M. SINGH et al.: The Band Structure of Cd,P, in a Magnetic Field 4H1

phys. stat. sol. (b) 114, 481 (1982)

Subject classification: 13.1; 22.3

Ddpartement de Gins'e Physique, Institut National des Sciences Appliqudes, Universitk Paul Sabatier, Toulouae

The Band Striicture of Cd8P, in the Presence of a Magnetic Field

BY M. SINGH~), J. CIS OW SKI^), P. R. WALL ACE^), J. C. PORTAL, and J. M. BEOTO

The band structure of Cd,P, is presented on the basis of Bodnar's model which is extended by Wallace in presence of magnetic field. The oscillations of the Fermi level with magnetic field are c,zlculated and it is found that the amplitude of the oscillations in Cd,P, is larger than in Cd& The effect of the so-called free-electron term is included in the calculation of the g-factor and cyclotron effective mass. The correction due to the free-electron term in the g-factor is smaller (gf< 2) than anticipated and depends on energy. Another interesting result of the calciilation is that there is linear relationship between cyclotron effective mass and energy. Auf der Grundlage des von Wallace fur ein magnetisches Feld erweiterten Modells von Bodnar wird die Bandstruktur von Cd,P, angegeben. Die Oszillation des Ferminiveaus im Magnetfeld wird berechnet und gefunden, daB die Oszillationsamplitude in C%P, groBer als in Cd,As, ist. Der EinfluB dea sogenannten Freie-Elektronen-Terms wird in die Berechnung des cJ-Faktors und der effektiven Zyklotronmasse eingeachlossen. Die Korrektur infolge des Freie-Elektronen-Terms zum g-Faktor ist kleiner (gf < 2) als vermutet und hiingt von der Energie ab. Ein anderes interessantes Ergebnis der Berechnung ist, daB eine lineare Beziehung zwischen der effektiven Zyklotronmasse und der Energie besteht.

1. Introduction I n the last few years, considerable attention has been given to the 113-V, semiconduct- ing compounds as a result of their interesting transport and optical properties (see the review paper of Zdanovicz and Zdanovicz [l]).

Among them, Cd3As, and Cd,P, are of particular interest as being examples of nar- row-gap semiconductors with tetragonal crystal structure ; the former is now believed to have an inverted band structure with the energy gap between s- and p-like levels of about -0.1 eV, whereas the latter has a simple band structure with the gap of about 0.5 eV.

The problem of a possible anisotropy in both compounds (really observed in Cd,As, by Rosenman [2] 13 years ago) has been theoretically considered by Bodnar [3, 41, who following Kildal[5] generalized the three-level Kane band model for the case of a tetragonal semiconductor with a small energy gap by taking into account the influ- ence of a tetragonal crystal field.

Subsequently, the Bodnar model for Cd,As, has been extended in the presence of a magnetic field by Wallace [6] and strongly confirmed by Shubnikov-de Haas (SdH) measurements by the Portal and Blom groups [7].

On the other band, the band structure of Cd,P, is much less known from both theo- retical and experimental points of view. As follows from recent SdH experiments by

l) Permanent address: Department of Physics, McGill University, Montreal, H3A 2T8 Quebec,

*) Permanent address: Department of Solid State Physics, Polish Academy of Sciences, Zabrze, Canada.

Poland.

482 M. SINQH, J. CISOWSKI, P. R. WALLACE, J. C. PORTAL, and J. M. BROTO

Arushanov et al. [8] Cd3P2 has a single nearly spherical conduction band, which makes i t possible to use the Bodnar model for this semiconductor in order to determine its band structure near the r-point.

The purpose of our work is to describe the band structure of Cd3P, in the presence of a magnetic field on the basis of Wallace’s extension of the Bodnar model. Special attention will be paid to the electronic g-factor where we will include the contribution of “free electron” term which has been calculated in a recent paper by Singh and Wal- lace [9]. The correction due to the free electron term is about 4% at E, = 20 meV and about 15% at E , = 250 meV.

Here, E is the energy of electrons from the bottom of the conduction band, E, the energy gap, A the spin-orbit splitting, 6 the crystal field splitting. P , and PI, are interband matrix elements. In the presence of a magnetic field Wallace has found the following Landau energy levels including spins :

where 1 is the cyclotron radius, 8 the angle between magnetic field and c-axis, k, the wave vector along the magnetic field, and n the Landau quantum number.

2.2 g-factor

The effective g-factor can be defined as

9 = 9w + 9f;

9 w =

1 [E(nt, kB = 0) - E(ni, kB = O ) ]

P B B (3)

where E ( m ) is the Landau energy in the absence of the free electron term and is ob- tained by (2). &(%a) is the correction to the noc Landau level due to the so-called free electron term and recently given in (9).

We write the following expression for gw as a function of energy: { ( E + E, + 6)a P? C O S ~ e + ( E + Eo)2 sin zee)

S 9w =

The Band Structure of Cd,P, in the Presence of a Magnetic Field 483

We will calculate here the g,-factor such that the Landau level falls on the Ferrni energy in question a t the appropriate value of magnetic field. Under this condition one can express gf as

1 Sf = (Fl - Fz) (Tl - Tz) ; = npo(n - 1) 9 pz = (n + 1) Fo(n + 1) , I

where Cai are normalization constants of Landau wave funct,ions which are given in cg, 101.

An expression for 1/m: as a continuous function of energy is written as

l/m: can be calculated drectly from ( 6 ~ ) .

3. Results and Discussions we have chosen from the literature the following parameters as most acceptable, which were obtained by optical and transport properties of Cd3P2 :

E,, = 0.58 eV [ l l ] ,

d = 0.23 eV 1113,

P , = PI, = 7 x eV cm [11], 6 = 0.08 eV .

The value of 6 has been taken from the work of Bodnar [3] and its positive sign has been recently confirmed by SdH experiments [S].

The E versus k, and E versus k, in Fig. 1 are obtained by using (1). One can see clear- ly that in Fig. 1 b the uppermost valence band is flat according to the present model, as in Cd3As2. In Fig. 1 a its shape a t the r-point is like "flat camel-back", but in Cd3As2 its shape is just opposite to Cd3Pz. It is evident that the simple Bodnar model, Like the Kane model for InSb, oversimplifies the treatment of this band giving much too large a value of the effective mass. The following improvements are possible: (i) taking account of the contribution of higher bands, (ii) taking account of the true structure, in which the distribution of vacant lattice sites will affect the band structure notably

4 84 M. SINGE, J. CISOWSKI, P. R. WALLACE, J. C. PORTAL, and J. M. BROTO

Fig. I . Dispersion relation for the four-level model of CqPz

in introducing Brillouin zone boundaries nearer to the I?-point, and (iii) including the effect of the "free electron term".

By using (2) we have calculated E versus kB curves for different Landau levels for the conduction band. Fig. 2a shows the Landau energy levels as a function of ks for B = 10 T for 8 = Oo and 8 = 90°. There is a small anisotropy which is increasing with increasing Landau level. The ddference in energy between two Landau levels is approximately constant for higher quantum numbers. The spin splitting is too sinall to be represented in the diagram.

Similarly Fig. 2b shows the Landau energy levels as a function of kB for B = 25 T

Here, the anisotropy is larger than in the previous figure but the nature of the curves is the same.

The variation of Fermi level with magnetic field has been calculated in the degener- ate case for the concentrations 1 x lo1', 5 x lo1', 1 x 10l8 The results are shown in Fig. 3 only for 8 = 0". We found that the oscillation amplitude in Cd3P, is larger than in Cd3As,. The arrows A, B, and C in Fig. 3 represent the values of Fermi levels in the absence of magnetic field. The spin splitting i s shown only for n = 0 and n = 1 and for higher n i t is too small to be drawn in the diagram.

The g-factor has been calculated from (4) and (5) as a function of energy for 8 = 0" and 8 = 90". The results are shown in Fig. 4. The free electron contribution is approx- imately 4.0% a t E , = 20 meV ( w 1.5 x 101' and about 15% a t E , = 250 meV

for 8 = 0" and 8 = 90".

Fig. 2. Landau energy levels of C4P2 as a function of kB for a) B = 10 and b) 25 T. - - - 8 = o", e=w0

The Band Structure of Cd,P, in the Presence of a Magnetic Field 485

Fig. 3. The Landau levels for kg = 0 and 0 = 0' 8 s a function of magnetic field; the oscillations of Fermi level are also shown for concentrations 1 x lo1', 6 x lo1', and 1 x 10l8 respectively

(1 x l O l e ~ r n - ~ ) . In other words, i t is much less than 1 and almost negligible for lower energies. Recently, Gulzman et al. [12] found a good agreement between theory and experiment in InSb (which is direct-band gap material like Cd,P,) without including the free electron term and higher bands, up to a concentration 4 x 10'' cmd3 (EB = = 0.06 eV) a similar conclusion was reached by Pidgeon and Brown [13]. Therefore, our calculation is consistent with the results of the above works.

We have also calculated the g-factor by using A = 0.1 eV which was estimated by Gelten et al. [14] from their optical measurements. For this case, the values of the g-factor are approximately half the previous ones. As one can see from these results the anisotropy is decreasing as the energy is increasing. Unfortunately, there are no experimental data on the g-factor available so far.

In the calculation of effective cyclotron mass with equation (7) we found that the cyclotron mass is a linear function of energy, i.e.,

where a = - (3.8 + 0.19 cos2 e) x 10-2, /? = 15.39 x lo-'.

Here the free electron contribution has been neglected. With the help of (8) we have also calculated the effective mass correction due to the free electron term, and it has been found that its contribution is approximately 6 to 15%, depending on the energy.

Fig. 4. The g-factor as a function of energy in C4P2. 0 = 0" and -- - 0 = 90". For (A) A = 0.23eV

and for (B) A = 0.10 eV I. 0

050 50 loo 1% zeo 250 E heV i -

486 M. SINGE et al. : The Band Structure of Cd,P, in a Magnetic Field

Acknowledgements

We wish to thank Prof. Askenazy for numerous helpful discussions. Two of the authors (M.S. and P.R.W.) also wish to acknowledge the support of the National Science and Research Council of Canada, for essential financial support in the form of a research grant. One of the authors (J.C.) is thankful to CNRS for financial support.

References

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Montpellier 1982.

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(Received June 2, 1982)