interplanar forces of black phosphorus caused by electron-lattice interaction

~ Solid State Communications, Vol.6],No.]0, pp.595-600, ]987. 0038-]098/87 $3.00 + .00 Printed in Great Britain. Pergamon Journals Ltd.

INT~RPLANAR FORCES OF BLACK PHOSPHORUS CAUSED BY ELECTRON-LATTICE INTERACTION

Naoshi SUZUKI and Masato AOKI

Department of Material Physics. Faculty of Engineering Science, Osaka University, Toyonaka 560, Japan

(Received 7 November 1986 by J. Kanamori)

The generalized electronic susceptibility of a narrow gap semiconductor, black phosphorus, is calculated by using electron-lattice coupling con- statnts derived microscopically on the basis of tight-binding calculations of the electronic band structure. Interplanar forces along the [I00] direction caused by the electron-lattice interaction are evaluated from the obtained generalized electronic susceptibility. Reflecting the narrow gap the interplanar forces take significant values even for far neighboring planes. Further, considerable decrease of the energy gap induced by pressure gives rise to a characteristic pressure dependences of the interplanar forces, which lead to qualitative explanation of the observed pressure dependences of the phonon dispersion curves.

Recent success in growing relatively large single crystals of black phosphorus (abbreviated BP hereafter) [I ,2] has stimulated many experimental and theoretical studies of its physical properties. BP is an interesting material from various points of view: (I) it is an elemental

semiconductor of narrow gap (Eg~0.3 eV) [3,4];

(2) it is a layered system consisting of puckered layeres [5,6]; (3) as pressure is increased, it shows a successive phase transition such as semiconductor (orthorhombic)to semimetal (rhom- bohedral) and to metal (cubic) [6-8]; (4) in its cubic phase it becomes a superconductor with a fairly high transition temperature T =6m13 K[9].

BP is an intersting system also f~om the lattice dynamical point of view. According to inelastic neutron scattering measurements[IO], for example, the whole of the longitudinal acoustic mode (LAx mode) along the [100] direction shows softening as pressure is increased, whereas the other phonon branches show hardening. This IAx mode corresponds to accordion motion of the puckered layer end affects strongly the bond angle of P atoms. Hence, this mode is expected to be strongly coupled with electrons. Thus, it is speculated that the softening of the LAx mode with increasing pressure is related with the electron-lattice interaction.

Thus far, lattice dynamical calculation at atmospheric pressure (hereafter denoted by 0 kbar) has been performed by Kaneta et al. [11 ] on the basis of the valence force model and by Kaneta and Morita [12] on the basis of the bond- charge model. The results obtained from the latter model can explain well the data obtained by inelastic neutron scattering [13], infrared absorption [14], Raman scattering [14,15] and specific heat measurements [16]. But there has been no attempt to explain the characteristic pressure dependneces of the phonon frequencies mentioned above. The aim of the present report

C " C V is to study micros opl all.~ the coupling between

the electrons and the lattice and to clarify the origin of the softening of the LAx mode caused by pressure. For this purpose we adopt the method based on tight-binding calculation of electronic structures. This method has been success- fully applied in explaining many aspects of structural phase transitions and phonon anomalies in transition metals and transition-metal compounds [17].

After band calculation by Takao et al.[18] we adopt the extended Huckel approximation (EHA) [19] and use Slater-type wave functions with Cleminti exponents [20] for 3s and 3P orbitals of P atoms. In the EHA transfer integrals are assumed to be proportional to overlap integrals:

~ab(la + T)S £p'£'~ (£p~£,~) -b ab ~,£ '~ ab = (I)

la~ab (£~=£ '~)

(a,b = 3s,3P)

Here ~P'i'~) S ~p'£'~) ab and ab denote the transfer and

overlap integrals between the a orbital of £~ atom and the b orbital of £'v atom where £(£') and ~(~) specify the unit cell and the atom in a unit cell, respectively. The unit cell of BP contains four P atoms as depicted in Fig.1. In

eq. (I) I a denotes the ionization energy and Kab

represents a parameter chosen appropriately. In actual calculations the ionization energies are taken from the values calculated by Herman and Skillman [21 ] (I3s=-1.2588 Ryd., I3p=-0.6139

Ryd. ) and as to the values of K and K we use ss pp

the commonly chosen value 1.75. The remaining parameter K may be chosen so as to reproduce the band ga~Po.3 eV at 0 kbar [18].

Since the overlap integral is calculated analytically for any pair of P stems, the energy

595

596

b12 - C

INTERPLANAR FORCES OF BLACK PHOSPHORUS

[Ryd] 0.£'

Vol. 61 , No. I 0

I z

-0..~ x

L U Z

-I .{

Fig.1. A portion of the crystal structure of BP.

band structure can be calculated if we give lattice constants a, b and c and structure cons- rants u and v. Using values of these crystal structure parmeters determined by Cartz et al. [22] we have calculated first the energy bands of BP at 0 kbar. In actual calculations the overlap integrals are taken into consideration up to tenth neighboring atom pairs. The value

of K has been determined to be 1.379, which is sp

slightly different from that determined by Takao et al. [18] who used the crystal structure para- meteres different from ours. The energy dispersion curves along the U-line (Z-L) in the first B.Z. are shown by solid curves in Fig. 2. The dispersion curves of the highest valence bands and the lowest conduction band as well as the overall features of the density of states (not shown here) are quite similar to those obtained by Asahina et al. [23] on the basis of the self- consistent pseudopotential method, which explain well many experimental results related to the electronic structure of BP [2£-27]. We have calculated also the energy bands at 15 kbar by using the crystal strucutre parameters at 15kbar [22J and the same value of Ksp. The dispersion

Fig.2. The energy band structure of BP along the U-line (Z-L). The minimum gap is located at the Z-point.

curves at 15 kbar are shown by dotted curves in Fig.2. As seen clearly, the gap between the highest valence band and the lowest conduction band decreases remarkably with increasing pressure. The minimum gap at the Z-point decreases from 0.3 eV to 0.174 eV.

In the tight-binding based method the electron-lattice interaction is expressed in terms of derivatives of overlap and transfer integrals with respect to the atomic displacements. Adi- abatic harmonic potential for the atomic motion is obtained by calculating the electronic free energy in terms of the atomic displacements up to the second order[17]. The result is expressed as folows :

AF =~ ~ ~ ~ (;v,q) _ (2) quv

where u (a=x,y,z) denotes the Fourier~transform of~he lattice ~ displacement and X (~v,q) is defined by

f(Enk) - f(En,k_ q) x ~(uv,q) =2Z I

nn'k Enk - En'k-q

x gu~ (nk,n,k_q)g~ (nk,n'k-q)* (3)

Here ~a(nk ,n 'k -q) denotes the e l ec t ron - l a t t i ce coupling which represents the strength of coupling between the two electronic states nk and n'k-q caused bY displacements of M-type atoms in the ~-direction. This coupling coefficients are expressed in terms of derivatives of overlap and

Vol. 6], No. lO

transfer integrals with respect to the atomic displacement. In the present calculation based on the extended ~dckel approximation the derivatives of transfer integrals as well as of overlap integrals are calculated analytically. ~The generalized electronic susceptibility

X a~ (Uv ,~ contributes directly to the dynamical matrix D ~ (~ ,q) :

D °d3 (Uv ,q) = X a6 (~v ,q) + R ~'lB (#v ,q) (A)

where K ~6 (l.tV ,q) is usually called the short- range repulsive part which includes all contri-

butions other than X ~ ~v ,q). Phonon frequen-

cies are obtained by diagonalizing ~ ~v ,q). The summation over k in eq.(3) has been car-

ried out by dividing the I/8 zone of the first B.Z. into 234 meshes. The numerical calculation

of X a6 ~ ,q) at T=O K has been performed for

seven wave vectors along the A-line (£-X), q = (m/6,0,0) (m=0-6), and eigenvalues and eigenvec-

tots of X a6 ~v ,q) have been determined for each

q. All of the obtained eigenvalues are uegative, which means that electron-lattice interaction has a role of lowering phonon frequencies. In Fig.3 we show for each q three eigenvalues of

X a~ (]JV ,q) which correspond to three acoustic

modes, LAx, TAy and TAz. The symmetries of LAx,

TAy and TAz are AI, AA, and A2, respectively.

The eigenvectors of the LAx mode consists mainly of a linear combination of x1+ x 4 and x2+ x3,

hut components of z 1- z& and z 2- z 3 are also

included a little. In Fig.3 the closed circles denote the results at 0 kbar and the open circles those at 15 kbar. As seen from the figure the magnitude of the eigenvalues become larger

as pressure is increased. Hence, X a6 (~v,q)

plays a role of softening the phonon frequencies with increasing pressure. The increase of the magnitude of the eigenvalue is largest for the LAX mode among three acoustic modes. This fact strongly implies a possibility that only the LAx mode shows softening as pressure is applied.

The large softening of the LAx mode with

Al-Symmetry is related with the electronic band

strucutre near the Fermi level. Because of the energy difference in the denominator of eq. (3) a main contribution to X arises from coupling between the highest valence band and the lowest conduction band near the Z-point. If we consi- der the wave vector q on the A-line and choose as the i%k state the bottom of the conduction

band at the Z-point, z4, then a large contribu-

tion arises if n'k-q state lies in the highest valence band with U2-symmetry along the U-line. In this case, there is a selection rule by group theoretical consideartion, and only the A l-mode

can connect these two electronic states, z[ and

U 2. As seen from Fig.2 the energy difference

E(z~)-E(U2) decreases considerably with increas-

ing pressure. This is the main reason why the


( c m 1 )

0

597

-I O 0

- 2 0 0

F

• % . . I

"~" 1 "

e Okb "~'~"~"

___..._._ 1 5 k b

A X

decrease of the eigenvalue of the LAx mode (A 1- -symmtry) under~ressure is the largest.

In general X ~ ~ ,q) can be expressed as the Fourier transform of interatomic forces [28] and their knowledge can bring us new insight into the nature of interatomic interaction and crystal bonding. In case we focus our attention on phonon dispersions along a particular direction in the q-space as in the present case, it is convenient to introduce interplanar forces [29]. It is noted that the lattice dynamics for wave vectors along a particular direction can be completely determined by giving forces between planes perpendicular to that direction because all of the same kind of atoms on one of such planes vibrate in the same phase. In terms of interplanar forces the generalized electronic

~ k Fig.3. Three eigenvalues of X~(~v,q) corres- ponding to three acoustic modes along the [100] direction (A-line). In this figure we have plotted -~(-eigenvalue)/M where M denotes the mass of P atoms. Note that the eigenvalues are negative. The numbers, I, 2 and 4, denote the symmetry of the accoustic modes: I; A I (LAx), 2;

A 2 (TAy), 4; A 4 (TAz). The open and closed

circles are obtained by diagonalizing X SB (~,q) calculated from eq.(3) whereas the solid and

dotted curves are obtained by using Xa6(~,q)

calculated from eq. (5).

598 INTERPLANAR FORCES OF BLACK PHOSPHORUS Vol. 61, No. |0

susceptibility XaB(Wv ,q) can be expressed as

follows:

X~B(wv'q) :[p ~xB(P~'P'v)exp(ilqiRp~,p'v)" (5)

where K ~ (p~ ,p'v ) represents an effective force X

constant between the p-th plane of ~ atoms and

the p'-th plane of v atoms, and Rpu,p,v denotes

the distance between the two planes. If we con- fine ourselves to the [100] direction, each atomic plane contains only one kind of P atom, 1,2, 3, or 4. Instead of performing inverse Fourier transform of eq.(5) we have determined the interplanar forces by a least-squares fitting pro-

cedure so as to reproduce ×aB(~v,q) for q = F,

(I/2)FX, and X. The form of interplanar forces is determined by the crystal symmetry and it takes the following form in the present case:

a, O, d

K~ (p~ ,p'v) = 0, b, 0 (6)

d' O, c

Off diagonal xy,yx,yz, and zy components vanish exactly. For both pressures, O and 15 kbar,

XaB(~v,q) have been reproduced almost completely

by taking account of interplanar forces up to the ninth neiboring planes. The full and dotted curves in Fig.3 represent the results of diago-

nalization of XaS(~v,q) obtained from eq.(5)

using such determined effective force constants. xx 0 In Fig.4 we show -K X (p,) which represents

the force acting on the plane p along the x-direction when the plane O is displaced by a unit length along the x-direction. Here the suffix ~, V for atoms are omitted for simplicity. A plane of atom I has been chosen as the plane O. The plane I is a plane of atom 2, the plane 2 is a plane of atom 3, the planes 3A and 3B are planes of atom 4, and so on. The eigenvalue of

the LAx mode is determined mainly by K.~XX(p,0). A

The nearest neighboring and the next nearest neighboring force constants take naturally large values, but it should be noted that the fourth neighboring interplanar force is also fairly large. Thus, the interatomic forces due to electron-lattice interaciton can be never described by near neighboring forces only. This long- range nature of the effective forces is ascribed to the small gap of BP. Other components of the force constants show distance-dependence similar

to those of KwXX(p,O). In Fig.A we show also the

difference between the force constants at O kbar

and at 15 kbar, AK..XX(p,O) ~ Kx(P,O)[O kbar] - A

K xx, 0 ~ ~X ~p' }[15 kbar]. As seen from the figure XX

the magnitude of K X (p,O) increases with

increasing pressure. In particlar, the fourth

interplanar force C(4,0) shows the neighboring

largest relative increase of the magnitude.

[nterpl anar Forces ~oooso [dyn/c.~

-K~ ~ ~p.o~ I

............ .,"::":-...>;."/.::..:. ..... :-.:..>.,5.:.>i.."::::,::::: ..... -AK X {p,O;'",.. '""2:2"'. ...... .. "'"";.')'... "'"'1:22"-.. '"":."

i ii !1 i! ii ii

Fig.g. Effective interplanar force constants

KXX(p,O) and short-range interplanar force ×

constants Ky(p,O) at 0 kbar. AK~.XX(p,O) A

represents the difference between the effective force constants at 0 kbar and at 15 kbar.

To see whether only the LAx mode can actually show softening as pressure is applied we have calculated the phonon dispersions along the [100] direction by taking account of short-range

repulsive part R~(wv,q). We have expressed

R~S(wv,q) in terms of short-range interplanar

~S(p~,p'v ) up to the third forces neighboring

planes. As mentioned before, phonon frequencies are determined by diagonalizing the total dyna-

mical matrix X ~B ~ ,q) + ~ (~ ,q). The short-

range interplanar force constants at O kbar have been determined by a procedure of least-squares fit so as to reproduce 13 phonon frequencies ( 8 at F, 2 at X and 3 at (I/2)FX) observed by Raman scattering [14], by infrared reflection [14] and by inelastic neutron scatte$ing [10]. The determined short-range forces K~(p,O) are illustrated in Fig.4 and the phonon-dispersion curves of the LAx and TAz modes at O kbar are shown by full curves in Fig.5. Since we have no way to evaluate the pressure dependence of the short-range force, we have assumed simply that

of K~RS(P,O) increase at the same rate all as

pressure is applied. The dotted curves in Fig.5 show phonon dispersion curves obtained by using

K~(p,O) at 15 kbar and short-range forces x larger by 5% than those at 0 kbar. As seen from the figure the LAx mode does show softening whereas the TAz mode shows hardening. Thus, experimental results by Yamada et al.[10] are qualitatively explained, though there is some ambi- guity in short-range force constants. Other phonon modes not shown in Fig.5 show hardening ex- cept the TAy mode whose frequency is almost un- changed. Experimentally there is no observation about the pressure dependence of the TAy mode.

It shoud be emphasized that far-neighboring

Vol. 6l, No. I0

--1 ( o m ) Phonon

1 0 0 r-


5 0

O it/

I

o Exp. (OKb)

• Exp.(15Kb)

L A x ( A 1 )

o

Energy

e

T A z ( A 2 )

o

A X

Fig.5. Phonon dispersion curves of the LAx and TAz modes. Solid curves -- at 0 kbar- dotted curves -- at 15 kbar. Experimental data are taken from ref.10.

599

the force constant model which takes account of interplanar forces only up to the third neighboring planes, but our trial was unsuccessful. In this connection it is noted that the valence force model used by Kaneta et al.[11] cannot explain either the pressure effect because their model takes into account interplanar forces essentially up to the third neighboring planes with respect to the [100] direction.

In summary: (I) Effective interplanar forces caused by elec-

tron-lattice interaction is fairly long- ranged in black phosphorus because of its small energy gap.

(2) These effective interplanar forces play a role of softening phonon frequencies as pressure is applied.

(3) The softening is the largest for the LAx

(At-symmetry) mode. This consequence comes

from: (i) the energy gap decreases consi-

derably by pressure and (ii) only the A 1-

phonon mode can couple the conduction band bottom (Z Z) with the highest valence band

states (U2-symmetry) near the Z point.

(4) These effective interplanar forces combined with short-range repulsive forces appropriately chosen can explain qualitatively the characteristic pressure dependences of phonon frequencies, i.e. only the LAx mode shows softening as pressure is applied whereas other modes show hardening.

(5) It is the effective force v~x(4,0) (fourth

neighboring interplanar force) that plays a most crucial role in pressure-induced softening of the LAx mode.

interplanar forces caused by electron-lattice interaction is indispensable to explaining the observed pressure-dependence of phonon frequencies. In particular, large increase of the mag-

nitude of KXX(4,0) caused by pressure has played ×

a vital role in pressure-induced softening of the LAx mode. In fact, we have tried to explain the observed pressure effects on the basis of

Acknowledgements We are grateful to Prof. K. Motizuki of Osaka

University for useful and fruitful discussion and for critical reading of the manuscript. This work is partly supported by the Kurata Foundation. The numerical calculations were performed with use of ACOS 1000 system at the computer center of Osaka University.

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interplanar forces of black phosphorus caused by electron-lattice interaction

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