IMPURITY-INDUCED SCATTERING RESONANCES AND LOCALIZED PHONON MODES IN INSULATING SOLIDS

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IMPURITY-INDUCED SCATTERING RESONANCES

AND LOCALIZED PHONON MODES IN

INSULATING SOLIDS

D. Mills

To cite this version:

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JOURNAL DE PHYSlQUE Colloque C 2, supplkment au no 2, Tome 28, Fkvr. 1967, page C 1-3

IMPURITY-INDUCED SCATTERING RESONANCES AND LOCALIZED PHONON

MODES IN INSULATING SOLIDS

Facult6 des Sciences,

Laboratoire de Physique des Solides, associC au C . N. R. S. 91

-

Orsay, France

Resume. - On consid2re l'influence d'une impureti5 isolCe sur les vibrations du reseau d'un cristal isolant. Pour des raisons de simplicit8, on se limite au cas d'une impurete differant par la masse seulement, situee dans un site de symetrie cubique. On discute la diffusion d'une onde ultrasonore par l'impurete, insistant particuli6rement sur le cas dans lequel se produit une resonance aigue de la section efficace de diffusion. On examine en detail le mouvement de I'impurete quand la frequence de l'onde incidente passe par la valeur correspondant a la resonance. On determine les conditions de l'apparition d'un mode localid en examinant les singularites de l'amplitude de diffusion en dehors de l'intervalle des frkquences permises pour le cristal h6te. Le changement de la densite d'etat du systBme est exprime en fonction du dkphasage dans la diffusion. On etudie l'influence des modes localis6s et des resonances de diffusion sur le spectre d'absorption infrarouge du cristal h6te, en pla~ant l'accent sur une description physique simple du mouvement de l'impurete par rapport a ses voisins.

Abstract. - We consider the influence of a single impurity on the lattice vibrations of an insulating crystal. For simplicity, we confine attention to the case of a mass defect impurity placed at a site of cubic symmetry. The scattering of an incident ultrasonic wave from the impurity is discussed, with particular emphasis on the case in which a sharp resonance in the scattering cross section is produced. We discuss in detail the motion of the impurity as the frequency of the incident wave passes through the resonance frequency. The condition for the occurrence of a localized mode is found by examining the singularities in the scattering amplitude outside the range of allowed frequencies of the host lattice. The change in density of states of the system is expressed in terms of the scattering phase shift. The influence of the localized modes and scattering resonances on the infrared absorption spectrum of the host is considered, with emphasis placed on a simple physical description of the motion of the impurity relative to its neighbors.

I. Introduction.

-

In this paper, we shall discuss the effect of impurity ions on the lattice vibrations of a crystal. We consider a n insulating crystal which has been doped with a small number of ions which differ in mass o r electronic structure from the ions of the host crystal. For simplicity, we assume the impurity concentration is sufficiently low that correlations between the impurities may be neglected. We also confine our attention to the case where the impurities occupy sites of the host lattice.

I n the absence of any impurity ions, one may in principle diagonalize the Hamiltonian of the crystal in a straightfoward manner, provided the terms in the potential energy of the lattice higher order than quadratic in the ionic displacements are neglected [I]. The

(*) National Science Foundation Postdoctoral Fellow.

presence of the impurity will introduce a perturbation into the Hamiltonian. If the impurity is a n isotope of the ion of the host crystal it replaces, the kinetic energy term will be changed. If the impurity differs in electronic structure as well as in mass, the potential energy term also will be affected, since the effective force constants coupling the impurity to its neighbors will be altered.

The perturbing terms in the Hamiltonian describe a process in which a phonon of wave vector k is scattered to another state k'. If one treats these terms in the first Born approximation, one finds the lifetime of an acoustical phonon of frequency 52 is proportional

to 0-", if the wavelength of the phonon is long compared t o the linear dimensions of the region affected by the perturbation. The presence of the impurity scattering affects the transports properties of the crystal in a

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C 1 - 4 D. L. MILLS

marked manner. A complete discussion of the effect of such scattering on the properties of insulating solids has been given by Ziman ['I.

Recent experimental and theoretical work has revealed that under certain conditions, a number of very striking effects may be associated with the presence of the impurity [3], [ 5 ] . One often finds that the cross section for the scattering of acoustical phonons from the impurity deviates strongly from the fi4

dependence mentioned above, even at frequencies low compared to the Debye frequency [6, 71. If the impu- rity is much heavier than the ion it replaces [6], or if the effective force constants which describe the coupling of the impurity to its neighbors are small compared to those of the host lattice [5,7] a strong resonance in the scattering cross section may occur for frequencies small compared to the Debye frequency. The presence of such a resonance may have strong effects on the properties of the system, even at low temperatures [9], and will produce a peak in the optical absorption spectrum [3].

In addition to scattering resonances in the continuum of phonon frequencies of the host lattice, the Hamiltonian (with neglect of the anharmonic terms) may possess eigenmodes with frequencies outside the range of allowed frequencies of the host lattice, i. e. in a crystal containing two atoms per unit cell, the impurity may resuIt in an eigenmode in the gap between the acoustical and optical branches, or above the optical branches. The displacement associated with such a mode is localized in the vicinity of the impurity. Such localized modes will also contribute peaks in the optical absorption spectra [4]. The effect of a localized mode about a paramagnetic impurity on the spin- lattice relaxation rate of an impurity center has also been observed [lo]. Localized modes can be produced when the impurity is lighter than the ion it replaces. In this work, we shall make no attempt to give a rigo- rous discussion of the impurity problem for the most general case, since this has been done elsewhere [3]. Rather we shall attempt to explore the influence on the properties of the crystal of the simplest type of impurity, in which the only perturbation is that produced by the mass difference.

11. The Effect of an Isotopic Impurity - General Considerations. - Let us first consider the pro-

perties of the undoped host crystal. We suppose the crystal is constructed cf N unit cells, each containing

z ions. We employ Born-von Karman boundary condi-

tions. The crystal may be described by a Lagrangian

where T is the kinetic energy, and V is the potential energy.

In V, we retain only terms quadratic in the displacements of the ions from their equilibrium position. For this system, the equations of motion may be solved exactly, in principle [I]. One finds 3 z branches in the phonon spectrum, with three acoustical branches, and 3 z - 3 optical branches. The frequency of the mode of wave vector k associated with the branch

l

will be denoted by

a,,.

If we excite the mode k l , then at any time the displacement v(na) from the equilibrium position of the ion at site a of the nth unit cell may be written in the form [l] :

v(n.) = (const.) e""" e x p ( i k . ~ ( n a ) )

.

(I)

JK

In eqn (I), m, is the mass of the ion at site a, and R(na) is a vector to the site na. The eigenvector ekI(a) is normalized so that

and satisfies the closure relation

In the last equation, I is the unit matrix, and on the left side ek,(a) ekI(/3) denotes the outer product of the vectors ek,(a) and ekn(p).

Since the set of eigenvectors of eqn (1) form a complete set, we may always describe the state of the crystal by an expansion of the form

Since v(na, t) must be real, P(: = (P - kil, in eqn. (2) ma

is the mass of the ion of the host crystal located at site a within the unit cell. We regard eqn (2) as a transfor- mation from the 3 z N coordinates ri(na, t) to the new set of normal coordinates (~,,(t). In terms of the new coordinates, the Lagrangian of the host crystal becomes.

Now let us introduce a single impurity into the lattice. We assume the impurity differs from the host ion it replaces only in mass. We choose the origin of the coordinate system so the impurity is located at the origin, and assume it is located at site i within the unit

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IMPURITY-INDUCED SCATTERING RESONANCES C 1 - 5

In the presence of the impurity, the Lagrangian of the crystal become L = Lo

+

6L, where

1

6L = - (m, - m,) i ( n ~ ) ~ =

2

where ak,,, ; kn(i) = ek,(i). eV,,(i). On the right hand side of eqn (4), we define E = m,/m,

-

1, where m, is the impurity mass, and m, the mass of the ion of the host crystal which has been replaced by the impurity. The equation of motion of cpk, may easily be obtained from

One finds

d2 2 E

- ( P k l

+

Q k l ( P k l = -

dt2 N k','

We will consider the scattering of a phonon from the impurity. In this way, we may study the scattering resonances mentioned above, and we will find the condition for the occurrance of localized modes by studying the poles in the scattering amplitude which may occur outside the bands of allowed frequencies. We assume qk, varies in time as exp(iQt). Then eqn (5) may be written in the form

The factor of i 6 has been inserted in the denominator of eqn (6) so the scattered wave will obey the outgoing wave boundary condition.

We assume the experimenter drives the mode of wave vector k , and polarization

A,,

and we find the form of the scattered wave. The solution of eqn (6) will then have the form (P,, = cp,

a,,,

+

(P',":, where (P',": is the amplitude of the scattered wave. Then

ti

satisfies

The first Born approximation may be obtained by neglecting the second term on the right hand side of eqn (7). If we consider an acoustical phonon with

wavelength long compared to the lattice constant, one may show in a straightforward manner that in the first Born approximation the scattering cross section is proportional to Q ~ .

Eqn (7) may be solved without difficulty for (P(,",,. In order to exhibit the result in a simple form, we assume the crystal to have cubic symmetry. We denote the amplitude of the scattered wave in the first Born approximation (the first term on the right of eqn (7)) by q g ) , and we define the function D(Q) by

Notice that D(Q) is a function only of the frequency of the wave.

Then (for a crystal with cubic symmetry) the solution of eqn (8) has the simple form

The effect of the higher order terms in eqn (7) is to multiply each Fourier component of the scattered wave q g ) by the factor (I

-

&D(Q))-l.

Since D(Q) depends only on the frequency of the incoming wave, upon substitution of eqn. (9) into eqn (2) we find

In eqn. (lo), r("(na) is the displacement of the ion at site a of cell n due to the scattered wave, while r(B)(na) is the displacement of the same ion computed in the Born approximation.

We note that if the frequency Q of the driving wave lies inside of one of the phonon bands of the host crystals, the function D(Q) is complex. We write

where

and

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phase as well as amplitude. The phase difference @(Q)

between v(")(na) and ~ ( ~ ' ( n c t ) is given by

@(Q) = tan-'

1 :I::52)

I

.

With the formulas of eqns (10)-(12), we are in a position to discuss the occurrance of scattering resonances and localized modes. Before proceeding with this discussion, it may be useful to make some comments about these results.

The physical assumptions underlying the first Born approximation have been discussed in many texts [ I I]. If one considers a wave which is scattered by some potential, then making the first Born approximation is equivalent to assuming the potential so weak that in the region where the potential is non-zero, the amplitude and phase of the wave is well approximated by that of the incident wave. In the problem considered here, we are sending a wave of wave vector

k,

and frequency Q,,,, down the crystal. The first Born approximation assumes that the motion of the impurity in response to the driving wave is well approximated by the motion of the host ion it replaces. We then compute the acoustical energy radiated to find the cross section. The phase angle (P(52) of eqn (12)

then describes the phase of the motion of the impurity relative to that executed by an appropriate ion of the host crystal placed on the impurity site, while

1

-

~D(52)

1

is a measure of the relative amplitudes. Let us examine the properties of eqn (10)-(12) for

the case E > 0 , when a heavy impurity has been placed in the lattice. As 52 + 0, both R(Q) and I(Q) -+ 0.

For small 52, R(Q) is positive, and will increase with Q

roughly as Q2. The function I(52) will vary as 52 n(SZ),

where n(52) is the density of states. For small 52, where

the phonon dispersion relation varies linearly with the wave vector, we have n(Q)

-

52' and I(C2)

-

Q3.

If 8 is sufficiently large and positive, then there will be

a frequency 52, small compared to the maximum frequency of the acoustical branches for which zR(S2,) = 1,

and I(52,) small. In the vicinity of this frequency, the

amplitude of the scattered wave will be strongly enhanced over the value computed in the first Born approximation. The phase angle @ ( a ) will increase

from zero, pass through 4 2 to a value approaching 71

in this region.

For frequencies near Q,, we may write

and

Since the scattering cross section o(Q) is proportio-

nal to the square of the amplitude of the scattered wave, we have in the region near Q,

where

dB)(SZ)

is the cross section computed in the first Born approximation.

Eqn (13) describes a peak in the cross section with

a Lorentzian shape. The half width of the peak at half maximum is

r

= I(SZR)/Rf(Q,), while

Eqn (13) is a satisfactory approximation to the cross

section throughout the resonance region only if

(r/52,) 4 1. Since for small 52, I(52)

-

523, and

R(52) 2: Q2, one has (TI52)

-

52 in this region. Thus

if the resonance occurs for 52, small, the criteria

(r/52,) & 1 may be satisfied. Notice that the half

width

r

of the peak is equal to (d@/dSZ

I,,)-'.

Thus if the resonance is sharp and well defined, the phase angle passes rapidly through n/2.

Thus if we place a heavy impurity in the lattice, we have the following picture of its behavior in response to an applied acoustical wave. At very low frequencies, the impurity will oscillate in phase with the applied wave, and the scattering cross section is well approximated by the value obtained from perturbation theory. As the frequency of the wave is increased, and if E is sufficiently large, there will be a narrow range of frequencies centered about a frequency 52,

--

1 / ~ ' / ~

within which the scattering cross section will be shar- ply peaked. As the frequency of the driving wave passes through this range, the phase of the motion of the impurity relative to the driving wave shifts rapidly from a value close to zero, through 4 2 and to a value

close to n, the scattering cross section will be a maximum when the impurity motion is out of phase with the driving wave by 4 2 .

In the preceeding paragraph, we stated that on the low frequency side of the resonance, the scattering cross section was well approximated by the value computed from perturbation theory. We should point out that on the high frequency side, this is not so. The impurity vibrates roughly 180° out of phase with the

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IMPURITY-INDUCED SC ATTERING RESONANCES C 1 - 7

In the next section, we shall evaluate R(52) and Z(52) for a crystal containing a single atom per unit cell, with a phonon spectrum of the Debye type. For frequencies Q

<

the Debye frequency a,, we shall find Z(9) smaller than R(52) by a factor of 52/52,. Then if &R(OR) = 1 for an 9, OD, for frequencies 52 > ft,

but still less than a,, we will have R(52) 1 (recall that for small 52, R(Q)

-

Q') and Z(O)

<

R(52). The cross section o(Q) will then be approximated by

Since R(9)

--

Q', on the high frequency side of the resonance, the cross section is roughly independent of frequency for some range of frequencies.

Impurity-induced resonances of low frequency have been observed by several workers. For example, Sievers and Takeno [5] have observed such a resonance at 16 cm-' in Li doped KBr. This observation is surprising, since the discussion above indicates that low lying resonances may be associated with heavy impurities. Sievers and Takeno attribute this resonance to a large decrease in the force constant associated with the Li ion.

Let us assume the frequency 52 lies outside of the allowed frequency range of the host crystal. Then the imaginary part Z(52) of D(52) vanishes identically, and

Recall that dB)(na) is proportional to the amplitude of the driving wave. If there exists a frequency outside the phonon bands for which

then the system will respond to an infinitely weakper- turbation. The frequencies which satisfy eqn (14) are the frequencies of the localized modes, which are eigenmodes of the crystal Hamiltonian. For the cubic crystal, each localized mode is three fold degenerate, with P-like symmetry.

If 52 is larger than the maximum frequency Q, of the crystal, we see R(52) < 0. Thus localized modes of frequency Q > 52, can occur only if r

<

0. In the three dimensional monatomic simple cubic lattice, Montroll and Potts [I21 have shown that the displacement associated with the localized mode falls off as exp(- BR)/R at large distances from the impurity, where B is a function of E .

We refer the reader to reference [3] for discussions

of the occurrance of localized modes in some simple models of crystals, such as the one dimensional chain, and the three dimensional simple cubic lattice.

Sievers, Maradudin and Jaswal [4] have observed peaks in the infrared absorption spectrum of C1 doped KI which they attribute to localized modes associated with the C1-ion, located in the gap between the acoustical and optical branches of the phonon spectrum. Studies of the solutions of eqn (14) by these authors indicate that the observed frequencies are well accoun- ted for by the simple mass-change model.

For a crystal of cubic symmetry perturbed by a single mass defect, it has been shown [13] that the change in density of state An(Q) which results from the presence of the impurity may by written in the form

where Q, is the phase angle defined in eqn (12). From this result, we see that a sharp in-band resonance of the type discussed above, in which @ increases rapidly through 7112, will give rise to a peak in the density of states centered about 52,.

Suppose we have a frequency interval A52 bounded by the frequencies Q1 and Q2. We assume 52, and B2 lie outside the allowed frequency bands of the host crystal. Let the presence of the impurity cause

P

localized states to split off from the bands, and move outside the interval AQ. Then upon integration of eqn (15), we find

We can draw a number of conclusions from this result. Suppose SZ1 = 0, and 9, is the maximum fre- quency of the acoustical phonon branches. If a loca- lized state emerges from the top of the acoustical band (recall the localized mode is three-fold degenerate for the cubic crystal), then @(a,) = n. We know @(Q) = 0, since I(52) -+ 0 as 52 + 0. Since the phase shift equals

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MILLS

may be sharp and well defined. We note that in addition to the localized mode between the optical and acoustical branches of C1- doped KI, Sievers et a1.141 have found a region of strong infrared absorption associated with C1- impurity for frequencies less than the maximum frequency of the acoustical branches. If no bound states are extracted from the region 1152, then necessarily @(a1) = @(52,). If we consider a simple monatomic cubic crystal with a heavy impurity, then from the discussion above we see it is not possible for a localized mode to occur, since R(52)

<

0 for 52

>

52,. Then eqn (16) implies @(52,) = 0. If the presence of the impurity results in a low frequency scattering resonance of the sort described above, we see that there must exist at least one more in-band frequency for which @(a) = n/2, so the possibility of two or more sharp in-band resonances occurs. If there are two frequencies for which @ = 4 2 , the high frequency intersection will be a region where Qi is decrea- sing with increasing. If the high frequency resonance is well defined, there will be a sharp dip rather than a maximum in the density of states.

We conclude this section by emphasizing the simi- larity between the work discussed in the present section and the work of Friedel and co-workers [14] on the electronic structure of dilute metallic alloys.

111. Application of the Theory to a Simple Model.

-

In this section, we apply the theory discussed above to a simple model for which we may easily obtain analytical expressions for R(52) and I(52). We consider an isotropic crystal with a phonon spectrum of the Debye type. Then

a,,

=

uk

for 0 c

k

< k,,

where

v

is the sound velocity (assumed independent of

A),

and

k,

is the Debye wave vector. If n is the

number of atomslunit volume, then

k;

= 6 z 2 n. One may easily show that [6]

and

A plot of R(Q) is given in figure 1.

Let us first consider the case of a heavy impurity, for which E

>

0. Then, as mentioned in the previous section, the equation ~R(52) = 1 can have no solutions for 52

>

OD, since R(Q)

<

0 in this region. Thus there are no localized modes, so we examine the possibility for the occurrence of scattering resonances. From the figure, we see that the maximum positive

value assumed by R is 0.63, for Q/Q, = 0.62. Then the phase angle cannot pass through n/2 for

FIG. 1.

-

The function R(Q) for the Debye model, plotted as a function of frequency.

If E is large compared to unity, a solution to eqn (14)

occurs when (52/52,)

--

l/J<. At this frequency, one finds

Thus, if E % 1, a well-defined scattering resonance of

low frequency is produced by the impurity. The existence of this resonance was first pointed out by Brout and Visscher [6].

When E is large and positive, the second solution

fo eqn (14) discussed in section II wili occur for SZ

near the zero of R(52). Here

@(a)

passes through 7112 from above, so the impurity-induced change in the density of states is negative. The zero of R(52) occurs for Q/QD

---

0.83. For this value of 52/QD, one finds T/52

-

0.12 z. Thus the resonances should be reaso- nably sharp. The value of r/52is not large because for this value of 52/52,, the derivative dRld52 is large. We next consider the light impurity case, where E < 0 Notice that as m, + 0, E-I -+

-

1 from below, so E is always less than

-

1. For 52 > a,, R(52) is negative, and for 52 & a,, R(52) -+ - 1 from below. Thus if we substitute a light impurity into our Debye solid, a localized mode with frequency greater than 52, will always be produced. If E is near - 1 (very light impurity), one finds the frequency 52 of the localized mode is (a/&) (11

41 -

I

E

I).

For E small, the frequency of the mode differs from the Debye frequency by an amount

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IMPURITY-INDUCED SCATTERING RESONANCES C 1 - 9

Our discussion of the previous section indicates that an in-band solution of eqn (14) must also occur. For

-

E close to unity, one finds'the resonance centered

about 52/S2,

--

0.9, with T/S2

--

0.7 TC, while for E

small, the resonance occurs when

52,

-

n

= 2 a D exp(- 213

1

E

I).

In the last-mentioned case,

In the Debye model, the in-band resonance appears to be sharp in the two limits considered. As E-'

varies from - 1 to - co, the resonance frequency increases from Q/Q, = 0 . 9 to 1 .O. Since Ir(S2) varies little in this region, and R(S2) increases monotonically with a singularity at 52 = Q,, one finds the high-frequency in-band resonance to be sharp at all frequencies In order to treat a more realistic model of a crystal in a quantitive way, it is necessary to ressort to numerical methods in order to evaluate R(0) and I(52). Summaries of such calculations, with reference to the literature may be found in the work cited in footnote [3]. Before concluding this section, we shall make a few comments concerning the results we obtained from the Debye model. We found that a localized mode with frequency 52, > 52, would always occur for E < 0.

This conclusion follows because of the logarithmic singularity in R(S2) as Q -+ Q,. From the work of

Friedel et al. we see that this behavior is a consequence of the jump discontinuity in the density of states for the Debye spectrum. In a real crystal, as the frequency Q approaches the maximum frequency of the band

5

the density of states goes to zero smoothly. If the frequency varies quadratically with the distance in k space from the band maximum of band

L

for small excur- sions from these points, then the contribution to the density of states from this band will vary as ( ~ ~ - 5 2 ) ' ~ ~ near Q,. Then (*) R'(S2) experiences a logarithmic singularity.

Now let us consider the case of a cubic crystal containing one atom per unit cell. As 52 -+ the maximum

frequency Q, of the host crystal from above,

-

R(Q) will increase to some value

-

R(QM), which will be a local maximum. If we place a light impurity in the crystal, a localized mode will occur only if

(*).

. .

from reference [14] we see that K(Q) is continuoils near

Q M , while

...

Thus for a realistic phonon spectrum, it is ncessary that Am satisfy

for a local mode to occur.

Because of the singularities in R'(Q) near band edges (or critical points in the phonon spectrum), in- band resonances wich occur near these frequencies may be well-defined and narrow.

IV. The Influence of Scattering Resonances and Localized Modes on Some Properties of the Crystal.

-

In this section, we discuss briefly the influence of the scattering resonances and localized modes induced by an impurity on some properties of the crystal. We consider the infrared absorption coefficient, the thermal conductivity, and the spin-lattice relaxation rate of a paramagnetic impurity.

A) THE INFRA-RED ABSORPTION SPECTRUM.

-

We consider a cubic crystal of the alkali halide structure, with two ions per unit cell of opposite charge. We desi- gnate the two sites by (+) and (-), and the masses by

m + and m-. Let a single mass defect impurity be located at a site normally occupied by an ion of mass m + .

We assume a plane polarized electric field is applied to the sample, with wave vector q directed parallel to the x axis, and the field parallel to the y axis. The Lagrangian L of the host crystal plus impurity is now augmented by the term

where E(na) = GEo exp(iq. R(na), and q(,, is the charge of the ion at site a within the unit cell. We assume

q+ =

+

e, and q- =

-

e.

From eqn (2) we find

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C 1 - 1 0 D. L. MILLS

I

eo,,(+)leo,,(-)

I

= ( n ~ - / m + ) ~ . In the limit as q + 0, eqn (17) becomes

where /i is the reduced mass of the unit cell.

From the Lagrangian L

+

L,, we may find the equation of motion for the amplitude p,, in the presence of the electric field. If one assumes the field has been switched on adiabatically from time t = - co,

so all quantities vary as exp(i52t

+

at), 6 > 0, then the equation of motion may be solved in the same manner as for the scattering problem of the second section. One finds

1 E

a2

eOtl(i). ek(i)

- - -

1 - E D ( ~

+

i d ) N

-

(52

+

I

where Q,,, is the frequency of the transverse optical mode of zero wave vector.

The energy loss of the wave (per unit time) as it passes down the sample is

We use the results of eqn (la), and notice that the first term describes the absorption by the host crystal, while the second term will give the impurity-induced absorption. If we divide dE/dt by the energy V E ; / ~ rc

stored in the volume of the crystal, we find the lifetime of the wave. We write the attenuation coefficient in the form a(0) = ao(0)

+

Aa(0), where [15]

and the contribution from the presence of the impurity (for 0 # Qo,)

4 nne2 E

Aa(0) = - -

"

1m

{-

m(+) N

[ ~ i t

-

o2l2

i

-

ED(Q

+

id) In eqn (19), n is the number of unit cells/unit volume.

From eqn. (19. b), one sees that in addition to the

absorption at associated with the host crystal, there is impurity induced absorption for all frequencies in the phonon spectrum where Z(O) is non-zero. Because of the denominator

there will be strong absorption for frequencies near the fundamental frequency Q,,. The presence of the impurity tends to smear out the sharp absorption line of the undoped crystal [I 61.

Since the form of eqn. (19. b) involves the same factor [l

-

&D(0

+

iL?)]-' encountered in our discussion of the acoustical attenuation coefficient, a well- defined resonance in the continuum will produce a peak in the infra red absorption coefficient. In the vicinity of such a resonance, Aa(0) may be written

where 52, is the frequency of the resonance, and

f ( 0

-

52,) is a Lorentzian function with halfwidth at half maximum of

r

= I(O,)/Rf(S2,). The function

f ( 0 ) is normalized so that its integral over all frequency is equal to unity. The maximum in the absorption occurs when r(")(na, t ) is out of phase with the driving wave by 4 2 , so that the current q(,,

{

dr(") (na)/at

}

induced by the impurity is in phase with the electric field.

A localized mode will also produce a peak in the absorption spectrum. The contribution from a localized mode a frequency 52, is easily seen to be

Studies of infra red absorption spectra have provided us with a considerable amount of information on localized modes and resonances associated impurities and defects in crystals. We refer the reader to the studies by Sievers and co-workers [4,5] on doped alkali halide crystals for examples of this type of work.

B) THERMAL CONDUCTIVITY.

-

The existence of a scattering resonance can affect the thermal conductivity of a solid in a pronounced way. A resonance will

produce a dip in the thermal conductivity when kTand

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IMPURITY-INDUCED SC ATTERING RESONANCES C 1 - 1 1

Schwartz and Walker [18] have observed dips in the thermal conductivity of KC1 which they interpret as resonant scattering of phonons from lattice vacancies,

as predicted by Krumhansl [19].

C) THE SPIN-LATTICE RELAXATION KATE.

-

Let us suppose we dope an insulating crystal with a number of paramagnetic impurity ions. Then in the presence of a magnetic field, we may measure the spin-lattice relaxation rate of the ion 1201. The principal mechanism by which an ion interchanges energy with the lattice was pointed out twenty-five years ago by Van VIeck[21] The motion of the nearest neighbor ions, which results from the thermal motion of the lattice, causes a fluctuating component to be present in the crystalline electric field. The electron spin feels this fluctuating field through the spin-orbit interaction. Thus the thermal motion of the nearest neighbors may induce a transition between states of the Zeeman multiplet associated with the ground electronic state.

One generally describes this interaction by means of an effective spin Hamiltonian, in which an expansion in powers of the displacement of the ions from their equilibrium position is employed [22]. For most applications, it is sufficient to discard terms of higher than second order in the ionic displacements.

In most standard calculations of the spin-lattice relaxation rate, one ignores the influence of the mass and force constant changes associated with the impurity on the dynamics of the lattice. We have seen in the discussion above, that in the immediate vicinity of the impurity, the motion of the lattice can be very strongly affected. A complete discussion of the spin- lattice relaxation rate requires the inclusion of effects of the impurity on the lattice motion.

The effect of a localized mode on the relaxation rate has been discussed by Klemens [22, 241. If we keep only terms quadratic in the ionic displacements in the Lagrangian of the lattice with the impurity, then we have seen in our earlier discussion that the localized mode is an eigenmode of the system, and hence has zero width in frequency. Since the frequency of the localized mode is very large compared to the spin resonance frequency, the spin is unable to interact with the localized mode, since energy-conserving spin transitions are not possible. It is necessary to include the effect of anharmonic terms in the potential energy in order to compute the relaxation rate. If 0, is the frequency of the localized mode, one finds a contribution to the relaxation rate proportional to

The effect of a localized on mode paramagnetic

relaxation has been observed by Feldman et al. [lo]. Calculations of the effect of a sharp, well-defined resonance in the continuum of phonon frequencies on spin-lattice relaxation have recently been performed [9]. The relaxation rate wich results from the terms in the spin Hamiltonian linear in the strain will not be greatly affected by the presence of the resonance, since these terms describe processes in which the spin flips, emitting or absorbing a phonon with a frequency equal to the spin resonance frequency (the direct process). The spin resonance frequency is much lower than the frequencies 52, for which scattering resonances may occur, so one does not expect the propecties of these low frequency phonons to be strongly affected. However, the two phonon Raman process, which with the neglect of the effect of the impurity on the lattice vibrations gives a contribution to the relaxation rate proportional to T7 (for a non-Kramers ion) or T9 (for a Kramers ion) at low temperatures, may be,strongly affected by a resonance since the Ramah''pr6cess allows the spin to sample the whole spectrum of thermally excited phonons. As we have seen, for frequencies in the resonance region, the amplitude of the motion ot the nearest neighbors relative to the impurity may be greatly enhanced when compared to the relative amplitude in the host lattice. Consequently, a resonance of frequency 0, may strongly affect the relaxation rate, even when kT @ ti0,.

One the basis of a simple model of a non-Kramers ion [25], one finds that (to within a constant multi- plicative factor) the matrix element M(k f + kfJ) for scattering a phonon from state k to state k', accom- panied by a spin flip, may be written in the form

where M,(k

5.

-, kt

1)

is the matrix element computed for the above mentioned process in the absence of force constant changes or mass changes.

At very low temperatures (kT

e

EL?,),

one finds in addition to the T~ term a term which varies expo- nentially with temperature as

Numerical estimates indicate that this term may domi- nate the relaxation rate at the upper end of the liquid helium temperature range.

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C 1 - 1 2 D. L. MILLS

excited electronic state which is separated from the ground state by an energy less than Q,, where Q, is the Debye frequency, then the spin may relax by making a real transition via phonon absorption to the excited state, and then decaying to the ground multiplet by phonon emission, with a spin flip occurring in the process. This process gives a contribution to the relaxation rate proportional to exp(- AlkT), where A is the excitation energy of the electronic state. The Orbach process plays an important role in the relaxation of rare earth ions, which often have such a low lying electronic excited state in the presence of the crystalline field. It may be difficult to distinguish between the contribution to the relaxation rate from the Orbach process, and the resonance induced relaxation, unless estimates of either Q, or the crystal field splittings are available.

APPENDIX

Relationship between the change in density of states and the phase angle

@(a)

In this appendix, we derive the relation between the change An(Q) in density of states resulting from the presence of the single mass defect and the phase angle @(Q) discussed in the text. Our discussion follows reference [3].

In the unperturbed crystal, the eigenfrequencies are found by solving an eigenvalue equation of the form

det

{

Mo(Q2)

)

= 0

.

We denote the roots of this equations by go', where

j runs from 1 to 3 zN.

In the perturbed crystal, the eigenvalue equation becomes

det

(

M(Q2)

)

= 0 ,

where M(Q2) = M,(Q')

+

d ~ ( 0 ~ ) . The eigenfrequencies of the perturbed crystal will be denoted by

~2;.

Define a matrix A(Q2) by the equation ~ ( ~ 2 2 ) = Mo(Q2) A(Q2)

.

Then

det

{

A(Q2)

)

= det

{

~ ( 0 ~ ) )/det

{

iV0(Q2)

)

( A . 2) = const. x

JJ

( a 2 - S Z ? ) / ~ (i12 - 5292). The change in density of states in the presence of the impurity is

An(Q) = C d ( Q 2 - 0;)

-

s(Q2

-

a ? ) .

j j

From eqn (A. 2), we may easily show that 1 d

An(Q) = -- - Im [In (det ~ ( 5 2 ~

+

id))]

.

( A . 3) n dQ

For the perturbed crystal, the secular equation may be derived from the matrix constructed from the coeffi- cients of cp,, in eqn (5) of the text. In a representation in which the rows and columns are labeled by (kd), one finds from the definition of eqn (A. 1) that

To compute the determinant of A, we transform the result of eqn (A.4) into a representation spanned by the vectors

U(na) =

Jm,

r(na).

Each element A(na, m p ; Q2) is then a 3 x 3 matrix, with subscripts referring ot the three coordinate axes. We find

The impurity is assumed located at site i within the unit cell 0 ; I is the unit matrix, and eka(a) eka(i) the outer product of the vectors eka(a) and eka(i).

In this representation, one easily sees that det { A(na, mp ; i12) ) = det ( A(oi, oi ; Q2) }

on the left hand side of the last equation, A(oi, o i )

refers to the 3 x 3 matrix constructed from eqn (A. 5) by setting n = m = 0, and a =

P

= i. For a cubic crystal, A(oi, oi ; Q2

+

i6) is diagonal with all djagonal elements equal to [I

-

&D(Q

+

id)].

det

{

~ ( 5 2 ~

+

id)

}

= [1

-

&D(Q

+

i6)I3

.

By inserting this last result into eqn ( A . 3), we find

the desired result

[I] See Chapter I of ZIMAN (J. M.), Electrons and Phonons

(Oxford University Press, London 1963).

[2] See Chapter VIII of reference [I].

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IMPURITY-INDUCED SCATTERING RESONANCES C 1 - 1 3

article by MARADUDIN (A. A.) in Astrophysics

and the Many-Body Problem Benjamin (W. A.,

Inc., New York 1963) ; see also the discussions by MARADUDIN (A. A,), MONTROLL (E. W.), and WEISS (G. H.) in Suppl. 3 of Solid State

Physics (Academic Press, New York, 1963).

[4] SIEVERS (A. J.), MARADUDIN (A. A.) et JASWAL (S. S.),

Phys. Rev., 1965,138 A 272.

[5] SIEVERS (A. J.) et TAKENO (S.), Phys. Rev., 1965, 140, A 1030.

[6] BROUT (R.) et VISSCHER (W.), Phys. Rev. Letters,

1962, 9, 54.

[7] VISSCHER (W.), Phys. Rev., 1963,129,28. [8] POHL (R. O.), Phys. Rev. Letters, 1962, 8, 481. [9] MILLS (D. L.), Phys. Rev., 1966, 146,336.

[lo] See FELDMAN (D. W.), CASTLE (J. G.), Jr., et MURPHY (J.), Phys. Rev., 1965, 138, A 1208.

[ l l ] For example, see Chapter VII of SCHIFF (L. I.),

Quantum Mechanics (McGraw-Hill Co New York

1955).

[12] MONTROLL (E. W.) et POTTS (R. B.), Phys. Rev.,

1955,100,525.

[13] See p. 219 of the article by MARADUDIN cited in reference [3].

[14] FRIEDEL (J.), GAUTIER (F), GOMES (A. A.) et LEN-

GLART (P.) (to be published).

[15] Note that in the present work, the mean free time is defined to be the time required for the energy density of the wave to decrease to

lle

of its initial value.

[16] A more elaborate calculation of the infra red absorption coefficient for a finite concentration of impurities has been discussed by MARADUDIN. We refer the reader to the discussion beginning on p. 253 of the first article cited in reference [3]. Our result of eqn. (19 b) agrees with his result, except for frequencies Q which differ from 520, by an amount less than

An,

= sZ, c

I

1 - ED(Qo,

+

is) I

where c is the impurity concentration. Maradu- din's more sophisticated analysis shows that the presence of a finite concentration of impurities

removes the delta function singularity associated with the host crystal at Q = Dot, and replaces it by a Lorentzian function peaked at a,,, with a width of the order of

[17] See also WALKER (C. T.) et POHL (R. O.), Phys. Phys.

Rev., 1963, 131, 1963.

[IS] SCHWARTZ (J. M.) et WALKER (C. T.), Phys. Rev. Letters, 1966, 16, 97.

[I91 KRUMHANSL (J.), Proceedengs of the International.

Conference on Lattice Dynamics. Copenhagen

1963, edited by WALLIS (R. F.) (Pergamon Press, Oxford 1965), p. 523.

[20] For a concise discussion of the theory of electron spin- lattice relaxation in insulating salts, and a discussion fo experimental measurements on a number of systems, we direct the reader to pages 33-69 of JEFFRIES (C. D.), Dynamic Nuclear

Polarization (interscience Publishers, New York

1963). This work contains many references to the literature.

[21] VAN VLECK (J. H.), Phys. Rev., 1940,57,426.

[22] For a discussion of the effective spin Hamiltonian for a transition metal ion placed in a diamagnetic lattice, see MATTUCK (R. D.) et STRANDBERG (P. W. P.), Phys. Rev., 1960, 119, 1204.

[23] KLEMENS (P. G.), Phys. Rev., 1962,125, 1795. [24] KLEMENS (P. G.), Phys. Rev., 1965, 138; A 1217. [25] From symmetry arguments, one may show the relaxation rate is unaffected by the resonance, if the impurity differs only in mass from the ion it replaces, and is also located at an inversion center of the crystal. The model of reference [9] may be used to represent, in a phenomenological way, the effects of force constant changes, or placing a mass defect at a site with no inversion symme- trv.