I'8 Y SI CA L RE VIEW B VOLUME 18, NUMBER 12

Infrared absorption in MNO microcrystals

15 DECEMBER 1978

Ronald FuchsAmes Laboratory-United States Department of Energy and Department of Physics, Iowa State University, Ames, Iowa 50011

(Received 30 May 1978)

The infrared-absorption spectrum of a rectangular MgO microcrystal can be calculated by lattice dynamicsif the crystal contains a small number of atoms, or by a macroscopic method that uses the bulk frequency-dependent dielectric function a(co). For a crystal containing 900 atoms, the results of these two methods arein qualitative agreement. It is concluded that the shape of the crystal has an important influence on theabsorption spectrum, and that a cubic crystal containing relatively few atoms has a spectrum characteristic of.a macroscopic cube.

A lattice-dynamical calculation of the vibrationalproperties of a MgO microcrystal containing 10x 10x 9 = 900 atoms has been performed recentlyby Chen, de Wette, andKleinman. ' The frequenciesand dipole strengths of the normal modes yieldinformation about the infrared absorption spec-trum. A dielectric continuum model has been usedby the present author' and by Langbein' to cal-culate the optical absorption spectrum of a cubemuch smaller than the light wavelength. For anionic crystal slab, a comparison of lattice-dy-namical and dielectric continuum calculations ofthe optical absorption indicates that a slab with athickness of 10-15 atomic layers can be con-sidered as a dielectric continuum. 4 It is of in-terest to ask a similar question for a cube: is a10x 10x 9 microcrystal large enough that itsoptical absorption spectrum can be obtained froma continuum theory&

Chen et al. ' used a rigid-ion model of MgO tocalculate the vibrational normal mode frequencies&u~ and eigenvectors $~(lkii), where f is a unit celllabel, k = 1, 3 denotes the ion type, and P =x, y, z.If an external field EB with frequency & is appliedto the crystal, the induced dipole moment in thedirection P is

m, = ~8(p) . E,(d~ —(d((d + 1,y)

where y is a damping parameter which gives afinite width to the absorption peaks at ~~, and

o'8(p) = + qaqa (~aiifa ) ' '

x $~(l'0' p) )p(f kp)

and (3) we get

&xg(~)& =v — Z~~ —td(~+ iy)

'

The sum rule (3) of Ref. 1 can be used to intro-duce normalized dipole strengths

o. i(P) = (&/Nq') o. 8(P),

which satisfy Z~o. 8(P) = 1. Here N is the numberof ion pairs in the crystal, q is the ionic charge,and p is the reduced mass. The susceptibilityof the particle, averaged over direction, is

&x(»=-~&x,(»=- 'Z. ' . . (4)1 V 1 Nq' V' me(P)3 g 3 pv 8 ~ e~ —e(&@+iy)

A continuum theory for the electric susceptibilityof a particle composed of a material with dielectricfunction ~ (e) = 1+4vX (~) gives'

C 8(m)&x ( )&=

where the n are depolarization factors for thepolarization modes labeled by m, and C 8(m) aredipole strengths which obey the sum rule Z Cs(m)=1. Ne can use the expressions

I

X 0(u'

valid for a rigid-ion diatomic lattice, and intro-duce the polarization mode frequency

&u= mr[1+ 4wn„X(0)]'~'.

The directionally averaged particle susceptibilitycan then be written

is the dipole strength. ' of the normal mode P. Inorder to make a connection with the continuumtheory of Ref. 2, it is convenient to define an elec-tric susceptibility for the particle,

&x(~)& = —Z (x g(~)&13 8

1 Nq' P Cq(m)3 pv 8 (u'„—(u((u i+y)' (7)

(X ,((u)& =3K,(vZ 8') ',where v is the particle volume. From Eqs. (1)

1

(3) Equations (4) and (7) are of the same form,showing that the normalized dipole strengths

18 7160 - 1978 The American Physical Society

INFRARED ABSORPTION IN MgO MICROCRYSTALS

o. 8(P) and frequencies &u~ in the lattice-dynamicaltheory correspond exactly to the dipole strengthsC8(m) and frequencies &o in the continuum theory.ln order to compare Eqs. (4) and (7) it is usefulto calculate the absorption A(&v) = (2 pv/~Nq')x~r~ Im(x(&u)). This dimensionless quantityis proportional to the absorption coefficient, withdimensions (cm '), of a dilute layer of particles, 'and it satisfies the condition f AO(a&) d(~/~r) = l.From Eqs. (4) or (7) we have

nIIIIIIIII I

IlI I

II

I IcuF

I I

I I

I

I

I

I

II

or

A, (~) =—~ n 8(P) lm2 ~ (al(d g

Sw 8 p cop —M(&d+ Xj')

A, ((o) =8

Z C8(m) lm8 m COm

—(d ((d + ZJ)

(8a)

(8b)

where A, (e) and A, (u&) denote the absorption ob-tained from the lattice-dynamical and continuumtheories, respectively.

The frequencies ~~ and dipole strengths a8(P)for the rectangular microcrystal have been ob-tained from Fig. 3 of Ref. 1. There are about150 modes with frequencies co~ in the range ~~«o~ & e~ and strengths o. 8(p) ~ 10 '. Although themicrocrystal is not precisely a cube, it will beconsidered as a cube for the purpose of compari-son with dielectric continuum calculations. Thereare, in fact, differences between a,'(P), u,'(P),and n,'(p); they arise because the shape is not acube and the axes are nonequivalent (see Fig. 1 ofRef. 1). The continuum calculation for a cubeyields a much smaller number of modes with ap-preciable strengths, only about six in eachcartesian direction. The x, y, and z directionsare equivalent, so the sum over p in Eq. (8b)

, immediately gives 3ZBC8(m) = C(m), where thesix values of C(m) are listed in Table I of Ref. 2.The frequencies + are obtained from the de-polarization factors n listed in the same tableby using Eq. (6). Rigid-ion model parameters forMgo are &sr= 7.51x10"rad/sec, &u~ = 16.74x 10" rad/sec, and 4wIt(0) = (w~/&ur)' —1 = 8.959.'The damping factor y, which controls the widthof the absorption peaks, must be chosen withcare. The width must be large enough that thevery detailed structure contained in the 150 normalmodes of the lattice-dynamical calculation dis-appears. However, it must not be so large that

08 IO I2 l4

FREQUENCY (IO'3 rad/s)I6

FIG. 1. Absorption A&(co) (solid curve) and A~(co)(dashed curve) as functions of frequency.

all information about the frequency dependenceof A, (+) andA, (u) is lost. A value y=0.04+r issatisfactory.

Figure 1 shows the absorption spectra A, (&u)

and A, (&u) for a cube. Although there is some dis-crepancy between the positions and heights ofthe absorption peaks, the main features of thespectra agree."A sphere would have a singlepeak at the frequency &u~ = 11.45x 10"rad/sec.Both theories predict a strong peak at. or slightlybelow this frequency, but there is also an im-

portantt

peak at about 10.2x 10" rad/sec, as wellas weaker peaks at higher frequencies. ' Thelattice-dynamical calculation of Ref. 1 thereforeconfirms the conclusion of Ref. 2 that there aremajor differences between the absorption spectraof a cube and a sphere, and that the shape of theparticles must be considered in comparing theorywith experiment.

ACKNOWLEDGMENT

This work was supported by the U. S. Depart-ment of Energy, Office of Basic Energy Sciences,Materials Sciences Division. I wish to thankProf. L. Kleinman and Prof. F. %. de bette forhelpful discussions.

~T. S. Chen, F.W. de Wette, and L. Kleinman, Phys.Rev. B 18, 958 (1978).R. Fuchs, Phys. Rev. B 11, 1732 (1975).D. Langbein, J. Phys. A 9, 627 (1976).

4R. Fuchs, Phys. Lett. A 43, 42 (1973).

~In Ref. 1 this quantity' is denoted as an "absorption co-efficient. " The terminology used here agrees arith thatof Ref. 2.

6T. P. Martin, Phys. Rev. B 7, 3906 (1973).~It shouM be noted that different continuum calculations

RONALD FUCHS

give somewhat different absorption spectra. The extra-polated values of C(m) and n~ listed in the Appendix ofRef. 2 give major peaks of heights 5.2, 5.", 1.23, 2.1,and 0.7 at the frequencies (in units of 10~3 rad/sec)10.0, 11.0, 12,05, 13.5, and 14.7. These peak heightsare in better agreement with those found by lattice dy-namics, but the peak positions do not agree as well.The continuum calculation of Ref. 3 gives peaks withheights 9.1, 1.1, and 3.8 at the frequencies 10.1, 11.5,and 14.1. The discrepancies between the various con-tinuum calculations arise from the slow convergenceof the methods used.

The discrepancies might be reduced if a continuumtheory based on a nonlocal bulk dielectric function&(q, co) were used, as nonlocal effects are expected to

&be important in very small particles.

In Ref. 2 a dielectric continuum model was used to cal-culate the absorption spectrum of a sample of cubic

MgO microcrystals. The resulting spectrum, shownin Fig. 3 of Ref. 2, shows almost none of the structurethat appears in Fig. 1 of the present work, and thereis also a shift of the absorption towards lower frequen-cy. This discrepancy is explained as follows. In thedielectric continuum calculation of A~(co) performed inthe present work, the values of (ol. and y have beenchosen so as to allow a comparison with the rigid-ionlattice-dynamical calculation. In real MgO the value of

.ul i.s sma'lier because of the electronic polarizabilityof the ions, causing a shift of the absorption peaks to-ward lower frequencies; moreover, the damping para-meter y, which simulates lattice anharmonicity' andinteractions between particles, depends on frequency andis larger on the average than the constant value y= 0.04arz, causing enough broadening that the absorp-tion peaks coalesce into a single asymmetric peak.