Theory of Burstein-Moss effect in semiconductors with anisotropic energy bands

We study the peculiarities of the Burstein–Moss shift employing two-band model with an anisotropic valence band. There is a long wave tail which has a convex or concave shape depending on the ratio between the longitudinal and transverse hole masses. The width of this anisotropy-induced tail is temperature-independent and increases with increasing electron concentration and difference between the hole masses. This width also does not depend upon the value of the energy gap. Having experimentally evaluated the tail width and the position of the break in the optical absorption curve, one can deduce the values of the reduced hole masses.


Introduction
It is well-known that in non-degenerate semiconductors the optical activation energy corresponds to the bandgap of the semiconductor.As the doping concentration is increased, electrons populate states within the conduction band which pushes the Fermi level to higher energy.This leads to the fact that the apparent band gap in optical experiments slightly increases.Such a phenomenon is known as the Burstein-Moss effect or the Burstein-Moss shift [1].
The difference between the optical and true band gaps has been studied in many bulk semiconductors and semiconductor devices [2][3][4][5][6].The last decades have been marked by intensive study of this effect in various types of nanostructures [7][8][9][10].Saleh and Soliman [11] have analysed the junction photon detectors based upon the Burstein effect.Turgut [12] has demonstrated that the Burstein-Moss effect will be crucial in the design of InN based lasers.Recently, Zhou et al [13] have proposed that the Burstein-Moss effect could be a potential mechanism to realize the multicolor tunable electrochromic materials phenomenon.
It should be noted that the existing theory of the Burstein-Moss effect [14,15] is based on the assumption that energy spectrum of both electrons and holes is isotropic.On the other hand, we have the right to expect that in anisotropic materials the Burstein-Moss shift has its own special features.

Theoretical analysis
In order to reveal a new feature of the optical absorption edge, we consider the simplest version of the two-band model in which the energy spectrum of the valence band (v) is assumed to be axially symmetric, while the spectrum of the conduction band (c) is isotropic: ( ) ( ) where ε g > 0 is the optical energy gap; ÿ is the reduced Planck's constant; m e > 0 is the conduction electron mass; m ∥ > 0 and m ⊥ > 0 are the longitudinal and transverse hole masses; θ is the angle between main crystalline axis of crystal and the wave vector k.
We take this approximation since, in all known direct band gap semiconductors the electron effective mass is isotropic or has a negligible anisotropy in a comparison with the hole effective mass which can have much more pronounced anisotropy [16].The reason is that the anisotropy of the hole bands is due to the 'interaction' of the various hole bands (light, heavy and spin-orbit) via anisotropic part of crystal field potential [17].The electron effective mass is isotropic because the band effectively does not 'interact' with other bands via anisotropic potential [17].
The well-known equation for the absorption coefficient α produced by direct transitions [18] has the following form in a spherical coordinate system: where e is the elementary charge; ò 0 is the vacuum permittivity; m 0 is the free electron mass; c is the velocity of light in free space; n is the refractive index of the semiconductor; ω is the angular frequency of the incident light; d cv is the optical matrix element; f (ε v ) and f (ε c ) denote the Fermi-Dirac distribution functions of the participating bands.Using the property of the delta function of a function, we write: where k 0 is the root of equation ε c − ε v − ÿω = 0. Taking into account the axial symmetry of the problem, equations (1)-( 4), and the law of conservation of energy, we get: is the characteristic absorption coefficient; β = ÿω/ε g ; r ∥ = m e /m ∥ ; r ⊥ = m e /m ⊥ .Usually, r ⊥ , r ∥ < 1 [16].
For a degenerate n-type semiconductor at low absolute temperature, the direct inter-band optical transitions are possible only on levels ε c (k 0 ) = ε v (k 0 ) + ÿω ε F , where ε F is the Fermi energy as measured from the top of the valence band in the vertically upward direction.Using equations (1) and (2)we can rewrite this inequality as Therefore, at m ⊥ < m ∥ the direct optical transitions are possible only for the values of θ smaller than some critical value of this angle satisfying the identity: At this condition, the square bracket in equation (9) turns to 1.At θ cr < θ < π/2 the bracket is equal to zero.If m ⊥ < m ∥ , the situation is exactly the opposite of that just discussed.Thus, we should perform the integration in equation (9) keeping in mind that the square bracket behaves like the Heaviside step function.The results are as follows: for m ⊥ < m ∥ and (γ − 1) or m ⊥ > m ∥ and β (γ − 1)(1 + r ∥ ) + 1 the result is also given by equation (15).

Results and discussion
In figure 1 we plot dependences α(β) constructed at different values of r ⊥ , r ∥ , and γ.All three curves exhibit steplike behavior.However, in the anisotropic cases, there is a blurring of the steps.In other words, we deal with a long wave tail in these two cases.If m ⊥ > m ∥ , then the tail is a convex curve.For m ⊥ < m ∥ the tail is a concave curve.Taking into account the finite temperature as well as long-range potential fluctuations (Urbach broadening [19]), the breaks on the absorption curve are smoothed out, that is, it becomes a smooth curve.The relative width of the tail is defined as: the value of this quantity is temperature-independent and increases with increasing γ and the relative difference between the hole masses (figure 1).As the electron mass increases relative to these masses, the width increases too.
The revealed tail may be observed by experiment even in heavily doped semiconductors with an isotropic valence band in the case of uniaxial or biaxial stress.Under these conditions, the effect is caused by straininduced anisotropy of hole masses [20,21].
Having experimentally evaluated the tail width (ξ) and the position of the break (β br , figure 1) in the optical absorption curve, we can deduce the values of the reduced hole masses.Indeed, for r ⊥ > r ∥ , Using these equalities and equation (13), we immediately get for the concave tail: The Fermi energy, that appears in equations ( 11)-( 14) and ( 16)- (15), in the case of strong degeneracy of n-type semiconductor and a sufficiently low temperature (k B T = ε F − ε g , where k B T is the characteristic thermal energy) can be estimated by the following expression [14]: where n e is the electron concentration (the net donor concentration).Taking into account equations (13) and (16) we conclude that the width of the tail does not depend on the value of energy gap.
We should note that the Burstein-Moss shift is often partially compensated by band-gap shrinkage [22].This renormalization of energy gap is considered as a result of mutual exchange and Coulomb interactions between the added free electrons in the conduction band and electron impurity scattering [23].The calculation of the renormalization of the band gap is a separate fundamental problem.At the same time, in our analysis, we can treat ε g as a quantity that has already been estimated from another experiment and properly renormalized.

The case of CdGeAs 2
Let us consider as an illustrative example the case of degenerate n-type tetragonal semiconductor CdGeAs 2 .Cadmium germanium arsenide is a promising non-linear optical material for use in infrared frequency  conversion devices.Doping CdGeAs 2 crystals with indium, selenium, and tellurium has resulted in n-type conductivity [24].The theoretical model [17] predicts the giant anisotropy of the heavy hole masses and a negligible anisotropy of the conduction electron mass in this compound.Wherein, the smaller of the hole masses is approximately equal to the electron mass.A completely similar situation occurs for the binary tetragonal semiconductor Cd 3 P 2 [25].
The experimentally observed tail in n-type bulk CdGeAs 2 crystals was previously attributed with only Urbach broadening [24].Many more careful experimental data provided especially at very low temperatures and very high electron concentrations are still needed to identify the above effect and isolate it from Urbach broadening.

Conclusions
The existence of the anisotropy-induced tail in the optical absorption spectrum allows one to explore in detail the anisotropy of the valence band.In particular, it can provide an independent information on the difference between the longitudinal and transverse hole masses.The possible non-parabolicity of the spectrum [14] and small anisotropy of the conduction band [17] can also be taken into account within the framework of this mathematical model to refine the position and width of the tail.

Figure 2 .
Figure 2. Optical absorption coefficient as a function of the photon energy for n-type CdGeAs 2 at T = 0.The electron concentration is 2 • 10 18 cm −3 .

Figure 3 .
Figure 3.The width of anisotropy-induced tail as a function of the electron concentration in n-type CdGeAs 2 .