surface coupling to collective and single-particle spin modes in normal 3he

Journal of Low Temperature Ph~stcs Vol 71. Nos 5/6. 1988

Surface Coupling to Collective and Single-Particle Spin Modes in Normal 3He

S. A. Bogacz* and J. B. Ketterson

Department of Physics and Astronomy, Northwestern UniversiO, , Evanston, Illinois

(Received November 30, 1987)

We propose a simple technique for probing the single-particle and collective excitations in normal 3He via a standing magnetic surface wave of arbitrary to and fixed k generated by a meander-line coil. An analytic treatment of the Landau-Sillin equation enables us to calculate the power absorption spectrum, which displays singularities associated with the I = 0 spin mode and a Doppler- shifted spin resonance (DSSR) of the single-particle excitations. Experimental measurement of the spectrum wouM determine the Landau parameters (F~, FT ) with spectroscopic precision. Furthermore, observation of the DSSR thresholds would provide an independent determination of the Fermi velocity v and F~. Finally, a possibility of exciting higher, l >- 1, spin modes is briefly mentioned in the framework of an extended Leggett-Rice equation.

1. INTRODUCTION

For the most part, magnetic resonance experiments are usually carried out in the presence of a spacially uniform rf excitation field. An important exception is light scattering studies I (where in-plane wave-vector conservation places restrictions on the wave vector of excitations generated in a medium). A second exception is the case of metallic samples where the skin effect concentrates the Fourier components of the wave vector of excitations generated in the medium to the vicinity of ~-~, where 8 is the skin depth; this property allows the generation of magnetostatic modes in ferromagnets 2 and Silin spin waves in pure metals 3 (at low temperatures). In the ease of NMR, spin diffusion is generally studied by applying an inhomogeneous static field, 4 rather than using an inhomogeneous rf field. 5'6 In this paper we explore rf excitation using a periodic meander line which has the property of having a specified wave vector lying in the plane of the surface of a sample. This situation is shown schematically in Fig. 1. By

*Permanent address: Accelerator Theory Department MS 345, Fermi National Accelerator Laboratory, Batavia, Illinois.

445 0022-2291.'88,.'0600-0445S06 00.'0 ~ 1988 Plenum Pubhshlng Corporation

446 S . A . Bogacz and J. B. Ketterson

I- l- I

Generator -]

L Top View

l l l l l l ° ° -

m

ilii~ iiiiii! ~iiiiiiiiiiil ~iiiil iiii'"~iii ~ iiiiiiiiiiiii ~iiiiiiiiii ~iii iiiiiiiiii~ iiiiiiii ~iiiiii ! "'" ~i i!i i!i :: ~ iii!!i :ii i!iil iiiiiiiil i iiiiill i ilii!i :: iiii~iiill iiiiii!i:~ ii i [iiiiiil" ""

Front View

Fig. 1. A schematic of the meander-line coil used to excite a standing magnetic surface wave.

symmetry , this structure restricts the in-plane c o m p o n e n t of the wave vector of excitat ions genera ted in the med ium to match that o f the meande r line; it places no restrictions on the normal component . Qualitatively, we expect the meande r line to behave as a diffraction grat ing (or, more precisely, left- and r ight-moving diffraction gratings). The k-vectors of the excitat ions genera ted in the m e d i u m would then be expected to satisfy the grating equation. In part icular , there will be a "cutoff f requency" for the grating, which can be p robed exper imenta l ly to gain informat ion on the excitat ion spectrum. A 3D ana log of the meande r line t ransducer has been p roposed by Corruccini 7 as a way of setting up a nonun i fo rm rf field to couple to the spin modes. The kinemat ics of this device would be governed by the Bragg law; as discussed above, the diffraction grating law appl ies to our 2D structure.

2. K I N E T I C E Q U A T I O N . S P E C U L A R B O U N D A R Y C O N D I T I O N S

In the f r amework of the Landau theory of a strongly interacting Fermi liquid one can describe the dynamics o f a normal 3He by the fol lowing kinetic equat ion:

a,~p+½{O~p, Opgp}-½{Opt~p, O~gp}- ih[gp, ~p] = Ip (1)

Surface Coupling to Collective and Single-Particle Spin Modes 447

Here ~p and ~p are two-by-two matrices in spin space representing the particle density and the ,quasiparticle energy and Ip denotes the collision integral. Furthermore, {A, B} represents the anticommutator of two spin matrices A and/~.

The various matrices can be decomposed into spin-symmetric and antisymmetric parts described by scalar and vector quantities, respectively. The following notation is introduced:

,~. = np~ + m ~ ~ = epl + hp cr (2)

where 1 is the unit spin matrix and ~ are the Pauli matrices. Furthermore, the vector mp describes the local spin density in phase space and hp represents the effective magnetic field (in energy units) incorporating both the external magnetic field and the induced molecular field in the liquid.

Using the notation introduced in Eq. (2), one can effectively separate the "spin-density" part of Eq. (1) by operating with Tr t~ ~ . . . and applying the simple properties of Pauli matrices. The resulting Landau-Sill in equation, s linearized with respect to a small fluctuation, can be written as follows:

Otlnp+OO~amp ~a o ~c~ o . o ~ a h p �9 " Op6p--OpHp

(3) - 2 [ h ~ a m p ] - 2 [ a h p x m ~ =Jp

Here

mp = m~ + ~mp

represents the nonequilibrium spin density and Jp is the spin-antisymmetric part of the collision integral. Furthermore,

hp = h ~ f dr'fa(p, p')am a, (4)

denotes the total magnetic field inside the liquid including the dc, rf, and o is the Fermi distribu- molecular field components, respectively. Finally, np

tion and e ~ =p2/(2m*) (m* is the effective mass of a -~He quasiparticle). For a weakly perturbed nonequilibrium state, the fluctuation amp occurs

only in the immediate vicinity of the Fermi surface. One can then introduce the energy-integrated fluctuation (depending only on the direction in p- space), which is defined as follows:

cr~ ~ J d,~p 6m~ (5)

where ~p = o ep-lS and /. is the chemical potential.

448 S .A. Bogacz and J. B. Ketterson

Substituting Eq. (5) into eq. (3), and using the fact that the spin antisymmetric part of the quasiparticle interaction f a is a slowly varying function of momenta near the Fermi surface, one obtains the final form of the Landau-Silin kinetic equation, 9

ato-~ + o~ ~ o, a . . . . Ox{o-~ + ( F (p,p )o'~,)~,+h '~ } + 2e~138{g ~ + (Fa(/3,/3')o-~,)~, + ht~}h~ = J~ (6)

Here

Fa(/~/~') = 2N(0)fa(P, P')IFs

ho = - ( h y / 2 ) H o

h = - ( h y / 2 ) H r f

where N(0) is the density of states on the Fermi surface, Y is the gyromag- netic ratio, and Ho and Hrf are dc and rf external magnetic fields, respectively. Averages bver the spherical angles in p-space are defined in a standard way by

1 I d f ~ ~ ('" ")~--- 4---~ "'"

The collisionless limit of Eq. (6) is used to describe the collective behavior of the spin degrees of freedom in a normal Fermi liquid at zero temperature. Its eigensolutions represent the so-called spin wave excitations of the system.

Equation (6) will be used as the starting point for the investigation of surface coupling to various spin modes in a normal Fermi liquid. However, in treating any problem involving a finite geometry, we must examine the appropriate boundary conditions. For the problems to be discussed later, this has been accomplished only for the case of specular reflection of the quasiparticles. This situation will now be examined in detail.

By summing Eq. (1) over the momentum (~ d~-...) and spin ( T r . . . ) degrees of freedom, one obtains the following continuity equation for the spin density:

O,(I d~'m~)+2e.~ f dzm~h~

(7)

where

Op = Op Ep

Surface Coupling to Col lect ive and Single-Particle Spin Modes 449

From the above equation one can easily identify ~ dr mp as the local spin density, while the expression

plays the role of a spin current density. We note in passing that the second term in Eq. (7) describes the change of the local magnetization due to spin precession around the local magnetic field direction. Using eq. (5), one can rewrite Eq. (8) in the more convenient form

S ~ f d~" (Op ep ~ '~ r = ~ �9 m p + O p h ~ � 9 ( 9 )

To fix the geometry, we consider a sample of a normal Fermi liquid bounded by the x - y plane and occupying the upper half-space; the dc magnetic field is directed along the z axis.

To discuss the specular reflection of quasiparticles from the interface illustrated in Fig. 2, one can introduce the particle current. This quantity is defined analogously to Eq. (8) by the expression

Since there is no net flow of particles through the interface (j-- = 0 for z = 0), the specularity condition, combined with the explicit formula for jz, Eq. (8), requires that at z = 0 the quantity

O~:,ep" np+Ophp .rap (11)

is antisymmetric with respect to reflection of the momentum in the z direction (p~-~ -p~) . Similarly, assuming that there is no "spin flip" at the interface

~:~~:~~i~:~::~:~ Normal Fermi Liquid ~:~:~:~:~:~:~::~:::~::~:::~:~:~:~:~- ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: iii.:i!i!iiiiiii!i!iiiiiiiii!i!i!!!i!iiiii!!i!!;iiiiii:iiiii!i!i!i!i!i•i!i!iiiiii!i!i!iiiiiiii!!i!•i:i:iiiiiiiiiiiiiiiii!!i!iiii!i!i!i!i:i!iiiii ~&~-~-2-~-~-~-~-2-~-/ . . . . . . . . . . . ~ g ~ - 7 ~ 2 ~ - ~ - i - ~ - ~ - ~ - - - ~ - - - ~ - ~ - ~ - 2 - ~ 2 ~ - 7 ~ i ~ - ~ .-_-.-_-_-.-.-_-~-.-.-~ ~ _-.-.-...-_-.-.-.-.-..-.-.-.-.-.-.-_-.-_-_ ~z-_-.-_-.- .- .- .- .- .- .-_.-_-

Fig. 2. S p e c u l a r re f lec t ion o f a 3He q u a s i p a r t i c l e f r o m a pe r f ec t ly s m o o t h in te r face .


during the specular reflection, we have that the net spin current S -'~ vanishes at the boundary. This, in conjunction with the specularity of particle reflection and Eq. (11), yields an additional symmetry requirement that at z = 0 the quantity

Opep" mp +0php �9 np (12)

is antisymmetric with respect to momentum reflection in the z direction. As was shown by Bekarevich and Khalatnikov, 1~ these two symmetries, Eqs. (1 i ) and (12), combined with energy-momentum conservation, require that O-p has mirror symmetry with respect to the Px-Py plane. This can be written symbolically as

trp-~ crp for p~_,_pZ (13)

at z =0, which is a particularly convenient formulation of the specular, no-spin-flip boundary conditions.

3. ENERGY FLOW THEOREM

The local energy density of a Fermi liquid is expressed by the formula tl

a / / = T r ( f dr ~p~p) (14)

Taking partial time derivative of this expression and applying the kinetic equation (1), one can express one of the terms, namely

T r ( f drO,~p. ~r,) (15)

as a divergence of the following vector quantity:

~ ~ -= Tr(�89 1 d~- ~{rSp, ~p} ) (16)

The last step involves a simple integration by parts combined with the properties of the spin matrix algebra. 12 Combining Eqs. (14)-(16), one obtains an energy flow theorem which has a structure of a continuity equation, namely

~

0,~//+ a ~ ~ = b ~ (17)

We have already defined an energy flux ~ by Eq. (16), and a source term (a volume force) 6e is given by the remaining term generated by the partial


time derivative of Eq. (14),

r e - - - T r ( f dr~p. O,~p) (18)

Again making use of the spin matrix algebra, one can easily extract the quadratic contributions of the fluctuation of the above-defined quantities,

= 2 j dr (Sr/p �9 SEp -1- t~mp. t~hp) (19)

I o o ~ = dT[(ep6np+np6ep+ho" 6rap (20)

o 3hp) " o o + rnp . Or~ep+(eprmp+mp3ep+horn p

+nOrhp) . ~ 4, . CptShp+vp (reprnp+rhp 6mp)]

5 p = 2 J d'r (6npO,(Sep + 6nlp. O,t3hp) (21)

In the case of a magnetic perturbation, to be considered here, 3np and 6ep are put equal to zero. Furthermore, by applying the specular boundary condition given by Eq. (13), one can see, after a little algebra, that ~z defined by Eq. (20) vanishes at the boundary, Therefore, the only contribution to the energy absorbed by a Fermi liquid arises from the volume force associated with the applied field, which is expressed in terms of the integrated source terms re of Eq. (21).

One can formulate the linear response of the liquid to some external perturbing field in terms of the total power absorbed per unit area of the sample interface. This in turn is easily expressed from Eq. (17) by the explicit formulas for ~ and 6 e, Eqs. (20) and (21). We also note that for a given external field the fluctuation 6rap is uniquely determined by the Landau-Sil in kinetic equations, Eq. (6), and the appropriate boundary conditions.

Therefore the problem of calculating the linear response of a Fermi liquid reduces to solving the inhomogeneous LS equations with the specular boundary conditions given by Eqs. (13).

The above program is carried out in the next section. We confine our discussion to the case of specular reflection of the quasiparticles from the interface. The other limiting case--diffuse reflection--leads to certain analytic pathologies. Due to the complexity of the applied driving field (a propagating surface wave), the method used by Bekarevich and Khalat- nikov 1~ for treating the Kapitza resistance could not be extended to our problem. One may be able to develop some numerical procedure to solve the diffuse case, since an analytic solution does not appear to be tractable.

452 S .A. Bogacz and J. B. Ketterson

4. T H E O R E T I C A L A P P R O A C H

Consider a sample of normal 3He occupying the z > 0 half-space with a uniform static magnetic field Ho directed along the z axis. One can excite the surface (x-y plane) with a standing magnetic surface wave, which can be decomposed into left- and right-running waves propagating along the x axis. The two remaining possible geometries (Ho directed along the x axis and along the y axis) were also examined and they do not lead to a qualitatively different picture; therefore we fix our attention on the first geometry in the presented model calculation.

As was shown earlier, 13 this kind of disturbance can be generated by an array of parallel, equally spaced wires carrying antiparallel currents I = Io e -'~t. The standing surface wave solution for the magnetic scalar potential has the following form:

c c

~b(t, r) = (4Io/c) cos wt ,v ( _ 1)m(2m + 1) -1 cos(k,,,x) e k,,.: (22) m = 0

where k,, = (2m + 1)kx. The value of kx is fixed by the geometry of the transducer and is given by kx = ~-/I, where I is the spacing between adjacent elements (carrying oppositely directed currents). Each Fourier component of the driving field can be written as a linear combination of four running waves and the problem reduces to finding the response of the system to the driving field

~b=(r, t) = ~b0 exp( . . . . . . k , - k :~ ( Z > 0 ) (23)

for both positive and negative w. As discussed earlier, the spin dynamics of a Fermi liquid are governed

by the LS kinetic equation. For normal 3He only the transverse component of the spin density o-,(r, t) is associated with propagating modes. The inhomogeneous form of the collisionless LS equation can be written as

Ac~ ::[: {O~+(vfi'~O~+il-l~)(l+Fa)}o'~=(vp O~ il~)O~b (24)

where the operator F a represents the spin-antisymmetrfc part of the molecular field interaction; its kernel is expressed as a series of Legendre polynomials Pt(fi, fi') with respective Landau amplitudes F~. Furthermore, o-~-= o-~ + io-~, ~ 1-10/(1 + F"), 1~0 is the Larmor frequency, and ~ is the magnetic scalar potential generated by the transducer.

This equation is solved for the potential given by Eq. (23) under the assumption that the 3He quasiparticles reflect specularly from the transducer interface. Since basically the same algebraic approach to the LS equation (as discussed in ref. 14) is used here, one can briefly present the steps of


our analytic solution. As discussed earlier, the assumption of specular reflection of the quasiparticles at the interface imposes the requirement that ~ro(r, t) be invariant under reflection (changing the sign of pZ at z = 0). We seek a solution of Eq. (24) with the potential given by Eq. (23), which satisfies the above boundary condition in the following form:

(r;(r, t) = exp ( . . . . . kxx~ f;oo [exp(-'k:-'~](r;(k--) dk~_ (25)

Substituting the above in Eq. (24), one can express ~r~(kz) in terms of its different moments in p-space as follows:

o ' i ( kz ) = - ( k v x + a ~ ) / ( k v x + f~; - ~o )

x [ F ; o - o + F ~ o - 2 ~ - i k x & ( k ~ ) ] , x = - ~ . ~c (26)

Here the moments of cr~ are defined by

: t : m z~ ~ • o- o = ( o - : ) : , o-,~ =-- ( o ' : p 5: (27)

where

1 f dlq~ ('")~4~ " '"

Using a method similar to that applied by Fomin, ~5 one can impose a specific boundary condition on the solution (26) by the appropriate analytic continuation of the driving potential ~b to the lower half-space. One can see by inspection of Eq. (26) that the specular reflection symmetry requires the Fourier transform of ~b be an even function of k_,; this, in turn, requires that the analytic continuation of the potential & to the lower half-plane must be odd. Summarizing the above, if we choose-

r k:, z > O (28) & ( z ) = _ r ek,: ' z < O

then

O(kz) = ( r k2z)

and the solution given by Eq. (26) automatically satisfies the specular boundary conditions (in real space). Of course, only the "upper half-sphere" part of cry(r, t) has physical meaning and can be expressed from the Bromwich inversion formula applied to Eq. (26). To close the solution for o-~(k_,), one can take the respective p-averages of Eq. (26). We express the

4 5 4 S . A . Bogacz and J . B. Ketterson

various o--moments explicitly as follows:

O-o(z)=l/(27ri)xf~:dk~e'k:Z2&o(G/k) 2

x (1 + F~/3)[o~A=(w, k)/kv - 1]/D • (29)

c r 2 ( z ) = - l / ( 2 ~ i ) x f:~, dkze'k:~2&ok~k~ k-3

(30) • a , • x(w/kv)[ ( D o - w ) A ( w , k ) / k v - 1 ] / D ~

Here

D==(l + F;)(l + FT/3)-oJ( • • 2F7 (31)

- { F ; ( 1 + F~/3) + (+ f~o- w)/( + f l ; - w )FT}w_A_~( w, k)/kv

where

i~-=- ( r k ) i a a �9 = f l n [ ( + P t o - w - k v ) / ( • (32)

k=(k~k,~)l. ,'2

We note that our solution represents the response of the liquid to a single running component of the full standing wave, Eq. (32). To obtain the full response, one has to add the three remaining solutions (for all m), which can be easily obtained from the analytic solution (29) by alternating the signs of o) and kx. We identify D ~= 0 with the dispersion relation for the l = 0 spin wave.

5. CONTOUR INTEGRATION

The remaining integration in Eq. (39) can be carried out explicitly using Cauchy's integral theorem. Since we are interested in the solution in the region z > 0, our contour of integration must be closed in the upper-half of the complex k= plane. To avoid multivaluedness of the integrand connected with the logarithmic term of 3_ i, we introduce a cut-line connecting the branch points of A i. Therefore, the contour, chosen in an analogous way to that in ref. 14, includes a detour part which excludes the cut-line and the branch points. Finally, the integrals evaluated will contain two contributions: a contribution from a simple pole arising from D = = 0 (given by the respective residue) and a contribution from the branch point of A = (represented by the integral along the cut-line). Both contributions can be expressed as a sum of plane waves with k-vectors given by the location in the complex k~ plane of the pole and the branch point, respectively. The result of the above contour integration can be summarized by listing the "residue" and "cut-line" contributions to the ~r-moments given by Eq. (39).


1. Residue contr ibut ion:

O-o I '~ ,= 2Cko(k,,/k)2(1 + F~/3)

~nd

Here

x [wA:~(w, k ) / k v - 1 ] e G : / U • k~ = k ~-(~e~ (33)

orS(z) = -2ckok'~k~.k-3(o)/kv)[( • f~; - o9)

x A~(~o, k ) / k v - 1]e~k:~/U • k: = k ~:t~)

U = = a D ~ - / a k ~

(34)

and the locat ion o f the residue is given by D~(kz = k ~re~) = O.

2. Cut-l ine contr ibut ions:

O-o ~"t' = 2~bo dk= eGZ(kx /k )2 ( l + F ~ / 3 ) ( w / k v ) / D ~- (35) k=lCut )

and

I ~a:, I k - c~ 9 - 3 a o,~ ..... ( z ) = - 2 ~ o d k : e =k k=k ( w / k t ) ( • (36) k•

Here the locat ion o f the b ranch point is given by

k ~ u ~ = [( • ~Q; - w)e /v - k~] ~''2 (37)

6. E N E R G Y FLUX E V A L U A T I O N

The energy flow theorem, Eqs, (4)-(9), allows us to evaluate the total power absorbed per unit area o f the liquid interface by time averaging the z integral o f the driving source term as follows:

(Q) = F;((O'o, ~ro})+ F~((o-~ +, o-~)) (38)

where

(CA, B)) =-- �89 dz {(O,A, B), + (A, O,B),} (39)

and the time average ( . . . ) , is defined as

CA, B), =-�89 R e ( A ' B ) (40)

Using expressions (38) and (40), one can easily prove the fol lowing identity. Let A and B be given by

A(r, t) = A exp( - iwt+ ikxx+ ik(A)z) (41)

B(r , t) = B exp( - iwt+ ik~x+ ik tmz)

456 S. A, Bogaez and J. B. Ketterson

where o~ ~ o~ + i /r and ~- is a relaxation time (which models a small dissipa- tion connected with the collision integral); we then have

. . . . . [�89 R e ( A 'B) 8AB(O~o/Ok:)lk~=k'*', k (a) real

((,% ~2;' = ~O(I/~') , k ~a), k tB~ imaginary (42)

Applying the above theorem to Eq~ (38) and substituting the explicit expressions for the o--moments, Eqs. (33)-(36), one obtains the final expression for (Q):

(Q) = (O)t~s) + (O), o.~) (43)

v~-here

and

( Q ) ~ = -~[(oD+/Ok)/(o.D+/ao~)t~] (44)

(Q)(Cuo_ D + z tu2 ~-5 t ~ + a 2 -(24~o/ ) k~v (o~- f l o) {~Fo(1 F~/3) a I I ~ 2 a 2 + F , D - ~ ( k ~ v ) / ( o ~ - f~o) ]}l~=k ..... , (45)

Obviously, accordingly to the theorem given by Eq. (42), both contributions to the energy flow, Eqs. (44) and (45), are nonvanishing only if the simple pole and the branch point are located on the real axis (k +~S; and k +(~"t' have to be real). Since both D + and A + are functions o f oJ, the locations o f the simple pole and the branch point depend on frequency. Certain critical values of to when the singularity moves from the real to the imaginary axis in the complex k= plane represent cutoff frequencies for the energy flow. One can identify those frequencies where the simple pole lies on the real axis with the allowed region for l = 0 spin-wave generation. Similarly, the frequency range in which the branch points of A + lie on the real axis defines the situation where a direct coupling between the single- particle excitations and the driving field occurs.

7. DISCUSSION AND NUMERICAL ANALYSIS

As a result o f the above analytic calculations, we see that there is a direct coupling between a magnetic surface wave and the I = 0 spin mode and also the particle-hole excitations in a normal Fermi liquid, The bulk dispersion relation for this mode is illustrated in Fig. 3 along with the boundaries of the particle-hole continuum as modified by the magnetic field. The numerically calculated power absorption spectrum is shown in Fig, 4. Note that it displays singularities associated with various thresholds for both the single-particle excitations and the (collective) 1 = 0 spin wave


~ / ~ o

0 - - ~ 0 1 2 3

kv/.O o

Fig. 3. Locations of the dispersion branch for the I = 0 spin wave and the particle-hole continuum in the dimensionless w-k plane. 1) o is the Larmor frequency. A value of k~/f~ o sets the probing profile.

mode . The loca t ions o f these s ingular i t ies (or cutoff f requencies) , i l lus t ra ted in Fig. 3, are a s soc ia t ed with the fo l lowing phys ica l p h e n o m e n a :

(a) ~ l ( k x ) is the t h re sho ld f requency for the p r o p a g a t i o n o f the / = 0 mode . (No te tha t at large k this m o d e crosses into the p a r t i c l e - h o l e c o n t i n u u m and deve lops a large imag ina ry par t as a resul t o f L a n d a u damping . )

(b) ~ • = ~ : t : k_~v defines the l imits o f a reg ion where the d i rec t gener- a t ion o f s ing le -par t ic le exc i ta t ions by the dr iv ing field is fo rb idden .

Physica l ly , (b) is re la ted to the so-ca l led D o p p l e r - s h i f t e d - s p i n - r e s o n a n c e (DSSR) p h e n o m e n o n . '6 To u n d e r s t a n d DSSR, cons ide r a 3He quas ipa r t i c l e moving pa ra l l e l to the sur face with the Fe rmi ve loc i ty + v and in terac t ing


(Q) [W/cm z]

10-9.~

] 0 "10

<Q) [r,W/~m 2]

"50 l

.3

0 1 2 3 4 5 6 co/s

Fig. 4. Power absorbed per unit area of the 3He/transducer interface as a function of the frequency of the driving field.

with a magnetic surface disturbance of phase velocity oo/k~; the quasiparticle will sense a Doppler-shifted frequency given by ~oo = w ( l • One can see clearly from our analytic result and Fig. 3 that when too matches the spin-enhanced Larmor frequency, a DSSR singularity will occur; this is equivalent to condition (b) above.

Experimental measurements of E~t(k) would allow one to probe the dispersion relation for the longitudinal (l = 0) spin mode and would determine the Landau parameters (F~, F~) with spectroscopic precision. Further- more, observation of the DSSRthresholds f~• would provide an independent determination of the Fermi velocity v and F~.

To complete the discussion, one has to account for dissipative effects. Phenomenologically, we modify our collisionless treatment by adding a small, imaginary part to the frequency in our final result, according to the substitution w ~ oJ + i / r . This is only a qualitatively correct picture since certain conservation theorems must be built into the true collision integral.

The power absorption spectrum was obtained numerically from Eqs. (43)-(45) for the following conditions; we use F ~ = - 0 . 7 and F ~ = - 0 . 5 5


for the Landau amplitudes 17 (p = 0 atm) and we a s s u m e 1 0 3 Gauss for the dc field. The wires in the meander line coil are assumed to carry a current of 0.01/~A and the spacing is assigned a value of 10 ttm. The spin diffusion relaxation time ~'D has the experimentally measured value of rD = 3.0• 10 -7 T -2 sec mK2, ~s and the value at 2 m K is used for our phenomeno- logical parameter r. Figure 4 shows the calculated result. One notes that singularities occur involving all of the kinematical features contained in our qualitative discussion. The peak power (absorbed by a 1-cm 2 meander line transducer) at the singularity associated with the I = 0 spin mode generation is equal to about 10 -1~ W. Similarly, the peak power at the edges, connected with the part icle-hole excitations, is about 5 x 10 - '~ W. Both magnitudes of power are readily detectable. In both cases we satisfy the condition that the dynamic spin density is much less than the static spin density (induced by the static external dc magnetic field); this is a necessary condition for our linear theory to apply.

In conclusion, we have presented the results of a detailed calculation concerning a new experimental technique which is capable of directly probing single-particle and collective excitations of an interacting Fermi liquid.

One may speculate how the absorption spectrum would change if the boundary condition were no longer specular. It is likely that the single- particle excitation part of the spectrum (DSSR edges) would be strongly affected and possibly the edges would "wash out" due to the diffusivity of quasiparticle reflections from the boundary. On the other hand, the collective mode part of the spectrum (! = 0 spin mode absorption peak) would probably not be affected by diffuse boundary scattering. It is known experimentally that the zero-sound collective mode is not severely attenuated on reflection from a boundary that is flat on the scale of a wavelength, but surely not flat on the scale of an inverse Fermi wave vector (e.g., a quartz transducer).

Finally we note that the qualitative features of the proposed experiment should also apply to the spin-polarized 3He. In particular, rather than one edge associated with the DSSR, there should be two, corresponding to the limiting points of the sp in-"up" and sp in-"down" Fermi surfaces. A Fermi liquid theory of polarized 3He, given by Quader and Bedell, w would serve as a starting point for generalizing the present theory to the spin-polarized case.

8. EXCITING OF H I G H E R M O D E S

It would be of much interest if one could modify the experimental conditions so as to excite higher (I-> 1) spin modes, which probably exist

460 S .A . Bogacz and J. B. Ketterson

in 3He. Introducing an additional gradient of the dc field would be one of the nontrivial generalizations. To examine this situation, one can follow the scheme of the Leggett-Rice equation 2~ one step further. Apart from the usual equations of motion for the spin density ~ro and the spin current cr ~, one would also include the time evolution of the dyadic ~-moment defined as follows:

~r ~ ~ (o'~/3~/3~)~ (46)

The third equation of motion can be easily obtained from the LS equation by multiplying by /3"/3 ~ and averaging over the /3 direction. Using spin conservation and spherical symmetry for the collision integral J~, one can represent its dyadic moment J"~ by introducing one additional relaxation time r r as follows:

J ~ = - (o ~t3 _ ~ o 0 ) / r r (47)

This approach is analogous to the modeling of J~ by the spin diffusion relaxation time to. On eliminating o" 0 and cr "~, one obtains a Schr6dinger- like equation for the spin current {r~--a higher analog of the Leggett-Rice equation.

However, in the presence of the dc field gradient this equation does not simply decouple into separate equations of motion for div{r ~ and curl {r ~, corresponding to the longitudinal (1 = 0) and transverse (l = 1) spin waves, respectively, which suggests mixing of both modes. Whether an r f -dc field combination with the requisite features to excite an 1 = 1 spin mode exists is still under study.

A C K N O W L E D G M E N T S

This work was supported by the National Science Foundation under grant D M R 86-02857. We thank Jim Sauls for stimulating discussions.

R E F E R E N C E S

1. P. C. Fletcher and C. Kittel, Phys. Rev. 120, 2004 (1960). 2. R. E. Camley and D. L. Mills, Phys. Rev. B 18 4821 (1978). 3. G. Dunifer, S. Schultz, and P. H. Schmidt, J. AppL Phys. 39, 397 (1968). 4. H. C. Torrey, Phys. Rev. 104, 563 (1956). 5. N. Mashuhara , D. Candela, D O. Edwards, R. F. Hoyt, H. N. Scholtz, and D. S Shernll,

Phys. Rev. Lett. 53, 1168 (1984). 6 G. Tastevin, P. J. Nacher, M. Leduc, and F. Lalo6, 3". Phys. Lett. (Parlsl 46, L249 (1985). 7 L. R. Corruccini, Ph.D. thesis, Cornell University (1971). 8. V. P. Silin, Sot,. Phys. JETP 6, 945 (1958). 9. R. S. Fishman and J. S. Sauls, Phys. Rev B 31,251 (1985).

10. I. L. Bekarevich and I. M. Khalatnikov, Soy. Phys. JETP 12, 1187 (19611.


11. G. Baym and C. J. Pethlck, in The Physlcs of Llquld and Solid Helium, Part II, K. H. Bennemann and J. B Ketterson. eds. tWiley, 1978), Chapter 1.

12. S. A. Bogacz, Ph.D. thesis, Northwestern University (1986). 13. S. A. Bogacz and J. B. Ketterson, J. AppL Phys. 58, 1935 (1985). 14. S. A. Bogacz and J. B. Ketterson, J. Low Temp. Phys. 60, 133 (1985). 15. A. I. Fomin, Soy. Phys. JETP 34, 1371 (1972). 16. H. N. Spector and J. B. Ketterson, Phys. Rev. B I1, 3724 (1971). 17 D. S. Greywall, Phys. Rev. B 27~ 2747 {1983). 18 L. R. Corruccini, D. D. Osheroff, D. M. Lee, and R. C. Richardson, J. Low Temp. Phys.

8, 229 (1972) 19. K. F. Quader and K. S. Bedell, J. Low Temp. Phys. 58, 89 (1985). 20. A. J. Leggett and M. J. Rice, Phys. Rev. Lett. 20, 586; 21, 506 (1968).

surface coupling to collective and single-particle spin modes in normal 3he

Documents