The projective representations of the wavevector group were applied to the analysis of exciton-phonon transitions in enantiomorphous tetragonal crystals ZnP₂ and CdP₂. Selection rules for indirect transitions were derived. Using pkrr-method, the different points in the Brillouin zone were examined to identify zero slopes and extremums of the dispersion curves E(k). Photoluminescence and edge absorption polarized spectra were interpreted taking into account the results of above theoretical analysis

Biliy, M.M.

Gubanov, V.O.

Koryakov, S.V.

Slobodyanyuk, O.V.

Yanchuk, Z.Z.

Semiconductor Physics Quantum Electronics & Optoelectronics

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

 Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 1. P. 19-24.  PACS: 02.20.–a, 78.55.–m, 78.40.Fy  Application of the method of projective representations to the analysis of exciton-phonon transitions in enantiomorphous tetragonal crystals ZnP2 and CdP2  S.V. Koryakov, V.O. Gubanov, M.M. Biliy, O.V. Slobodyanyuk, Z.Z. Yanchuk  Taras Shevchenko Kyiv National University, Department of Physics 2, prospect Academician Glushkov, 03022 Kyiv, Ukraine Phone: +38 (067) 465-64-18 E-mail: sergiy_koryakov@univ.kiev.ua  Abstract. The projective representations of the wavevector group were applied to the analysis of exciton-phonon transitions in enantiomorphous tetragonal crystals ZnP2 and CdP2. Selection rules for indirect transitions were derived. Using pkrr-method, the different points in the Brillouin zone were examined to identify zero slopes and extremums of the dispersion curves )(kEr. Photoluminescence and edge absorption polarized spectra were interpreted taking into account the results of above theoretical analysis.   Keywords: projective representations, exciton-phonon transitions, α-ZnP2, CdP2.  Manuscript received 12.11.04; accepted for publication 18.05.05.  Symmetry analysis helps to predict some basic features of electronic transitions that appear as bands in optical spectra. The transition probability is defined by the product of the squared transition matrix element and combined density of states. Symmetry analysis gives selection rules, i.e., it shows what matrix elements are non-zero. The high density of states is expected in the vicinity of points with zero slope and/or extremums of the energy dispersion curves )(kEr. In this paper, we represent the method of selection rule calculations for exciton-phonon transitions in the tetragonal enantiomorphous crystals α-ZnP2 and CdP2. These crystals belong to P41212 or P43212 space symmetry groups. Earlier projective representations were used for analysis of one- and two-phonon Raman spectra of the same crystals [1, 2]. The tetragonal crystals of zinc and cadmium diphosphides are indirect semiconductors, so it is necessary to examine interband transitions between different points of the Brillouin zone (Fig. 1). Here we examine the high symmetry points Г ( 0=Γkr), Z (21Zbkrr −= ), А (222321Abbbkrrrr −−−= ), М (2232Mbbkrrr −−= ), Х (22Xbkrr −= ) and Х' (23Xbkrr −=′ ), R ( 2221Rbbkrrr −−= ) and R' (2231Rbbkrrr −−=′ ). Points Г, Z, А, and М possess fullsymmetry of the point group 422 as well as X, X', R, and R' ones belong to the point group 222. To calculate selection rules and to ascertain whether the mentioned points are those of band extremums, it is necessary to know irreducible representations of wavevector groups. The irreducible representations of the wavevector groups can be found using the method of projective representations [1, 3]. To derive the selection rules for exciton-phonon transitions, we need to know representations both for phonon and electron states. The electron states are transformed according to the representations Γ6, Γ7, (Z1 + Z2), (Z3 + Z4), Z5, (А1 + А3), (А2 + А4), ((A5))·(M6 + М7), (X1 + X3), (X2 + X4), (X'1 + X'4), (X'2 + X'3), ((R5)), and ((R'5)). The phonon states are transformed according to the representations Γ1 – Γ5, Z6, Z7, (A6 + A7), (M1 + M4), (M2 + M3), М5, X5, (R1 + R4), (R2 + R3), (R'1 + R'3), and (R'2 + R'4). Parentheses are for merging of complex conjugated representations. Double parentheses are for duplication of two-dimensional representations. The merging and duplication of representations are caused by the invariance with respect to the time inversion and governed by the Herring criterion [4]. The projective representations for Λ, S, V, Σ, Δ, W, and U points depend on the wavevector [3]. Their evolution in certain direction can be pictured graphically as shown in Fig. 2 for Γ – Σ – M.   © 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine  19  Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 1. P. 19-24.    Fig. 1. Schematic drawing of the Brillouin zone that corresponds to tetragonal crystals of space symmetry groups Р41212 and Р43212.    Fig. 2. Evolution of representations in Г – M direction for groups P41212 (solid line – electron states, dashed line – phonon states).  The selection rules for exciton-phonon transitions follow from the formula: )()()()()(1 *phex0 ggggghN qkGgskkην∈κμ′− χχχχχ= ∑ rrrr, where  and  are characters of representations of hole and electron states in valence and conduction bands, respectively;  is the character of representation of the dipole transition operator;  is the character of the hole and electron moving relatively to their center of gravity,  is the character of the phonon state representation. For the allowed transition , and for the forbidden one . If any external magnetic field is absent, the characters of merged representations have to be used (selection rules are shown in Table 1). )(gk∗μ′χ r )(gkνχ r)(gsχ)(gκχ)(gqηχ r0phex0 ≠−N0phex0 =−NFor the point 0kr with zero slope of energy dispersion curves, the condition 0)( 0 =αdkkdEr should be fulfilled. In addition, in the point of extremum the components of inverse effective mass tensor αβm1     Fig. 3. Photoluminescence spectra of tetragonal crystals ZnP2 (a) and CdP2 (b).   should not be equal to zero. We have analyzed using the pkrr-method [4] whether examined points are the critical ones (results are summarized in Tables 2 and 3). From Table 3, it is easy to see that there are non-zero components of the inverse effective mass tensor αβm1  in every examined point of the Brillouin zone. It means that some dispersion curves of )(kEr may have the extremum in these points. But not all points are those of zero slope (see Table 2). There is no point in the Brillouin zone that has all zero components αdkkdE )( 0r for both electron and phonon states. Photoluminescence and absorption polarized spectra of α-ZnP2 and CdP2 (Figs 3, 4) were obtained at the temperature 1.7 K using a spectrometer DFS-24 with the resolution less than 1 cm–1 at λ = 550 nm. Photoluminescence was excited by radiation of Ar+ laser (λ = 514.5 nm) with the output power 900 mW. The main feature of photoluminescence spectra in both polarizations (Fig. 3) is the equal number of bands with approximately the same phonon energies. It means that all the observed interband transitions are allowed in both polarizations.   © 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine  20  Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 1. P. 19-24.  Table 1. Selection rules for exciton-phonon transitions between different points of the Brillouin zone in tetragonal crystals ZnP2 and CdP2.  Type of transition Allowed transitions Forbidden transitions Г2⊂Г6×(Z1+Z2)×Z7,Г6×(Z3+Z4)×Z6, Г6×Z5×Z6,Г6×Z5×Z7, Г7×(Z1+Z2)×Z6,Г7×(Z3+Z4)×Z7,Г7×Z5×Z6,Г7×Z5×Z7 Г2⊂Г6×(Z1+Z2)×Z6, Г6×(Z3+Z4)×Z7, Г7×(Z1+Z2)×Z7, Г7×(Z3+Z4)×Z6 Г – Z Г 5⊂Г6×(Z1+Z2)×Z6, Г6×(Z1+Z2)×Z7, Г6×(Z3+Z4)×Z6, Г6×(Z3+Z4)×Z7,Г6×Z5×Z6,Г6×Z5×Z7,Г7×(Z1+Z2)×Z6, Г7×(Z1+Z2)×Z7,Г7×(Z3+Z4)×Z6,Г7×Z5×Z6, Г7×Z5×Z7 Г 5⊂0 Г 2⊂Г6×(А1+А2)×(А6+А7), Г6×(А3+А4)×(А6+А7), Г6×2А5×(А6+А7), Г7×(А1+А2)×(А6+А7), Г7×(А3+А4)×(А6+А7), Г7×2А5×(А6+А7) Г 2⊂0 Г – А Г 5⊂Г6×(А1+А2)×(А6+А7), Г6×(А3+А4)×(А6+А7), Г6×2А5×(А6+А7), Г7×(А1+А2)×(А6+А7), Г7×(А3+А4)×(А6+А7), Г7×2А5×(А6+А7) Г 5⊂0 Г2⊂Г6×(М1+М2)×(М6+М7), Г6×(М3+М4)×(М6+М7), Г6×М5×(М6+М7), Г7×(М1+М2)×(М6+М7), Г7×(М3+А4)×(М6+М7), Г7×2М5×(М6+М7) Г2⊂0 Г – М Г 5⊂ Г6×(М1+М2)×(М6+М7), Г6×(М3+М4)×(М6+М7), Г6×М5×(М6+М7), Г7×(М1+М2)×(М6+М7), Г7×(М3+А4)×(М6+М7), Г7×2М5×(М6+М7) Г 5⊂0 Г2⊂Г6×(X1+X3)×X5, Г6×(X2+X4)×X5, Г7×(X1+X3)×X5, Г7×(X2+X4)×X5 Г2⊂0 Г – Х Г 5⊂Г6×(X1+X3)×X5, Г6×(X2+X4)×X5, Г7×(X1+X3)×X5, Г7×(X2+X4)×X5 Г 5⊂0 Г2⊂Г6×(X'1+X'4)×X'5, Г6×(X'2+X'3)×X'5, Г7×(X'1+X'4)×X'5, Г7×(X'2+X'3)×X'5 Г2⊂0 Г – Х' Г 5⊂Г6×(X'1+X'4)×X'5, Г6×(X'2+X'3)×X'5, Г7×(X'1+X'4)×X'5, Г7×(X'2+X'3)×X'5 Г 5⊂0  © 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine  21  Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 1. P. 19-24.  Г2⊂Г6×((R5))×(R1+R4), Г6×((R5))×(R2+R3), Г7×((R5))×(R1+R4), Г7×((R5))×(R2+R3) Г2⊂0 Г – R Г 5⊂Г6×((R5))×(R1+R4), Г6×((R5))×(R2+R3), Г7×((R5))×(R1+R4), Г7×((R5))×(R2+R3) Г 5⊂0 Г2⊂Г6×((R'5))×(R'1+R'3), Г6×((R'5))×(R'2+R'4), Г7×((R'5))×(R1+R'3), Г7×((R'5))×(R'2+R'4) Г2⊂0 Г – R' Г 5⊂Г6×((R'5))×(R'1+R'3), Г6×((R'5))×(R'2+R'4), Г7×((R'5))×(R'1+R'3), Г7×((R'5))×(R'2+R'4) Г 5⊂0     Table 2. Number of components of momentum for different points in the Brillouin zone of crystals ZnP2 and CdP2.  Number of components 0)( 0 ≠αdkkdEr Representation || z ⊥ z Г1, Г2, Г3, Г4 Г5  Г6, Г7 0 1 1 0 0 1 (Z1+Z2), (Z3+Z4) Z5  Z6, Z7 2 diagonal; 2 non-diagonal 1 0 0 diagonal; 0 non-diagonal 0 0 (М1+М4), (М2+М3) М5 (M6+M7) 0 diagonal; 0 non-diagonal 1 2 diagonal; 1 non-diagonal 0 diagonal; 0 non-diagonal 0 2 diagonal; 3 non-diagonal  in Г - A direction (A1+A3), (A2+A4) ((A5))  (А6+А7) 0 diagonal; 0 non-diagonal 4 diagonal; 3 non-diagonal 6 diagonal; 2 non-diagonal  || z || x | | y (X1+X3), (X2+X4) X5 0 diagonal; 0 non-diagonal 0 2 diagonal; 2 non-diagonal 1 0 diagonal; 0 non-diagonal 0  (X'1+X'4), (X'2+X'3) X'5 0 diagonal; 0 non-diagonal 0 0 diagonal; 0 non-diagonal 0 2 diagonal; 2 non-diagonal 1  (R1+R4), (R2+R3) ((R5)) 0 diagonal; 0 non-diagonal 4 diagonal; 1 non-diagonal 0 diagonal; 0 non-diagonal 4 diagonal; 1 non-diagonal 2 diagonal; 2 non-diagonal 4 diagonal; 2 non-diagonal   (R'1+R'3), (R'2+R'4) ((R'5)) 0 diagonal; 0 non-diagonal 4 diagonal; 1 non-diagonal 2 diagonal; 2 non-diagonal 4 diagonal; 2 non-diagonal 0 diagonal; 0 non-diagonal 4 diagonal; 1 non-diagonal       © 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine  22  Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 1. P. 19-24.  Table 3. Number of components of inverse effective mass tensor for different points in the Brillouin zone of crystals ZnP2 and CdP2. Number of components αβm1 Representation || z ⊥ z Г1, Г2, Г3, Г4 Г5  Г6, Г7 1 1 1 1 3 1 (М1+М4), (М2+М3) М5 (M6+M7) 2 diagonal; 1 non-diagonal 1 2 diagonal; 1 non-diagonal 4 diagonal; 3 non-diagonal 2 6 diagonal; 3 non-diagonal (A1+A3), (A2+A4) ((A5))  (А6+А7) 6 diagonal; 2 non-diagonal 16 diagonal; 6 non-diagonal 12diagonal; 6 non-diagonal (Z1+Z2), (Z3+Z4) Z5  Z6, Z7 2 diagonal; 2 non-diagonal 2 3 (X1+X3), (X2+X4) X5 8 diagonal; 3 non-diagonal 2  (X'1+X'4), (X'2+X'3) X'5 8 diagonal; 3 non-diagonal 2 (R1+R4), (R2+R3) ((R5)) 8 diagonal; 3 non-diagonal 24 diagonal; 16 non-diagonal (R'1+R'3), (R'2+R'4) ((R'5)) 8 diagonal; 3 non-diagonal 24 diagonal; 16 non-diagonal     Fig. 4. Edge absorption spectra of tetragonal crystals ZnP2 and CdP2. Comparing the spectra of exciton-phonon transitions, phonon Raman scattering [1, 2], and taking into consideration the results of our symmetry analysis (see Tables 1, 2, and 3) we can suppose that the minimum of conduction band may be located in M point. There are two considerations for this conclusion: 1) all Г – M transitions are allowed in both polarizations, but some of Г – Z transitions are forbidden in Er||z polarization (it was mentioned that the main feature of the obtained photoluminescence spectra (Fig. 2) is the equal number of bands with approximately the same phonon energies in both polarizations); 2) for electrons in M point, there are minimal number of non-zero components αdkkdE )( 0r in Г – M direction and M point is the point of zero slope for all the dispersion curves of phonon states in the same direction. The points Z and R are not those of zero slope for all dispersion curves of electron states in the corresponding directions. In the point A, there are non-zero components αdkkdE )( 0r both for electron and phonon states. Additional criteria to further selection of a candidate for the conduction band extremum can be obtained from the measurements of free-exciton luminescence in magnetic fields when the degeneration of energy states will be removed as caused by merging the states. For instance, the energy dispersion curves in Г – M direction that corresponds to non-merged states M6 and M7 will have zero slope in M point.  © 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine  23  Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 1. P. 19-24.  References  1. V.O. Gubanov, S.V. Koryakov, R.A. Poveda, Z.Z. Yanchuk, The analysis of dispersion of phonon states in tetragonal crystals ZnP2 and CdP2 by method of projective representation // Zhurn. Fiz. Doslidzhen’ 6 (1), p. 109-117 (2002). 2. S. Garasevich, V. Gubanov, P. Korenyuk, S. Koryakov, A. Slobodyanyuk, Z. Yanchuk, Two- phonon Raman spectra of tetragonal crystals ZnP2, CdP2, and TeO2 // SPIE Proc. 5507, p. 26 (2004). 3. V.O. Gubanov, S.V. Koryakov, Projective representations construction for different points of Brillouin zone. Application for space symmetry groups P41212 and P43212 // ArXiv: cond-mat. 0406519 (2004). 4. G.L. Bir, G.E. Pikus, Symmetry and deformation effects in semiconductors. Nauka, Moscow (1972).    © 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine  24 

Application of method of projective representations to the analysis of exciton-phonon transitions in enantiomorphous tetragonal crystals ZnP₂ and CdP₂

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Application of method of projective representations to the analysis of exciton-phonon transitions in enantiomorphous tetragonal crystals ZnP₂ and CdP₂

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Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)