density of states and specific heat of elastic vibrations in layer structures

Superlattices and Microstructures, Vol. 3, No. 4, 1987 445

DENSITY OF STATES AND SPECIFIC BEAT OF ELASTIC VIBRATIONS IN LAYER STRUCTURES

R. Hots and R. Siems Fachbereich Physik, Universitat des Saarlandes,

Federal Republic of Germany

Received 1 July 1987 by Gottfried Dohler Accepted 4 August

On the basis of a determination of normal modes in materials consisting of periodic arrangements of macroscopic layers (of period d), the low frequency density of states and the corresponding low temperature specific heat were calculated numerically. The average temperature dependence of the latter changes in the vicinity of a characteristic temperature TO (proportional to l/d), from a low temperature (cr0T3)- to a higher temperature (~T~+ST*)- law. Depending on the material parameters, B may be positive (especially if Stoneley waves are present) or negative. The coefficients a and a0 can differ by a factor larger than 2. Characteristics of thick and thin layers and the implications of the results on the interpretation of experimental data are discussed.

1. Introduction

In the course of the present growth of interest in layered structures, experimental data on the specific heat at low temperatures have layers if atomisticbYe: E1i'iE:edlariZZ widths , which show characteristic deviations from the behavior of homogeneous materials. For these systems, pronounced dependences of the specific heat vs temperature curves on the layer thicknesses (for fixed ratios of the total masses of the constituent materials) have been measured explicitely. A theoretical determination of the density of states at k=O, projected to basic vibrational states, has performed 5

recently been for the alloy supperlattice

GaAs-AlxGal_xAs. The numerical treatment of the system (which contains disordered AlxGal_xAs layers) made use of the k-space version of Haydock's recursion method. The problem of large computing times led to a restriction to zero wave vector components perpendicular to the layers.

In the present paper densities of states and specific heats in macroscopi-

tally stratified systems are determined within the frame of the linear elastici- ty theory of inhomogeneous materials. The only limitation on frequencies and wave vectors are that they are small enough for the continuum theory to be applicable. With this restriction states with all k-vectors and all polarizations are taken-into account. They determine the lattice contribution to the low temperature specific heat. We rn&; use of previously obtained results on normal modes in layer structures.

Systems are considered, which consist of periodic sequences of 2 isotropic layers n=1,2 with thicknesses *a,, Lame's constants X,, +,, and densities p, (cf. Fig.1). n=2 labels the material with the lower value of the bulk transverse sound velocity ct (below, the subscript 2 is occasionally omitted for brevity, e.g. in the figures; U, for example, represents ~2). We use the abbreviations M="1/"2, R= P,/ p2, A=al/a2, d=2(al+a2). A Cartesian coordinate system is used whose xl-axis is oriented perpendicular to the layers.

We present both analytical approximations and numerical results, the

0749-6036/87/040445+ 10 $02.00/O 0 1987 Academic Press Limited

446 Superlattices and Microstructures, Vol. 3, No. 4, 1987

Fig.l: Geometry of the layer structure. Odd (shaded) and even layers are described by different sets of material parameters.

latter for two systems with the following sets of parameters:

System A: x~=J+,, M=4.0, R=1.5, A=1.3 System 8: Xn=pn, M=.18, R=.14, A=1.3

These constants were choosen in such a way that Stoneley waves, i.e. propagating waves localized near the interfaces of adjacent layers, occur in System B but not in System 'A.

Starting from a determination of normal modes and their frequencies (Section 2), the density of states is calculated (Section 3) and used to determine the specific heat (Section 4). The results are summarized and discussed in Section 5, and conclusions are presented in Section 6.

For comparison we first recapitulate some results of the continuum- (Debye-) theory for homogeneous elastically isotropic specimens of arbitrary shape and with arbitrary boundary conditions: The number A(S) of normal modes with frequencies below n follows an asymptotic law lo

',l A(Q)/Q3 = D (1) m

with D = (V/6n2)(2/ct3+l/c13). (2)

Here V is the volume, ct and cl are ve- locities of transverse and longitudinal sound waves. The resulting Debye law for the low temperature specific heat

cD(T)=aT3, with a=(lkB 4n4/5h3),D .(3)

is actually measured for macroscopic samples at temperatures below eD/50. For highly dispersed materials (powders) the

specific heat is larger. This is in agreement with the fact, that the surface contribution to the specific heat of plates with free (or clamped) surfaces are described by positive '1 (or negative 12) T2-terms respectively.

2. Normal Modes

The normal modes can be characterized by their frequency fi , the component &=(0,k2,k3) of their wave vector paral- lel to the layers , and the polarization index. In any layer m the mode displacements are superpositions of 6 plane waves described by the respective coefficient vectors &(E,fl) (with 6 components). The latter determine the mode not only in the mth layer but in the whole system since the boundary conditions at the interfaces of neighbouring layers are equivalent to linear equations expressing & in terms of L-1. Depending on whether m is even or odd, these equations are represented by, two matrices S or TS-lT (with Tik=(-)l'ik) respectively. The relation between x2k+l and Y2k_l isldescribed by the transfer- matrix M=TS TS. For an infinite material only such vectors x are allowed, which are solutions of the eigenvalue problem

M(K,~)Y(K,o) = ~(K,~)Y(K,Q) (4) -_

with eigenvalues A of modulus 1. The system of equations (4) decom-

poses into a system of 4 equations for the coefficients Y1...Y4 which describe modes with displacements in the plane of propagation (spanned by K and [loo]) (p-modes) and an independent system of 2 equations for Y5 and y6 describing modes with displacements perpendicular to that plane (a-modes).

Equations (4) can be satisfied only for special regions ("bands") of the K-n plane corresponding to propagating modes (cf 11,12). Examples for a- and p- modes . are shown in Figs. 2.

Within the frame of the theory of periodic structures, by the way, the normal modes can be described and clas- sified as follows: The assumed geometry corresponds to a periodic arrangement of unit cells of (macroscopic) dimensions d along the xl-axis and of atomistic dimensions (which are practically zero

Superlattices and Microstructure& Vol. 3, No. 4, 1987

0 2

WAVE VECTOR K (l/a) Dispersion curves S(K). L.h.s.: The 16 corresponding dispersion curves

System A, r.h.s.: System B. Top: a-modes, belong to the lowest bands. Curves with bottom: p-modes. For a periodicity volume with 16 double layers

kl and -kl coincide; for every band there are there are, thus, only 9 different

16 allowed kl-values in the first Bril- dispersion curves visible. louinzone: kl=nn1/8d, nl=0,+1,..,+7,8.


as compared to the wave lengths considered) in the other two directions. Only such modes are considered, which are acoustical with respect to the 3 atomistic lattice constants. With respect to d they may be acoustical or optical. The normal mode displacements of the vth branch then have the Bloch form

. . u(k,v;r,t) = v(k,";xl)el~-ln(k'~)t -- -- (5)

with v(xl+d) = x(x,). -

They are characterized by & and v . The number v specifies, to which branch (the acoustical or one of the many optical ones) and to which of the three polarizations the mode belongs. With the transfer matrix M, the Bloch theorem assumes the form

My = e ikld

Y,

e ikld

corresponds to the eigenvalue A of Eqn.(rl). The solubility condition

detlM(K,g) - e ikld

11 = 0 (6)

represents the dispersion relation between a, kl, K and v.

As a basis for a convenient method of counting states (cf. Sec.3) we choose periodic boundary conditions with period Ll=Nd, L2' L3 with Li>> d. The permissible wavenumbers are then ki=2nni/Li with integer and nl=O,+l,.., +(N/2-l),N/2.

"2,3

3. Density of States

A) Numerical results

To count the number A(B) of states with frequencies below a given n we no- tice that in an K-a-diagram all states with given values of kl and given polarizations lie on dispersion curves like those presented e.g. in Fig. 2. Denoting (cf. Fig.31 the intersections of these curves with the line R =const by K,, n=l,2,3..., and the sign of their slopes at these intersections by s,, one has for large L2, L3 the approximation

AN(n) = (L2L3/4r2) I: sn8Kn2. n

Approximate densities of states Z(B) for p-modes, obtained by forming AAN/an, are shown in Figs.4a,b,c for system A and

0 K, KS Ka K,

Fig.3: Scheme for counting the number of states with frequencies below 0 as explained in the text.

in Figs.Qd,e for system B. The curves exhibit characteristic van Hove singularities superimposed on roughly parabolic frequency dependences. Espe- cially in Figs.lc,e one notices a cross- over of the global density of states from one parabola (at low frequencies) to another one (at higher frequencies). This behavior will be discussed in Sections 3B and 3C.

The determination of the specific heat for temperatures below eD/5o requires only a knowledge of the density of states up to a frequency of about QD/5. For d=200 1, n,=2.1013,-1

ct=2000 m/s, and one has e.g. nD/5=act/a,

which lies at the r.h. boundary of Fig.4a. A decrease of d to 50 8 brings "D/5 down to 2ct/a (at the r.h. boundary of Fig.4c,e).Furthermore, the deviations of Z(n) from the parabola describing the average density of states are larger, the smaller the lattice constant d is.

In the rest of Section 3 upper and lower bonds for Z(B) and analytical approximations valid in different G-ranges are derived.

B) Debye type density of states function and interface correction

First we make use of the theorem l5 that the eigenfrequencies of an oscilla- tory system do not increase if the con- straints are relaxed and do not decrease if they are stiffening. For any coherent layer structure, A(R) is thus smaller (or larger) than the corresponding num- bers of states in reference systems consisting of the 2N separated layers of

Superlattices and Microstructures, Vol. 3, No. 4, 1987

60 -

1234567 J

15 - /

10

5

1 2 3

L

20 1

15 1

FREQUENCY (et/a) Fig.4: Density of states for p-modes. 1.h.s.: System A, r.h.s.: System B. Fat full lines: exact (numerical) results. Broken and dotted lines: high approximations ZD=3Dn2

fre uency and 9 ?=3Dn +~FR,

resp. Thin full lines: low 2

frequency

the periodicity volume with free (F) (or Since for each of the separated homoge- totally clamped, C) surfaces respective- neous single layers of the reference ly: system the asymptotic law (1) is valid,

2N 2N II AC(a) 'I A(O) 5 1 AF(Q) (7) n=l n n=l n

this yields

lim A(0.)/n3 = Z D, =:D (8) Qt-tm n=l

449

parabola Z0=3DOi' . In top, middle and bottom parts of the figure different scales are used for the abszissae. Vertical lines indicate frequency of the van Hove sigularity QO.

450 .%werlattices and Microstructures, Vol. 3, No. 4, 1987

for arbitrary coherent layer systems. - A detailed proof based on Courantfs maxi- mum-minimum-theorem will be published in

16 another paper . For the special (two-layer) structure

considered in the present paper one has 2

D = (V/6a2) t (2an/d)(2/ctn3+l/cln3).(9) n=l

The corresponding Debye-parabola

ZD = 3DG2 (10)

for the density of states is represented by the dashed lines in Fig.4. One ob- serves a systematic deviation of the numerical values from 3Dn2. From calculations for a number of systems in addi- tion to those described in Fig.4 we deduce that 2(n) fluctuates about an average (represented by the dotted lines in Fig.4)

an amplitude different from that of thd Debye law. This difference is especially pronounced in our example B (Fig.4d,e).

The approximation for the low frequency range is derived by expanding the dispersion relation (6) for small K, s, and kl, assuming the number N of double layers in the periocicity volume to be large. This yields explicit expressions for the dispersion curves. Proceeding in the same way as in the numerical analy- sis and transforming the sums for the number of states into integrals (for N-t-) one derives l6 for j-modes (j=a,p)

lim lim zNj( a)/vfi2 = iToj. (13) n-+0 N-+-

The a-mode constant is given by

Z(a) = 3Dn2 + 2Fn (11)

whose second term is interpreted as an interface contribution. Depending on the system parameters it may have either sign. For all systems which exhibit Sto- neley waves for p-modes, F is positive (Figs.4d,e). A similar result viously obtained for a free was "E- plate . This increase of Z(n) is, however, not only due to the contribution of the Sto- neley waves themselves: For a suitable set of parameters, also the average density Za(n) of a- modes may contain a positive linear term (Montroll correction), though there are no Stoneley waves with a polarization perpendicular to the plane of propagation. Furthermore one may obtain a positive linear term for systems which (just) do not permit any Stoneley waves at all. Apart from these exceptional cases near the existence limit of Stoneley waves, however, F is negative for layer structures without Stoneley waves (cf. Figs.4a-c).This corresponds to a similar result for clamped plates.

- a- Do - tl(l+AR)/2*2(1+A)(l+AM)ct3, (14)

with t 12=(l+AR)(l+A/M). The analytical expression for the p-mode constant is more complicated:

tl ;op=[(tl+ll)(t+l)+j W(x)dx]/4 n2(l+A)ct3,

11

with

_ - W(x)=

[(l+t-x2ml)2-41t+4x2m2-4x4m3]112 '

t = t12/t2 1 = 112/12

ml= (12+t2+t0)/12t2 m2= (l12+t12 )/12t2 m3= 1/12t2

11 2= (~+AR) l2 = (l+A)2+

t2 = (1+A)2

t0 = 4A(M-1

q=p/(2p+h),

q+Aq'/M), 4A(M-l)(qq'/M-qq'+q-q'/M)

2(l-q)(l-q')/M.

C) Parabolic density of states near n=O

In the low frequency interval OLR<Q~ one obtains, thus, the approximation

At very small frequencies, below the van Hove singularity at no in the vicinity of

n/2(al/ctl+a2/ct2) (12)

the density of states is nearly quadratic in R, however, as stated above, with

zo(n) = 3Don2, with - Do=DoV/3 (15)

In Table 1 the constants Do in low frequency region are compared the respective constants D valid at frequencies.

the with high


TABLE 1. Comparison of the constants DO and D in the low and high frequency approximations resp. of the density of states (for Systems A and B, cf. Sec:l) Superscripts a,p: a-, p-modes; v= Va/6nct3d.

DOa/v Da/; DOp/y Dp/;;;

A 0.60 0.83 0.90 0.99 B 1.90 1.20 3.60 1.44

For System A there is, at K=O, no forbidden fi interval in the K-n-plane between the lowest pair of p bands and the next, and only a moderate gap for the a-modes (cf. Fig.2); furthermore, no Stoneley waves exist. For this system the amplitude of the parabola describing the low frequency density of states is a bit smaller than that of the high frequency parabola.

For System B, on the other hand,there is, at K=O, a forbidden 0 range above the first pair of p-bands and a large gap for the a-modes (cf. Fig.2), and Stoneley waves exist. In this case the amplitude of the low R parabola is larger than that of the high fi parabola by more than a factor 2.

For frequencies in the vicinity of the edge of bands at K=O (cf. Fig.2), the existence of the forbidden region leads to a levelling (in kl-direction) of the dispersion surface and to corresponding van Hove singularities in the partial densities of states for a- as well as for p-modes. For p-modes there are additional van Hove singularities at frequencies corresponding to saddle- points in the dispersion surfaces, e.g. at "o (cf. Eq.12) where the character of the vibrations changes from longitudinal to transverse.

4. The Specific Heat

At low temperatures the specific heat

R D

c(T) = j Z(n) ; ( hn

0 hn/kT )dn e B -1

is determined by the small frequency part of the spectrum. The limitation of the integration to an interval _ _ OCQCnq

!il = hn /12k

he exzct t&perature dependence of c, obtained by numerical integration using the numerically determined density of states is represented by the fat full lines in Fig. 5.

The specific heat as derived from the simple Debye-parabola ZD(R)=3DQ2 is given by the T3-curve indicated by the dashed lines in Fig. 5. An inclusion of the linear interface term in Z(n) leads to the approximation

c(T) = .T3 + BT2 (16)

with

a=(4kg 4"4/5A3) 'D, 8=(6kB 3C(3)/h2)*2F (17)

represented by the dotted lines in Fig. 5.

The latter two approximations are valid at temperatures above about 5To=hQo/2kB. All these results are limited, of course, to temperatures so low (e.g. 5 eD/50) that the continuum approximation is applicable for the compact materials. The position of eD/50 on the abcissae of Fig.5 depends on the layer dimensions, viz. d. For materials with ct=2000 m/s and QD=2*1013s-l it coincides with the r.h. boundaries of Figs. 5a; 5b,d; 5c,e for d=2001; 1OOEi; 50A respectively.

The Debye frequency, obtained from the parabola 3(D1+D2)R2 for the density of states (lo), is given by

';i 3= (DlnD13+D2"D23)/(Dl+D2), D (18)

with the bulk Debye frequencies ciDn (n=1,2) of the two materials. QD is independent of the supperlattice period d. By making use of the relation

F/D=S'ct/d (19)

between the coefficients of Eq.(ll) (with /S!a2.1 for the examples presented in this paper), it can be proved l6 that the error encountered when RD is used instead of fiD is smaller than 1% for d>70a and smaller than 10% for d:,71.

At very low temperatures

T<TO = Qo-h/12k, (20)


1 TEMPERATURE

2

(0.1 Act/kga>

Fig.5: Specific heat for p-modes (cf. the corresponding line type. Vertical T‘ig.4) 1.h.s.: System A, r.h.s.: System lines indicate temperature TO below B. Curves were calculated from the which the low temperature cubic parabola density of states curves of Fig.4 with approximates c(T).

finally, the specific 5. Discussion of Results pure T3-law

heat is given by a

Q(T) = aOT3, For suitable values of the elastic

(2.1) constants, the densities, and the layer dimensions, the

with "S= (4kB 4n4/5h3).D0. calculations presented

above show three distinct intervals in the Z(n) and the c(T) dependences:

This is represented by the thin full simple power laws with different lines in Fig.5. coefficients for very small and for


(b)

/

I 0.5 I

TEMPERFITURE T (fic,/Akg)

Fig.6: Specific heat of System B. Plot of C/T2 vs T for p-modes. The bilayer periods are (a) d= A, (b) d=2A . Fat full lines: exact results. Thin full lines: low T approximation cO=aOT3. Broken and dotted lines: proximation cD=aT3 and

hiqher T c=crT3+BT2

ap-

respectively.

larger values of the variables 51 or T and a transition region in between. For the specific heat this can visualized if c/T2 is plottedbEs ~l~~r~~ Fig.6. For System B the slope of the dotted line, i.e. the coefficient a of the higher temperature T3 term, is much smaller than the slope a0 of the low T approximation. (The actual c(T)- represented by the full curved line - approaches the dotted higher T curve beyond the right margin of Fig.6b). The constant 13 of the interface contribution is given by the dotted line's intersec- tion with the ordinate.

If thus T3

a specific heat following a curve is measured, it is important to

know which of the two T3-laws is actually observed. For measurements in a certain temperature range at systems consisting of periodic arrangements of

two materials present in given total quantities, this depends on the layer dimensions, as can be seen by a comparison of Figs.6a and 6b. For systems with marked difference between CL and a0 this leads to the following classification of layer thicknesses near a given reference Temperature TE:

Thick layers

With respect to TE we define thick-layer structures those for wh at TE, the absolute value of quadratic term in the specific heat small as compared to the cubic term Eqs.16, 17), i.e. for which e.g.

as ich, the

IRTR2 I < aTR3/100 or lRl/ a<TR/lOO.

Layers are, thus, thick if (cf. Eqs. 17, 19)

d ' dl = S'1500 r;(3)hct/(~4kBTR) (22)

For TE=lK, ct=2000ms-1, S=2.1 one has, e.g., dl=60004.

For such structures the density of states for modes contributing substan- tially to the heat capacity at TE is well represented by a parabola

zD= 3Dn2

whose constant D is obtained by summing the respective Debye constants for the single layers (cf. Eq.2)

The specific heat is approximated by

cD="T 3

where a is the sum of the respective simple layer constants (cf. Eq.3)

Medium Layers

For a given temperature TE we define as medium-layer structures those, for which the layer widths are such that (cf. Eqs. 22,231

5dO cd<d 1' The density of states - for the

relevant frequencies - varies about the average

Z(n) = 3DQ2+2FR

and the specific heat in the vicinity of the reference temperature is given by

; = aT 3 + BT2.


Thin layers

For a given temperature TR,thin-layer structures are characterized by layer widths for which (cf. Eqs.l2,20)

TR<TO

of the coefficients a0 and (I corresponds to that of Do and D and may assume, as the latter, values near 1 or values as large as 2 or more.

or d < do=( nod/ct)ctfi/12kBTR. (23)

For TR=lK, for example,

nOd/ct=4.6, and ct=2OOOm/s the limitinq width is

Acknowledgement - This work w=s performed within the frame of the Sonderforschungsbereich "Ferroelek- trika", which is supported by the Deu- tsche Forschungsgemeinschaft.

approximately d0=60i. - For thin layers the density

in the n-range relevant for capacity near TR is given by

and the Zo( n2)=3DOR2,

specific heat by co(T)=aoT3.

6. Conclusions

of states the heat

In the present paper it was shown that the frequency dependence of the density of states of layer materials follows an a 2-law at very low frequencies, then passes through pronounced van Hove singularities and approaches another n2-law -

finally combined

with an interface term which is linear in n - at higher frequencies. The sign of the interface term depends on the parameters describing the system; if (but not only if) these permit Stoneley waves, it is positive. It is important, also for the evaluation of experimental data, the f12

that the coefficients Do and D of -terms in the two frequency ranges

may differ considerably: Both the numerical calculations and the analytical results show the influence of the parameters on DO/D: This ratio may be approximately 1 for some sets of parameters and about 2 for others. The latter situation is likely to occur if there is a pronounced gap between the lowest bands, i.e. a forbidden frequency interval, for phonons propagating perpendicular to the layers, and if Stoneley waves are possible.

The existence of these frequency ranges with different simple z(n)-dependences leads to corresponding temperature ranges with different simple T-dependences of the specific heat c. Star- ting f5om T=O the latter is described by an aoT law with an accuracy better than l%upto a characteristic temperature

TO' Between To and about ST0 there is a transition to an aT3+BT2 law. The ratio

1.

2.

3.

4.

5.

6.

7.

8.

9.

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density of states and specific heat of elastic vibrations in layer structures

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