1. Introduction

Nowadays, the nitride semiconductors such as GaN, AlN and InN attract a considerable attention due to their outstanding physical, chemical and mechanical properties and also because of the recent progress in the technology that allowed to produce high quality nitride films with help of MOVPE and MBE (for a recent review, see Ref. Reference Strite, Lin and Morkoc[1]). The attractive properties of the nitrides include high heat conductivity, hardness, chemical stability and high luminescence intensity. These wide gap semiconductors are very promising materials for LEDs and semiconductor lasers in wide spectral interval from ultra-violet to green and even orange Reference Nakamura, Senoh, Iwasa, Nagahama, Yamada and Mukai[2] Reference Nakamura, Senoh, Iwasa and Nagahama[3] because their solid solutions may have the energy gap varying from 2eV in InN to 6.2eV in AlN.

Important characteristics of materials used in luminescence devices are the rates of different electron—hole recombination processes. However, there is no information at the moment about the intensities of these processes in the nitrides. In this work, we concentrate on the calculation of the radiative recombination rate in GaN, InN, AlN and their binary alloys Ga_xAl_1−xN and In_xAl_1−xN on the base of the experimental absorption data that is present in the literature Reference Yoshida, Misawa and Gonda[4] Reference Dingle, Sell, Stokowski and Ilegems[5] Reference Perry and Rutz[6] Reference Bloom, Harbeke, Meier and Ortenburger[7] Reference Tansley and Foley[8], the calculated electron energy band dispersion laws Reference Jones and Lettington[9] Reference Rubio, Corkill, Cohen, Shirley and Louie[10] Reference Foley and Tansley[11] Reference Huang and Ching[12] and the temperature dependence of the band gaps in the pure materials Reference Guo and Yoshida[13] Reference Teisseyre, Perlin, Suski, Grzegory, Porowski, Jun, Pietraszko and Moustakas[14] Reference Manasreh[15].

2. Energy Spectra of the Nitride Semiconductors

We consider more common and popular hexagonal phase of the nitrides. All of them belong to the crystal class C_6v. Their conductivity bands are non-degenerate, and their electron states originate from atomic s-functions. In the Γ point of Brilluine zone they transform according to Γ₁, the unite representation of C_6v. The valence band is complicated and consists of two branches. One of them transforms according to Γ₁ whereas the other is degenerate and form the two-dimensional representation Γ₆. If spin—orbit interaction is taken into account, Γ₆ farther splits into two bands, Γ₇ and Γ₉ Reference Manasreh[15]. However, the latter splitting manifest itself only along k_x and k_y, but it equals zero in the very Γ point and along k_z, z being the direction of the hexagonal axis c which usually coincides with the normal to the film. We will neglect this splitting and consider Γ₆ as a degenerate band.

According to the results of Ref. Reference Bloom, Harbeke, Meier and Ortenburger[7] Reference Jones and Lettington[9] Reference Rubio, Corkill, Cohen, Shirley and Louie[10] Reference Foley and Tansley[11] Reference Pastrnak and Roskovcova[16] Reference Bloom[17], the order of levels in the valence band of the nitrides is different: in GaN and InN, Γ₆ branch lies above Γ₁, whereas in AlN it lies below Γ₁. The symmetry of the electron wave functions in different bands and band branches leads to the following selection rules for the radiative transitions:

• • For the transition from Γ₁ ^c to Γ₁ ^v, only z component of the transition matrix element differs from zero, and correspondingly, the emitted photon is polarized along z axis.

• • On the contrary, for the transition from Γ₁ ^c to Γ₁ ^v, z component of the transition matrix element equals zero, and the emitted photon polarization is perpendicular to z axis.

3. Radiative Recombination Rate: The Method of Calculation

Usually, Shockley-van Roosbroeck formula is used for the calculation of the radiative recombination intensity Reference Bloom[17], which allows one to calculate the transition rate if the spectral dependence of the absorption coefficient is known. However, this is not the case in the nitrides where the optical absorption has been measured only in a small vicinity of the band edge, and another method is needed that would allow to express the recombination rate through the material parameters.

A straightforward quantum-mechanical calculation similar to given in Ref. Reference Landsberg[18] leads to the following expression for the spontaneous radiative recombination rate:

where n is the refractive index, m₀ is free electron mass, μ_x = (m_e,x ⁻¹ + m_h,x ⁻¹)⁻¹ is the reduced carrier mass in x direction, and similar for y and z, and

where e _kλ is the polarization direction of the photon with the momentum {\bf k}, the sum is over two polarizations, the averaging is over all photon momentum directions, and P _cv is the interband transition matrix element at Γ point of the Brilluine zone.

Defining the radiative recombination coefficient according to the equality

where n and p are carrier concentrations, one comes to the following expression:

where

4. Temperature and Alloy Composition Dependence of the Energy Gap

To derive the accurate temperature dependence of the radiative recombination rate, one needs to know how the gap varies with the temperature in the materials under consideration. In many semiconductors, including the nitrides, the empiric Varshni formula Reference Varshni[19] approximates well the observed temperature dependence of the gap:

where γ and β are parameters. Their values for nitride thin films were found out in Ref. Reference Guo and Yoshida[13] Reference Teisseyre, Perlin, Suski, Grzegory, Porowski, Jun, Pietraszko and Moustakas[14] Reference Manasreh[15]:

There are data (for example, Ref. Reference Manasreh[15] for GaN) pointing out on a dependence of Varshni's formula parameters on the different methods of fabrication of the nitride films with the same wurtzite structure. In our calculation we have taken values of these parameters found in single-crystal films produced by MBE.

For the gap value in the binary alloys with the components A and B, we used the expression

with d=1.0 for Ga_xAl_1−xN and d=2.6 for In_xAl_1−xN Reference Yoshida, Misawa and Gonda[4] Reference Koide, Itoh, Khan, Hiramatu, Sawaki and Akasaki[20] Reference Phillips[21].

5. Calculation of the Interband Matrix Element

To calculate the interband transition matrix element, we made use of the formula for the absorption coefficient that also contains P _cv (see, for example, Ref. ∼ Reference Landsberg[18]):

where the integration is over all polarization directions in the plane perpendicular to the incident photon momentum.

Extracting the coefficient b from the measured frequency dependence of α, one can find P _cv components in the xy plane assuming that the light beam was perpendicular to the film surface. However, this method does not allow us to find P_cv,z. This is not important for the transitions from Γ_1,c to Γ_6,v as in InN and GaN (see selection rules above), but in AlN where Γ_1,v band lies higher than Γ_6,v, this may cause problems. We still have calculated B for AlN, too, assuming that Γ_1,v - Γ_6,v splitting in AlN is rather small Reference Yamashita, Fukui, Misawa and Yoshida[22], and the hole population of Γ_6,v, and hence the transition probability to this band, may be higher than that in Γ_1,v due to much higher density of states in Γ_6,v. The values of P_cv,x=P_cv,y found this way together with other material parameters used in calculations are listed below:

The material parameters for the alloys were found using a simple linear interpolation between the values for the alloy components.

The calculated temperature dependence of the radiative recombination coefficient B is shown in Figures 1 and 2. One can see that B is highest in InN and the compositions close to it, and lowest in GaN and the alloys reached in this material.

Figure 1. The radiative recombination coefficient in the binary alloys (In,Al)N. x is the InN fraction in the alloy.

Figure 2. The radiative recombination coefficient in the binary alloys (Ga,Al)N. x is the GaN fraction in the alloy.

6. Conclusion

We calculated radiative recombination coefficients for three nitride semiconductors and their binary alloys. To do it, we extracted values of the interband matrix elements from the absorption data. The matrix elements do not differ considerably in these semiconductors, and the difference between the recombination coefficients is connected also with the difference in the band gap values and carrier masses.