(24)
Our task is to find a suitable trial function +(k) leading to simple conditions on B which make \<+*H,#) > 1 for some real q between 0 and w/try.
III.
APPLICATION TO q ALONG A SYMMETRY DIRECTION
Gor’kov and Dzyaloshinskii emphasized that q . v(k) will have more than one maximum when q is along a symmetry direction and u,(k) is maximum at a k not parallel to q. In general one must know both the shape of the Fermi surface and the magnitude of v(k) over the surface, in order to determine whether this condition can be satisfied in a given metal. However there is an important case in which the condition is guaranteed to hold regardless of the magnitude of v(k): Suppose we display each branch of the Fermi surface in a primitive cell with the full point group symmetry of the lattice (in practice we shall always use the first Brillouin zone), and suppose that when the surface is so displayed, there is no Fermi surface in any of the branches in the direction of q from the origin. Then regardless of the values of u(k) along the Fermi surface, the maximum of q . v(k) cannot occur for k parallel to q for the trivial reason that there are no occupied Fermi surface states in that direction. Thus q . v(k) must be maximum for k not along q. If q happens to be an axis of symmetry then other values of k with the same value of q . v(k) can be produced by symmetry operations. Because the cells in which the branches have been displayed are primitive, these other values will correspond to physically distinct states, and the Gor’kov-Dzyaloshinskii condition will be satisfied. Among the currently known Fermi surfaces, there is no surface along the [l 1l] direction of the noble metals [17], [Fig. l(a)], and along the direction of the c axis of beryllium [18] and graphite [19], [Figs. l(b) and l(c)]. In all of these cases, when q is along the required symmetry direction, the symmetry group of the kinetic equation is CsV. Since these are the only simple cases we have discovered, we will discuss only this group in detail. The generalization of the following sgs/6dI-3
34
CHENG
AND
IvERMIN
(b)
lectrons
FIG. 1. Fermi surfaces and the direction of q which makes q * v have multiple maxima. (a) Copper, silver, or gold. (b) Beryllium (after Loucks and Cutler [IS]). When reduced to the tit zone, the “coronet” must be sliced in half by a horizontal plane and placed at the top and bottom of the zone. The (two) “cigars” are sliced vertically into six pieces that go to the corners of the zone. There is therefore, as required, no Fermi surface along the c axis. (Most hexagonal metals fail to meet this criterion, because the “lens” is intersected along the c axis.) (c) Graphite [after Williamson and Foner [19]).
discussion clear. When q depending or not. In
to other symmetry groups, should it turn out to be necessary, should be is along the symmetry axis of the group C,, , there are two possibilities on whether vy occurs on the vertical reflection symmetry planes general, there will be six k on the Fermi surface where vu is maximum
ZERO
35
SOUND
[Fig. 2(a)], but if o,, is maximum for k on the vertical reflection symmetry planes, we will have only three k [Fig. 2(b)]. We derive the conditions for zero sound for the case of six maxima, from which we can easily deduce the conditions for the case with three.
FIG. 2. The relative positions of k”‘s on the Fermi surface which have the maximum value of 4. v(k) in the symmetry group C,, . The direction of q (taken as the z axis) is pointing out of the page and the dotted lines represent the vertical reflection symmetry planes. (a) When k”s are not on the vertical reflection symmetry planes. (b) When k”‘s are on the vertical reflection symmetry planes.
Suppose the maxima of u, occur at k,O ‘0. k,s on the Fermi surface. Their relative positions are shown in Fig. 2(a). We pick as our trial function Ci
$04 =
1
0
-
for
k E AS,
(P,,lW)
i = 1, 2,..., 6; (25)
otherwise,
where the AS, are small regions on the Fermi surface around the ki” maintaining the C,, symmetry. The Ci are constants to be chosen to make the condition for the existence of zero sound as weak as possible. Unless the coupling is quite strong, the best choice turns out to require that trial functions of the form (25) have no net charge and longitudinal current densities. Since both (1 - B,) and vu belong to the identity representation of the group C,, , any function belonging to the other irreducible representations will automatically be orthogonal to (1 - Bo) and t‘, , and hence have no charge and longitudinal current densities. We therefore can choose the trial function according to the irreducible representations of the symmetry group, and for each representation we have a condition for the existence of zero sound. In addition to the identity representation 1201 (AI), C,, has one one-dimensional (A,) and one two-dimensional (E) irreducibIe representation:
36
CHENG
AND
MERMIN
1. Representation A, : Our trial function will transform under this representation, if we choose the constants Ci in (25) as follows: Ci = fC
for
i = 1,3, 5,
c,=-c
for
i = 2,4, 6,
and
(26)
where C is a normalization constant such that (#*#> = 1. Since the trial function, (25) and (26), has been chosen to satisfy the condition (#*6, ) = (zj*p) = 0, we have Qt,4 = #. The condition (23) becomes
1 = ’ + 6( l/U - (~v,,/~)]~>As
1
1 ’ B(ki ’ k5) 1 - (qv u&)/a.~) 1
-6 1 -
(qv&)
)I AS
* 1 - (qv,,(kJ/o) I
’
where ( )As is an integration over one of the dS only. Note that the term in v, vanishes due to the symmetry of the trial function. At this point we assert that if B,, +
242
-
42
- Bu - B,, c=-0
(28)
zero sound exists. Here Bi, = B&O, ki”) = B(kjo, k$‘).
If (28) holds, we can always choose dS so small that MW%
,4) + 2B&,
k$ - Bk, k2) - WI9 kJ - WI, bJ1 > b. > 0
for all ki E O& . Then, the inequality
(29)
(27) can be rewritten as
&>I+( l:F)Aj((l 19:,).)Asp(l ‘qz,l),,-I].t30) The coefficient of the bracket exceeds zero for every q < w/vy, while the coefficient of b, within the bracket becomes arbitrarily large as q approaches o/eaK from below. This establishes (22) for some q. < w/v, and hence (28) is
ZERO
37
SOUND
a sufficient condition for the existence of zero sound in the representation A, . Note that the condition (28) does not depend on the strength of the coupling, but only the sign. 2. Representation E: This is a two-dimensional representation. When we choose a trial function #,(k), there will correspond another trial function 4,(k): $J, and 4, are transformed into each other like x and y under the operations of C,, . We choose the trial functions Z& and &, in the form (25) and denote their constants by Cix and Ciu, respectively. We have:
C,” = C,” = c,
c,y Z --c,p = ac >
C,” = csx = (- l/2 + %&/2)C,
c,y = -coy = (-a/2
C,” = c5x = (- l/2 -
C,y = -C,y
Here C is a normalization
1/301/2)C,
= (0112-
-
%6/2)C,
(3 1)
43/2)C.
constant such that
<~,**a!> = <*Y**Y) = 1 and Q:is a constant to be chosen to make the condition possible. We again have
of zero sound as weak as
and
and therefore Q& = Either trial function Following exactly the that the condition for
& and Q/J, = &, . #Z or #y must lead to the same condition for zero sound. same analysis as we used for the representation A, , we find the representation E is:
Here we have taken the x axis on the vertical reflection symmetry plane between k,O and kZo, and u,(l) and u,(l) are the x and y component of the velocity at k,O; x is to be chosen to maximize the left side of (32). Equation (32) is of the form am2 + ba + c > 0, which can be satisfied for some real a, if and only if a + c + / l/b2 + (a - cy 1 > 0.
38
CHENG
AND
MERMIN
This leads to the condition, WI,
+ u,2(1)) + 1[d/5 (BIB - B14) + 12 k,2V$~2a”(1)]2 - 4s) - 6 ko2(v0V)q2c2 + [(2B,, - B,, - BIB) - 6 ko2(vg2(;2;
vw2@))]‘/“’
> 0.
(33)
Note that the left side of (33) is an increasing function of q2. Therefore (33) will hold independent of q if it holds in the q + 0 limit. In this limit the condition becomes
(34) Even if (34) fails we will have zero sound for some value of q2, if (33) is satisfied in the large q limit. To leading order in qc/kovF , (33) becomes WI,
- 43
+ [3(B1e - Bd2 +
which is the weakest form our condition
(242
- 44 - BIS)~I~‘~> 0,
(35)
can assume.
Gor’kov and Dzyaloshinskii distinguish two regimes: the radio frequency region, qc < kovp and the infrared, qc > kOuF. In terms of this distinction we can say that zero sound will exist in both regimes if (34) holds, and only in the infrared, if (35) but not (34) holds. In the latter case the value of qc/kovF , above which we are assured of zero sound, is given by setting the left side of (33) to zero. Now we turn to the case when VT= occurs for k on the vertical reflection symmetry planes. In that case, k,O and k20 will merge as one point klo on the dotted line in Fig. 2(a), as will k,O and kao, and k,O and kao, Fig. 2(b). The trial function (25) in the representation A2 will then be zero, and our condition for zero sound cannot be satisfied by that representation. In the representation E, we must pick a trial function like (31) with (Y= 0 and therefore C, = C, , C, = C, , and C, = C, as required. The condition for the existence of zero sound can then be obtained from (32) by setting a = 0:’ B,, - B13 - 3 k”;;yl)
> 0.
* Note that in both cases (Aa and E) these results also follow directly from the forms taken by conditions (28) and (33) in the case B12 = B,, , I?,, = B14 = B,, and v,,(l) = 0.
39
ZERO SOUND
This is again most easily satisfied in the infrared regime, where it becomes simply:
B,, - B,, > 0.
(37)
Comparing (37) with (35) we see that the case of six maxima is more favorable to the existence of zero sound than is the case of three. Unfortunately the actual sign of the left sides of (28), (39, and (37) cannot be reliably estimated, and we can do no more than repeat Gor’kov and Dzyaloshinskii’s oracular comment that the likelihood of their being positive “is, roughly speaking, 50 per cent.”
IV. APPLICATION
TO q NOT ALONG A SYMMETRY
DIRECTION
For an arbitrary direction of q, there is only one maximum of q . v(k). But if the Fermi surface consists of several electron (or hole) pockets, it may be possible for q . v to have two maxima for certain nonsymmetry directions of q. We consider here by way of illustration a Fermi surface consisting of ellipsoidal pockets. Consider an ellipsoidal constant energy surface: f(c) = +(k - k,) a-l(k
- k,),
(38)
where a-l is the constant inverse mass tensor. In the simplest cases one might have f(c) = (E - constant) for particles and (constant - E) for holes. In some cases such as the electron ellipsoids in bismuth [21,22], more general forms have been used. From (38), we have for k on the Fermi surface, v(k) = ti-l(k
- k,)/f’(+).
(39)
It is easily shown that q * v(k) will be maximum at constant E = + , when k - k,, is parallel to q. The value of q . vmax(k) is then the value at the maximum of q . fi-l
. (k - k,,)
f’(+)
29 .f(EF) . = I k - k, I f’(c~)
(40)
Now suppose we have two ellipsoids: fl(c) = g(k - k$‘) l6l:;‘(k - kt’), fi(c) = $(k - kf’) tii,‘(k
- kt’).
(41)
40
CHENG
AND
MERMIN
The condition that q . v(k) has the same maximum k(l) and kc21 requires: I kt2)
-
kb2’
I
on ellipsoids 1 and 2 at points
fi(+)fZ’(4
1k(l) - kb” 1 ’ h’(+)&(q) Since
k(2)
-
kh2) and
k(l)
k(2)
The condition
J$) _
= ‘*
(42)
are both parallel to q, we have the two possibilities kt’
=
f
fiIfi m
(k”’ - k:‘).
that k(l) lies on ellipsoid 1 requires fi(+)
= i&(l)
- kt)) ti;‘(k(‘)
If we substitute (43) into the corresponding restriction on k(l): f2(+) = ; (#)l
- k;‘).
(44 equation for kc2), we find a second
(k(l) - k$‘) a;l(@)
_ ky),
(45)
or
fi(+) = ; (k”’ - k;)) ($)” $ fi,‘(@’ - kt’), Solutions will therefore exist if, when ellipsoid 2 is scaled in its linear dimensions by 151.
(-L,“”
(47)
and translated to the same origin as ellipsoid 1, the two intersect. Any vector q pointing from the common center to a point on the curve of intersection will then give equal values of (q . v)max on the two ellipsoids. This analysis can be applied to the Fermi surface of bismuth [21,22], which has three equivalent electron ellipsoids and a single hole ellipsoid. From the data given in [21], we find that each of the three electron ellipsoids has intersections with the hole ellipsoid, when it is scaled by (47), and they are translated to the same origin. Therefore for certain directions of q, q . v(k) can have the same maximum value at the hole ellipsoid and one of the electron ellipsoids. However, two of the components of the mass tensor of the electron ellipsoid are much smaller than the third component and those of the hole ellipsoid, and u(k) will be smallest when (k - k,) is parallel to the third axis. Due to the relative orientations of the three electron ellipsoids and the small components of the mass tensor, we find that when q is in the directions which make the maximum of q * v on the hole ellipsoid and one of the electron ellipsoids equal, there will be a larger maximum of q . v on one of the other two electron ellipsoids (see Fig. 3). Thus for such a direction of q we have only one q * vmaXon the whole Fermi surface.
ZERO
41
SOUND
FIG. 3. The intersections of one of the electron ellipsoids and the hole ellipsoid when it is scaled by (47) in bismuth. The central ellipsoid is the hole ellipsoid which is an ellipsoid of revolution aboutthez axis(perpendicular to the page); the threeelectronellipsoids canbetransformed into each other by 120” rotations about the z axis. The dotted circle represents the hole ellipsoid which is translated to the origin of electron ellipsoid 1. A and B are two points on the curve of intersection, When q is parallel to OA(II O’A’), q. vmnx will occur on electron ellipsoid 3, and when q 11OB on electron ellipsoid 2.
However there is another possible way to give q . v two maxima: q * v can have equal maxima on two of the electron ellipsoids, which are larger than its maxima on the third and on the hole ellipsoid. The relative orientations of the three electron ellipsoids are shown in Fig. 4. The z axis points out of the page and the three ellipsoids can be transformed into each other by 120” rotations about the z axis. The ellipsoid has a long axis and short axes. The long axis tilts out of the page at an angle of 6”. Since the angle of tilt is very small we shall neglect it for the
C
3 .
>
FIG. 4. The intersections of two of the electron ellipsoids in bismuth. The dotted ellipsoid represents electron ellipsoid 2 which is translated to the origin of electron ellipsoid 1. The curves of intersection will be on two perpendicular planes OA and OB.
42
CHENG
AND
MERMIN
moment. When Q - k,,) is along the short axes, the velocity u(k) will be large. Now we translate ellipsoid 2 to the same origin 0 as ellipsoid 1, as shown in Fig. 4. Since these two ellipsoids are equivalent there is no need to scale. The intersections will be on two perpendicular planes OA and OB. When q is on the plane OB, the maximum of q . v for the whole Fermi surface will be on ellipsoid 3. However when q is on the plane OA, q will be more or less along the direction of the long axis of ellipsoid 3 and therefore the maxima of q * v on ellipsoids 1 and 2 will be larger than the maximum of q . v on ellipsoid 3. Finally, the mass of the hole ellipsoid turns out to be much larger than the mass components of the short axes of the electron ellipsoid, which permits us to conclude that q * v is not maximum on the hole ellipsoid. Therefore when q is on the plane OA, q * vmax for the whole Fermi surface will be on ellipsoids 1 and 2. We have discussed the above with nontilted ellipsoids, in which case the two perpendicular planes are planes of reflection symmetry for the structure consisting of ellipsoid 1 and the translation of ellipsoid 2. However the plane OA remains a plane of reflection symmetry when the long axis is tilted. Therefore we can conclude in general that when q lies within a plane that contains the z axis and is parallel to the long axis of one of the electron ellipsoids, then q * v will assume its maximum value at two points, lying in each of the other two ellipsoids. Note that in general q will satisfy this condition even though it may be far from any symmetry direction. The Fermi surfaces of arsenic [23] and antimony [24] both consist of several equivalent electron pockets which can be approximated as ellipsoids and several nonellipsoidal hole pockets. Because of the complex structure of the hole pockets, we can draw no definite conclusion. If, however, the mass tensor for the holes is sufficiently large, the conclusion we reached for bismuth will also hold for arsenic and antimony. Finally we return to the variational principle to extract a condition for the existence of zero sound when q * v has two maxima. If there is no symmetry at all, the general analysis we used in Section III is very complicated. We therefore specialize our discussion to the case of bismuth. Since the dimensions of the electron ellipsoids and the hole ellipsoid are very small compared with the size of the zone [22], we make the approximation that B(k, k’) depends only on to which ellipsoids k and k’ belong, but not the detailed positions of k and k’ on the ellipsoids. Because of the symmetry between the electron ellipsoids, we then have only four independent values of B(k, k’): if k and k’ are on the same electron ellipsoid, if k and k’ are on different electron ellipsoids, if one of k and k’ is on the electron ellipsoid, and the other on the hole ellipsoid, if both k and k’ are on the hole ellipsoid.
(48)
ZERO
43
SOUND
Suppose the maxima of q . v are at k,O and k,O on the Fermi surface. We choose the trial function to be for
k E AS,,
for
k E AS,,
(49)
otherwise, where AS, is a small region around kio on the Fermi surface and C is a normalization constant determined by (z,/J*$) = 1; 01 is a constant to be chosen to make the condition for zero sound as weak as possible. The constant energy surface of the electron ellipsoid is in the form of (38) with [21,22] f(~) = ~(1 + E/Q) w h ere eg is the small energy gap between the valence and conduction bands. Using the form of the electron ellipsoid (38) and trial function (49), we can calculate the integral over the Fermi surface with logarithmatic accuracy: (v,,~)) = C(1 + a) W h In (1 - qj,
max
= CtB, + 4%
h ln (1 -
+j,
where B is defined in (21) and (16), and pi = ,&kio), and5
Using the approximate form (48) of B(k, k’) it is easily shown that fil = flZ . Therefore we can choose 01 = -1 to make the trial function #(k) have no charge and longitudinal current densities, as we did in the symmetric case. Condition (23) therefore becomes (52) ’ Here M, , M., , and M, are the components of the mass tensor referring to the principal axes of the ellipsoid.
44
CHENG
AND
MERMIN
By the same reasoning that led from (27) to (28), we may conclude that if (53)
zero sound will exist in bismuth for directions of q which have two q * v,, on the Fermi surface. Here we have used the fact that when q satisfies the above condition v,(l) = -v,(2), because the plane OA in Fig. 4 is a plane of reflection symmetry. From the data given in [21], it is easy to calculate ~~(1) when q lies on the plane OA. It is in the range from 1 to 6 x lo7 cm set-l. Since k, is about 1.0 x 10’ cm-l, kg,(l) is on the order of lOi set-l. Zero sound is then possible for small values of B if q > lo* cm-l. We also require q < +/fit+ , which is in the order of lo6 cm-l for bismuth. This restriction is due to the fact that Landau’s theory is only applicable to inhomogeneities with “macroscopic” wave vector. Since B& and Bz2 are probably of order unity or less, zero sound can occur in bismuth only in the infrared. Whether it exists will depend on the sign of B& - B,e , which unfortunately is beyond reliable theoretical estimation.
ACKNOWLEDGMENT We have Dzyaloshinskii
had
some theory.
valuable
discussions
with
J. S. Clarke
on the structure
of the Gor’kov-
REFERENCES 1. L. P. GOR’KOV AND I. E. DZYALOSHINSKJI, Zh. Eksp. Teor. Fiz. 44 (1963), 1650; English transl.: Sov. Phys. JETP 17 (1963), 1111. 2. V. P. SILIN, Zh. Eksp. Teor. Fiz. 35 (1958), 1243; English transl.: Sov. Phys. JETPS (1959), 870. 3. L. D. LANDAU, Zh. Eksp. Teor. Fiz. 30 (1956), 1058; English transl.: Sov. Phys. JETP3 (1956), 920. 4. V. P. SILIN, Zh. Eksp. Teor. Fiz. 33 (1957), 495; English transl.: Sov. Phys. JETP 6 (1958), 387. 5. P. M. PLATZMAN AND W. M. WALSH, JR., Phys. Rev. Left. 19 (1967), 514; 20 (1968), 89. 6. N. D. MERMIN AND Y. C. CHENG, Phys. Rev. Lett. 20 (1968), 839. 7. S. C. YING ANO J. J. QUINN, Phys. Rev. Lett. 20 (1968), 1007. 8. Y. C. CHENG, J. S. CLARKE, AND N. D. MERMIN, Phys. Rev. Lett. 20 (1968), 1486. 9. P. M. PLATZMAN, W. W. WALSH, JR., AND E-NI Foo, Phys. Rev. 172 (1968), 689. 10. S. C. YING AND J. J. Qunq Phys. Rev. 173 (1968), 473. 11. B. J. MCINTYRE, S. C. YING, AND J. J. QUINN, Phys. Rev. Lett. 21 (1968), 1244. 12. G. A. BARAFF, C. C. GRIMES, AND P. M. PLATZMAN, Phys. Rev. Lett. 22 (1969), 590. 13. S. C. YING ANLI J. J. QUINN, Phys. Rev. 180 (1969), 193. 14. S. C. YING AND 3. J. QUINN, Phys. L&t. A 26 (1968), 347. 15. N. D. MERMIN, Phys. Rev. 159 (1967), 161. 16. P. Nozmars, “Theory of Interacting Fermi Systems,” p. 16. Benjamin, New York, 1964.
ZERO
SOUND
45
17. D. SHOENBERG, Phil. Trans. Roy. Sot. London 255 (1962), 85. 18. T. L. LOUCKS AND P. H. CUTLER, Phys. Rev. A 133 (1964), 819. 19. S. J. WILLLAMSON AND S. FONER, Phys. Rev. A 140 (1965), 1429. 20. M. HAMERMESH, “Group Theory,” p. 126. Addison-Wesley, Reading, Mass., 1964. 21. G. E. SMITH, G. A. BARAFF, AND J. M. ROWELL, Phys. Rev. A 135 (1964), 1118. 22. D. SHOENBERG, “The Physics of Metals, 1,” (J. M. Ziman, Ed.), p. 109. Cambridge Univ. Press, 1969. 23. P. J. LIN AND L. M. FALICOV, Phys. Rev. 142 (1966), 441. 24. L. M. FALICOV AND P. J. LIN, Phys. Rev. 141 (1966), 562.
OF PHYSICS:
60,
27-45 (1970)
Zero
Sound
in Anisotropic
Metals*
Y. C. CHENG AND N. D. MERMIN+ Laboratory
of Atomic
and Solid State Physics, Ithaca, New York 14850
Cornell
University,
Received March 18, 1970
The Gor’kov-Dzyaloshinskii theory of zero sound in anisotropic metals is generalized, and applied to specific metals. The main generalizations are: (a) sufficient conditions for the existence of zero sound are derived without the assumption of weak coupling: (b) examples are discussed in which these conditions are satisfied even when the propagation vector does not lie along or close to a symmetry axis. A survey of metallic Fermi surfaces indicates that zero sound may be possible in the noble metals along directions close to the [ill] direction, and in beryllium and graphite along directions close to the c axis. It may also be possible in the semimetal bismuth (and perhaps antimony and arsenic) along certain nonsymmetry directions. In all these cases sufficient conditions on the Landau F-function are derived for zero sound modes to exist at infrared frequencies (and, in some cases, at radio frequencies.) The conditions can be satisfied in the weak coupling limit, provided Fhas the appropriate sign. Unfortunately this is beyond reliable theoretical calculation, and the final question of whether the modes exist is best settled by experiment.
I. INTRODUCTION
In 1963 Gor’kov and Dzyaloshinskii [l] pointed out that Silin’s [2] analysis of zero sound in a charged isotropic Fermi liquid [3,4] required substantial modification in suitably anisotropic metals. In particular, they found that Silin’s conclusion, that the existence of zero sound required unrealistically large values of the Landau F-function, could be replaced by conditions that could be realized, for special directions of propagation in suitable materials, by an F-function of arbitrarily small magnitude, provided only its sign satisfied the proper condition. Their paper has since received surprisingly little attention, in part, no doubt, due to their conclusion that such modes would be extremely difficult to detect. We see no reason to disagree with their pessimistic conclusion, but nevertheless feel that it is appropriate to reformulate, make rigorous, and extend their analysis, for several reasons: * Work supported by the National Science Foundation + Alfred P. Sloan Foundation Fellow.
27
through Grant GP-9402.
28
CHENG AND MERMIN
(i) Starting with the paper of Platzman and Walsh [5] in 1967 there has been a considerable body of work on the detection of Fermi liquid modes and the theory underlying their interpretation [6-131. Most of this work has concentrated on measurements in the alkali metals, considered as isotropic electron Fermi liquids, and relies on the presence of a static magnetic field for the modes on which it concentrates. However, more recently [ 13, 14],l efforts have been made to extend the analysis, still in a magnetic field, to metals with anisotropic Fermi surfaces. Such investigations might go seriously astray if some of the spectacular consequences of anisotropy discovered by Gor’kov and Dzyaloshinskii in the absence of a magnetic field are not properly taken into account. In particular, the features responsible for allowing the propagation of zero sound in weakly interacting Fermi liquids, in the absence of a magnetic field, can materially alter the conditions for propagation of the modes in a magnetic field, provided similar anisotropies are present. The detection of the field dependent modes is within the range of present experimental technique, so the extension of Gor’kov and Dzyaloshinskii theory to the presence of a magnetic field, though difficult, is not without practical interest. (ii) The analysis of Gor’kov and Dzyaloshinskii applies only in the weak coupling limit. Although this is probably the case of practical interest, we have been able to extend their analysis to the case of a general F-function of arbitrary strength, deriving sufficient conditions for the existence of zero sound, that reduce to those of Gor’kov and Dzyaloshinskii in the weak coupling limit. (iii) Knowledge of metallic Fermi surfaces has been considerably extended since the paper of Gor’kov and Dzyaloshinskii. We can now discuss the Gor’kov and Dzyaloshinskii analysis in the context of particular metals. (iv) We point out the possibility that zero sound may propagate in certain semimetals along special directions which are not, however, directions of symmetry, as in the analysis of Gor’kov and Dzyaloshinskii. In this paper we take up points (ii), (iii), and (iv). Since the analysis is substantially more difficult in a magnetic field, we shall defer discussion of that case to a subsequent paper, in which it is investigated in some simplified models. The basic point underlying the analysis of Gor’kov and Dzyaloshinskii as well as our own is quite simple: zero sound with wave vector q in a charged Fermi liquid with small F-function is impossible if q . v(k) is maximum at only a single k on the Fermi surface. (Here v(k) is the electronic velocity.) This is, of course, always the case for an isotropic Fermi liquid, where q . v(k) is maximum at the single value of k parallel to q. The reason for this limitation is that the zero sound oscillations 1 This paper [14] is concerned with spin oscillations, but the general consequences of anisotropy we wish to emphasize are equally relevant in that case.
29
ZERO SOUND
must have negligible charge and current oscillations or they will be completely dominated by macroscopic electromagnetic interactions, being negligibly influenced by many body interactions. Now if the charge and current densities in a mode are negligible, then the kinetic equation (see next section) will have the form: (w - q . v) v(k) = q . v J’ -$$+k,
k’) I&‘).
When f is very small then v can be large only when OJis very close to q * vmax . If there is only a single maximum, then Y will be appreciable only at a single point of k-space and there will necessarily be a charge density associated with the mode. If, however, q . v has more than one maximum, it is possible to find solutions with zero charge (and, if necessary, current) density. In Section II, we formulate a variational principle that makes these crude considerations rigorous for F-functions of arbitrary strength. From this we can derive sufficient conditions for the existence of zero sound by a suitable choice of trial functions. We consider only cases with q . v(k) maximum for more than one k on the Fermi surface, since f is almost certainly not so large that zero sound can exist with a single maximum. Multiple maxima can occur in some metals when q is along a symmetry direction (as pointed out by Gor’kov and Dzyaloshinskii) but also in certain others when q is along special directions which are not directions of symmetry. We will discuss these two cases in Section III and Section IV respectively.
II. THE VARIATIONAL PRINCIPLE
In this section we generalize to the charged anisotropic case a variational argument 1151 used to discuss the existence of zero sound in neutral isotropic Fermi liquids (He3). We start with the kinetic equation [2, 61:
(As is customary in investigating the existence of modes, we make the (generally unrealistic) assumption that the relaxation time can be made large enough to satisfy the condition WT > 1.) Here w and 4 are the angular frequency and wave vector of the zero sound mode, v is related to the deviation from equilibruim 6n of the quasi-particle distribution function by an
=
_
($)
yei(a+-d
30
CHENG AND MERMIN
and the angular brackets represent an average over the Fermi surface:2 dS deF)
=
1
4p3u
(4)
*
The quantity V is defined by
8k) = (1 + F>40 = 44 + 1 $$+k = v(k) + -
1
dEF)
s
k’) 40 $$$
t’(k, k’) v@‘),
(5)
where F(k, k’) = g(+) f(k, k’) and f(k, k’) is the spin-independent part of the Landau Fermi liquid interaction function. Finally, k,2 = 4rrezg(+) and 11and 1 refer to the direction of q. Equation (2) is derived in the form used here for isotropic charged Fermi liquids in [6]; the procedure in the anisotropic case is exactly the same. For zero sound to exist it is necessary that (2) have a solution for a real value of w that exceeds the largest value assumed by q * v(k) as k ranges over the Fermi surface: ” > 4uYBx@).
(6)
If this condition is not met, the mode will merge with the continuum and undergo catastrophic Landau damping. In addition, we are only interested in modes with frequencies well below the plasma frequency. This amounts to the condition:
k&‘F3 w,
(7)
where up is a typical Fermi velocity and k,vF is on the order of the plasma frequency. In deriving the variational principle we must also exploit the stability condition?
> 0,
(8)
for any disturbance v(k) whatsoever, provided only that
(9)
(v) = 0.
(10)
and Conditions
(9) and (10) mean that the disturbance v(k) creates no net charge and
%We have set fi = 1. AIL integrals are over the Fermi surface.
SSee, for example, 1161 for the isotropic case. It can be extended to the anisotropic case.
ZERO
31
SOUND
current densities and hence no macroscopic electromagnetic energy. For such disturbances the stability condition (8) is the same as in the neutral Fermi liquid. It is convenient to express the stability condition in terms of 17= (1 + F)v. Equations (81, (9), and (lb) become (G*(l - B)C) > 0
(11)
(VV) = 0
(14
((1 - B,)lQ = 0.
(13)
for any V, provided only that
and
Here B(k, k’) is defined in operator notation as
B=
1F
’
(14)
i.e.,
B(k k’) = Fe, k’) - -&
1 4$;;!t) B(k, k”) FW, k’),
(15)
B(k, k’).
(16)
and 1 BoW = g(EF) .r dS(k’) 4dv(k’)
Equation (2) is to be solved in the space of all functions #(k) with k varying over the Fermi surface. The scalar product of 4(k) and 4(k) is defined as (17)
The condition that the frequency of zero sound be much smaller than the plasma frequency requires the solution of (2) to have negligible projection on u,, , for if the solution has an appreciable projection on z1,,, the term in v I will dominate all the other terms in the equation when k,,vr > W. As discussed in [6] and [8], we can drop the term containing vII in (2) provided we project the resulting equation out of the vfl subspace: k,b P wPS - qv,,PC - wBPC -j- v . &PC>] [ q2c2 L
= 0,
32
CHENG
AND
MERMIN
where P=
1 - 16,,)(6,, 1;
6, =-,
v II (v2IIy2
(19)
Here we have also anticipated that w/q will be of order VFfor the modes of interest, and therefore have neglected w2 in comparison with q2c2. Note that it follows from Eq. (18) that since PG has no component in the v,, subspace, it also has no component in the (1 - B,) subspace, i.e., we have ((1 - B,,) PF) = 0. (This is just the assertion that zero longitudinal current implies zero charge density.) We can therefore rewrite (18) as V=
[Q (+)
Q + QBQ - -$$
Q I v,) *
(20)
where Q is the projection operator which projects out both the v, and (1 - B,) subspaces. This is convenient to do, since the explicit appearance of the projection operator Q insures that two of the restrictions (12) and (13) appearing in the stability condition (11) will be automatically satisfied. For simplicity, we only consider crystals with inversion symmetry, so that (v,,(l - B,)) = 0, and therefore 1 - B,, Q = 1 - I %X6,, I - I i&b I; B= ((1 - Bo)2)1’2 * (21) In Eq. (20), we consider w as a fixed parameter and q, a variable. The condition for zero sound is that H, have eigenvalue 1 for some real q between 0 and o/v:“. (We take w to be positive. The range of q between 0 and -cu/v~~ need not be considered, since the spectrum of (2) is symmetric about 0 for a crystal with inversion symmetry.) Since H, is Hermitian for real w and q, and since B is bounded, the maximum eigenvalue Aa of H, is a continuous function of q. Now it follows from the stability conditions (1 l), (12), and (13) that all the eigenvalues of Hg are less than 1 when q = 0: for functions with no component in the v, subspace, H,,=, = QBQ and all the eigenvalues of QBQ are less than 1 in this subspace because of (1 l), (12), and (13); for functions with components in the vI subspace, Hp.+, is dominated by the term
- +$ QI v,>*(v, I Q, the eigenvalues of which approach - co as q -+ 0. Therefore, A,,, < 1. If for some q. between 0 and o/v? we can show that h, > 1,
(22)
ZERO
33
SOUND
then by continuity Xql will equal 1 for some q1 between q. and 0, and zero sound will exist with wave vector q1 . To derive a sufficient condition for the existence of zero sound, we can therefore use the variational principle that tells us that b > (#*H&j
= (#*
[Q (+j
Q + QBQ - $$
for any trial function #(k) satisfying the normalization
Q I VA . (v, I Q] #),
(23)
condition
cldr*+> = 1.
(24)
Our task is to find a suitable trial function +(k) leading to simple conditions on B which make \<+*H,#) > 1 for some real q between 0 and w/try.
III.
APPLICATION TO q ALONG A SYMMETRY DIRECTION
Gor’kov and Dzyaloshinskii emphasized that q . v(k) will have more than one maximum when q is along a symmetry direction and u,(k) is maximum at a k not parallel to q. In general one must know both the shape of the Fermi surface and the magnitude of v(k) over the surface, in order to determine whether this condition can be satisfied in a given metal. However there is an important case in which the condition is guaranteed to hold regardless of the magnitude of v(k): Suppose we display each branch of the Fermi surface in a primitive cell with the full point group symmetry of the lattice (in practice we shall always use the first Brillouin zone), and suppose that when the surface is so displayed, there is no Fermi surface in any of the branches in the direction of q from the origin. Then regardless of the values of u(k) along the Fermi surface, the maximum of q . v(k) cannot occur for k parallel to q for the trivial reason that there are no occupied Fermi surface states in that direction. Thus q . v(k) must be maximum for k not along q. If q happens to be an axis of symmetry then other values of k with the same value of q . v(k) can be produced by symmetry operations. Because the cells in which the branches have been displayed are primitive, these other values will correspond to physically distinct states, and the Gor’kov-Dzyaloshinskii condition will be satisfied. Among the currently known Fermi surfaces, there is no surface along the [l 1l] direction of the noble metals [17], [Fig. l(a)], and along the direction of the c axis of beryllium [18] and graphite [19], [Figs. l(b) and l(c)]. In all of these cases, when q is along the required symmetry direction, the symmetry group of the kinetic equation is CsV. Since these are the only simple cases we have discovered, we will discuss only this group in detail. The generalization of the following sgs/6dI-3
34
CHENG
AND
IvERMIN
(b)
lectrons
FIG. 1. Fermi surfaces and the direction of q which makes q * v have multiple maxima. (a) Copper, silver, or gold. (b) Beryllium (after Loucks and Cutler [IS]). When reduced to the tit zone, the “coronet” must be sliced in half by a horizontal plane and placed at the top and bottom of the zone. The (two) “cigars” are sliced vertically into six pieces that go to the corners of the zone. There is therefore, as required, no Fermi surface along the c axis. (Most hexagonal metals fail to meet this criterion, because the “lens” is intersected along the c axis.) (c) Graphite [after Williamson and Foner [19]).
discussion clear. When q depending or not. In
to other symmetry groups, should it turn out to be necessary, should be is along the symmetry axis of the group C,, , there are two possibilities on whether vy occurs on the vertical reflection symmetry planes general, there will be six k on the Fermi surface where vu is maximum
ZERO
35
SOUND
[Fig. 2(a)], but if o,, is maximum for k on the vertical reflection symmetry planes, we will have only three k [Fig. 2(b)]. We derive the conditions for zero sound for the case of six maxima, from which we can easily deduce the conditions for the case with three.
FIG. 2. The relative positions of k”‘s on the Fermi surface which have the maximum value of 4. v(k) in the symmetry group C,, . The direction of q (taken as the z axis) is pointing out of the page and the dotted lines represent the vertical reflection symmetry planes. (a) When k”s are not on the vertical reflection symmetry planes. (b) When k”‘s are on the vertical reflection symmetry planes.
Suppose the maxima of u, occur at k,O ‘0. k,s on the Fermi surface. Their relative positions are shown in Fig. 2(a). We pick as our trial function Ci
$04 =
1
0
-
for
k E AS,
(P,,lW)
i = 1, 2,..., 6; (25)
otherwise,
where the AS, are small regions on the Fermi surface around the ki” maintaining the C,, symmetry. The Ci are constants to be chosen to make the condition for the existence of zero sound as weak as possible. Unless the coupling is quite strong, the best choice turns out to require that trial functions of the form (25) have no net charge and longitudinal current densities. Since both (1 - B,) and vu belong to the identity representation of the group C,, , any function belonging to the other irreducible representations will automatically be orthogonal to (1 - Bo) and t‘, , and hence have no charge and longitudinal current densities. We therefore can choose the trial function according to the irreducible representations of the symmetry group, and for each representation we have a condition for the existence of zero sound. In addition to the identity representation 1201 (AI), C,, has one one-dimensional (A,) and one two-dimensional (E) irreducibIe representation:
36
CHENG
AND
MERMIN
1. Representation A, : Our trial function will transform under this representation, if we choose the constants Ci in (25) as follows: Ci = fC
for
i = 1,3, 5,
c,=-c
for
i = 2,4, 6,
and
(26)
where C is a normalization constant such that (#*#> = 1. Since the trial function, (25) and (26), has been chosen to satisfy the condition (#*6, ) = (zj*p) = 0, we have Qt,4 = #. The condition (23) becomes
1 = ’ + 6( l/U - (~v,,/~)]~>As
1
1 ’ B(ki ’ k5) 1 - (qv u&)/a.~) 1
-6 1 -
(qv&)
)I AS
* 1 - (qv,,(kJ/o) I
’
where ( )As is an integration over one of the dS only. Note that the term in v, vanishes due to the symmetry of the trial function. At this point we assert that if B,, +
242
-
42
- Bu - B,, c=-0
(28)
zero sound exists. Here Bi, = B&O, ki”) = B(kjo, k$‘).
If (28) holds, we can always choose dS so small that MW%
,4) + 2B&,
k$ - Bk, k2) - WI9 kJ - WI, bJ1 > b. > 0
for all ki E O& . Then, the inequality
(29)
(27) can be rewritten as
&>I+( l:F)Aj((l 19:,).)Asp(l ‘qz,l),,-I].t30) The coefficient of the bracket exceeds zero for every q < w/vy, while the coefficient of b, within the bracket becomes arbitrarily large as q approaches o/eaK from below. This establishes (22) for some q. < w/v, and hence (28) is
ZERO
37
SOUND
a sufficient condition for the existence of zero sound in the representation A, . Note that the condition (28) does not depend on the strength of the coupling, but only the sign. 2. Representation E: This is a two-dimensional representation. When we choose a trial function #,(k), there will correspond another trial function 4,(k): $J, and 4, are transformed into each other like x and y under the operations of C,, . We choose the trial functions Z& and &, in the form (25) and denote their constants by Cix and Ciu, respectively. We have:
C,” = C,” = c,
c,y Z --c,p = ac >
C,” = csx = (- l/2 + %&/2)C,
c,y = -coy = (-a/2
C,” = c5x = (- l/2 -
C,y = -C,y
Here C is a normalization
1/301/2)C,
= (0112-
-
%6/2)C,
(3 1)
43/2)C.
constant such that
<~,**a!> = <*Y**Y) = 1 and Q:is a constant to be chosen to make the condition possible. We again have
of zero sound as weak as
and
and therefore Q& = Either trial function Following exactly the that the condition for
& and Q/J, = &, . #Z or #y must lead to the same condition for zero sound. same analysis as we used for the representation A, , we find the representation E is:
Here we have taken the x axis on the vertical reflection symmetry plane between k,O and kZo, and u,(l) and u,(l) are the x and y component of the velocity at k,O; x is to be chosen to maximize the left side of (32). Equation (32) is of the form am2 + ba + c > 0, which can be satisfied for some real a, if and only if a + c + / l/b2 + (a - cy 1 > 0.
38
CHENG
AND
MERMIN
This leads to the condition, WI,
+ u,2(1)) + 1[d/5 (BIB - B14) + 12 k,2V$~2a”(1)]2 - 4s) - 6 ko2(v0V)q2c2 + [(2B,, - B,, - BIB) - 6 ko2(vg2(;2;
vw2@))]‘/“’
> 0.
(33)
Note that the left side of (33) is an increasing function of q2. Therefore (33) will hold independent of q if it holds in the q + 0 limit. In this limit the condition becomes
(34) Even if (34) fails we will have zero sound for some value of q2, if (33) is satisfied in the large q limit. To leading order in qc/kovF , (33) becomes WI,
- 43
+ [3(B1e - Bd2 +
which is the weakest form our condition
(242
- 44 - BIS)~I~‘~> 0,
(35)
can assume.
Gor’kov and Dzyaloshinskii distinguish two regimes: the radio frequency region, qc < kovp and the infrared, qc > kOuF. In terms of this distinction we can say that zero sound will exist in both regimes if (34) holds, and only in the infrared, if (35) but not (34) holds. In the latter case the value of qc/kovF , above which we are assured of zero sound, is given by setting the left side of (33) to zero. Now we turn to the case when VT= occurs for k on the vertical reflection symmetry planes. In that case, k,O and k20 will merge as one point klo on the dotted line in Fig. 2(a), as will k,O and kao, and k,O and kao, Fig. 2(b). The trial function (25) in the representation A2 will then be zero, and our condition for zero sound cannot be satisfied by that representation. In the representation E, we must pick a trial function like (31) with (Y= 0 and therefore C, = C, , C, = C, , and C, = C, as required. The condition for the existence of zero sound can then be obtained from (32) by setting a = 0:’ B,, - B13 - 3 k”;;yl)
> 0.
* Note that in both cases (Aa and E) these results also follow directly from the forms taken by conditions (28) and (33) in the case B12 = B,, , I?,, = B14 = B,, and v,,(l) = 0.
39
ZERO SOUND
This is again most easily satisfied in the infrared regime, where it becomes simply:
B,, - B,, > 0.
(37)
Comparing (37) with (35) we see that the case of six maxima is more favorable to the existence of zero sound than is the case of three. Unfortunately the actual sign of the left sides of (28), (39, and (37) cannot be reliably estimated, and we can do no more than repeat Gor’kov and Dzyaloshinskii’s oracular comment that the likelihood of their being positive “is, roughly speaking, 50 per cent.”
IV. APPLICATION
TO q NOT ALONG A SYMMETRY
DIRECTION
For an arbitrary direction of q, there is only one maximum of q . v(k). But if the Fermi surface consists of several electron (or hole) pockets, it may be possible for q . v to have two maxima for certain nonsymmetry directions of q. We consider here by way of illustration a Fermi surface consisting of ellipsoidal pockets. Consider an ellipsoidal constant energy surface: f(c) = +(k - k,) a-l(k
- k,),
(38)
where a-l is the constant inverse mass tensor. In the simplest cases one might have f(c) = (E - constant) for particles and (constant - E) for holes. In some cases such as the electron ellipsoids in bismuth [21,22], more general forms have been used. From (38), we have for k on the Fermi surface, v(k) = ti-l(k
- k,)/f’(+).
(39)
It is easily shown that q * v(k) will be maximum at constant E = + , when k - k,, is parallel to q. The value of q . vmax(k) is then the value at the maximum of q . fi-l
. (k - k,,)
f’(+)
29 .f(EF) . = I k - k, I f’(c~)
(40)
Now suppose we have two ellipsoids: fl(c) = g(k - k$‘) l6l:;‘(k - kt’), fi(c) = $(k - kf’) tii,‘(k
- kt’).
(41)
40
CHENG
AND
MERMIN
The condition that q . v(k) has the same maximum k(l) and kc21 requires: I kt2)
-
kb2’
I
on ellipsoids 1 and 2 at points
fi(+)fZ’(4
1k(l) - kb” 1 ’ h’(+)&(q) Since
k(2)
-
kh2) and
k(l)
k(2)
The condition
J$) _
= ‘*
(42)
are both parallel to q, we have the two possibilities kt’
=
f
fiIfi m
(k”’ - k:‘).
that k(l) lies on ellipsoid 1 requires fi(+)
= i&(l)
- kt)) ti;‘(k(‘)
If we substitute (43) into the corresponding restriction on k(l): f2(+) = ; (#)l
- k;‘).
(44 equation for kc2), we find a second
(k(l) - k$‘) a;l(@)
_ ky),
(45)
or
fi(+) = ; (k”’ - k;)) ($)” $ fi,‘(@’ - kt’), Solutions will therefore exist if, when ellipsoid 2 is scaled in its linear dimensions by 151.
(-L,“”
(47)
and translated to the same origin as ellipsoid 1, the two intersect. Any vector q pointing from the common center to a point on the curve of intersection will then give equal values of (q . v)max on the two ellipsoids. This analysis can be applied to the Fermi surface of bismuth [21,22], which has three equivalent electron ellipsoids and a single hole ellipsoid. From the data given in [21], we find that each of the three electron ellipsoids has intersections with the hole ellipsoid, when it is scaled by (47), and they are translated to the same origin. Therefore for certain directions of q, q . v(k) can have the same maximum value at the hole ellipsoid and one of the electron ellipsoids. However, two of the components of the mass tensor of the electron ellipsoid are much smaller than the third component and those of the hole ellipsoid, and u(k) will be smallest when (k - k,) is parallel to the third axis. Due to the relative orientations of the three electron ellipsoids and the small components of the mass tensor, we find that when q is in the directions which make the maximum of q * v on the hole ellipsoid and one of the electron ellipsoids equal, there will be a larger maximum of q . v on one of the other two electron ellipsoids (see Fig. 3). Thus for such a direction of q we have only one q * vmaXon the whole Fermi surface.
ZERO
41
SOUND
FIG. 3. The intersections of one of the electron ellipsoids and the hole ellipsoid when it is scaled by (47) in bismuth. The central ellipsoid is the hole ellipsoid which is an ellipsoid of revolution aboutthez axis(perpendicular to the page); the threeelectronellipsoids canbetransformed into each other by 120” rotations about the z axis. The dotted circle represents the hole ellipsoid which is translated to the origin of electron ellipsoid 1. A and B are two points on the curve of intersection, When q is parallel to OA(II O’A’), q. vmnx will occur on electron ellipsoid 3, and when q 11OB on electron ellipsoid 2.
However there is another possible way to give q . v two maxima: q * v can have equal maxima on two of the electron ellipsoids, which are larger than its maxima on the third and on the hole ellipsoid. The relative orientations of the three electron ellipsoids are shown in Fig. 4. The z axis points out of the page and the three ellipsoids can be transformed into each other by 120” rotations about the z axis. The ellipsoid has a long axis and short axes. The long axis tilts out of the page at an angle of 6”. Since the angle of tilt is very small we shall neglect it for the
C
3 .
>
FIG. 4. The intersections of two of the electron ellipsoids in bismuth. The dotted ellipsoid represents electron ellipsoid 2 which is translated to the origin of electron ellipsoid 1. The curves of intersection will be on two perpendicular planes OA and OB.
42
CHENG
AND
MERMIN
moment. When Q - k,,) is along the short axes, the velocity u(k) will be large. Now we translate ellipsoid 2 to the same origin 0 as ellipsoid 1, as shown in Fig. 4. Since these two ellipsoids are equivalent there is no need to scale. The intersections will be on two perpendicular planes OA and OB. When q is on the plane OB, the maximum of q . v for the whole Fermi surface will be on ellipsoid 3. However when q is on the plane OA, q will be more or less along the direction of the long axis of ellipsoid 3 and therefore the maxima of q * v on ellipsoids 1 and 2 will be larger than the maximum of q . v on ellipsoid 3. Finally, the mass of the hole ellipsoid turns out to be much larger than the mass components of the short axes of the electron ellipsoid, which permits us to conclude that q * v is not maximum on the hole ellipsoid. Therefore when q is on the plane OA, q * vmax for the whole Fermi surface will be on ellipsoids 1 and 2. We have discussed the above with nontilted ellipsoids, in which case the two perpendicular planes are planes of reflection symmetry for the structure consisting of ellipsoid 1 and the translation of ellipsoid 2. However the plane OA remains a plane of reflection symmetry when the long axis is tilted. Therefore we can conclude in general that when q lies within a plane that contains the z axis and is parallel to the long axis of one of the electron ellipsoids, then q * v will assume its maximum value at two points, lying in each of the other two ellipsoids. Note that in general q will satisfy this condition even though it may be far from any symmetry direction. The Fermi surfaces of arsenic [23] and antimony [24] both consist of several equivalent electron pockets which can be approximated as ellipsoids and several nonellipsoidal hole pockets. Because of the complex structure of the hole pockets, we can draw no definite conclusion. If, however, the mass tensor for the holes is sufficiently large, the conclusion we reached for bismuth will also hold for arsenic and antimony. Finally we return to the variational principle to extract a condition for the existence of zero sound when q * v has two maxima. If there is no symmetry at all, the general analysis we used in Section III is very complicated. We therefore specialize our discussion to the case of bismuth. Since the dimensions of the electron ellipsoids and the hole ellipsoid are very small compared with the size of the zone [22], we make the approximation that B(k, k’) depends only on to which ellipsoids k and k’ belong, but not the detailed positions of k and k’ on the ellipsoids. Because of the symmetry between the electron ellipsoids, we then have only four independent values of B(k, k’): if k and k’ are on the same electron ellipsoid, if k and k’ are on different electron ellipsoids, if one of k and k’ is on the electron ellipsoid, and the other on the hole ellipsoid, if both k and k’ are on the hole ellipsoid.
(48)
ZERO
43
SOUND
Suppose the maxima of q . v are at k,O and k,O on the Fermi surface. We choose the trial function to be for
k E AS,,
for
k E AS,,
(49)
otherwise, where AS, is a small region around kio on the Fermi surface and C is a normalization constant determined by (z,/J*$) = 1; 01 is a constant to be chosen to make the condition for zero sound as weak as possible. The constant energy surface of the electron ellipsoid is in the form of (38) with [21,22] f(~) = ~(1 + E/Q) w h ere eg is the small energy gap between the valence and conduction bands. Using the form of the electron ellipsoid (38) and trial function (49), we can calculate the integral over the Fermi surface with logarithmatic accuracy: (v,,~)) = C(1 + a) W h In (1 - qj,
max
= CtB, + 4%
h ln (1 -
+j,
where B is defined in (21) and (16), and pi = ,&kio), and5
Using the approximate form (48) of B(k, k’) it is easily shown that fil = flZ . Therefore we can choose 01 = -1 to make the trial function #(k) have no charge and longitudinal current densities, as we did in the symmetric case. Condition (23) therefore becomes (52) ’ Here M, , M., , and M, are the components of the mass tensor referring to the principal axes of the ellipsoid.
44
CHENG
AND
MERMIN
By the same reasoning that led from (27) to (28), we may conclude that if (53)
zero sound will exist in bismuth for directions of q which have two q * v,, on the Fermi surface. Here we have used the fact that when q satisfies the above condition v,(l) = -v,(2), because the plane OA in Fig. 4 is a plane of reflection symmetry. From the data given in [21], it is easy to calculate ~~(1) when q lies on the plane OA. It is in the range from 1 to 6 x lo7 cm set-l. Since k, is about 1.0 x 10’ cm-l, kg,(l) is on the order of lOi set-l. Zero sound is then possible for small values of B if q > lo* cm-l. We also require q < +/fit+ , which is in the order of lo6 cm-l for bismuth. This restriction is due to the fact that Landau’s theory is only applicable to inhomogeneities with “macroscopic” wave vector. Since B& and Bz2 are probably of order unity or less, zero sound can occur in bismuth only in the infrared. Whether it exists will depend on the sign of B& - B,e , which unfortunately is beyond reliable theoretical estimation.
ACKNOWLEDGMENT We have Dzyaloshinskii
had
some theory.
valuable
discussions
with
J. S. Clarke
on the structure
of the Gor’kov-
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ZERO
SOUND
45
17. D. SHOENBERG, Phil. Trans. Roy. Sot. London 255 (1962), 85. 18. T. L. LOUCKS AND P. H. CUTLER, Phys. Rev. A 133 (1964), 819. 19. S. J. WILLLAMSON AND S. FONER, Phys. Rev. A 140 (1965), 1429. 20. M. HAMERMESH, “Group Theory,” p. 126. Addison-Wesley, Reading, Mass., 1964. 21. G. E. SMITH, G. A. BARAFF, AND J. M. ROWELL, Phys. Rev. A 135 (1964), 1118. 22. D. SHOENBERG, “The Physics of Metals, 1,” (J. M. Ziman, Ed.), p. 109. Cambridge Univ. Press, 1969. 23. P. J. LIN AND L. M. FALICOV, Phys. Rev. 142 (1966), 441. 24. L. M. FALICOV AND P. J. LIN, Phys. Rev. 141 (1966), 562.