Superlattices and Microstructures, Vol. 7 7, No. 2, 1992 167 OPTICAL PROPERTIES OF SEMICONDUCTOR NANOCRYSTALS WITH DEGENERATE VALENCE BAND Al. L. Efros Department of Physics E16, Tech. Univ of Miinchen, 8046 Garching, Germany (Received 19 May 1991) The energy spectrum and wave functions of holes in CdSe and CdS nanocrystals have been calculated. It is shown that in CdS nanocrystals the lowest quantum size level (QSL) of holes has p-type symmetry and optical transitions between this state and the lowest QSL of electrons are forbidden. The radiative lifetimes in CdSe nanocrystals are estimated to be of the order of 23 ns. The real structure of the valence bands have been shown to increase the electron-hole pair - polar phonons coupling with decreasing nanocrystal size. 1. Introduction It is well known that the qualitative description of the optical properties of semiconductors needs taking into account its real band structures. In bulk semiconductors and 2D semiconductor structures such consideration usually leads to the shifts of the optical transitions energies and/or to the appearing of some additional transitions which are forbidden in the framework of a simple parabolic band model. However, in OD semiconductor systems the real band structure gives some qualitatively new effects. We present here a theoretical analysis of the degenerate valence band effect on the structure of absorption spectra and luminescence, on the radiative life time of non equilibrium electron-hole pairs (e-h’s), and on the resonance Raman scattering in small size microcrystals (nanocrystalsl. 2. Energy Spectrum and Wave Functions of Holes We have found analytical equations for the energy of the quantum size levels (QSL’sl of holes in the framework of six-band Hamiltonian describing the energy spectrum of cubic semiconductors with a finite value of spin-orbital splitting A Ill. Analysis of these expressions shows that the ground state of holes for A=0 is the even state with momentum j=1/2. Its energy is described by the following expression EI,2 = h2Vp(2mha21 where m h is the heavy hole effective mass, a is the micro- crystal radius, Vl= 4.49 is the first root of the Bessel function j,fxl. The corresponding wave functions are formed only by spherical function Yl m and have p-type symmetry. For A = m the ground state of holes is the the odd state with j=3/2. The system of equations ill is reduced in this case to j,(khaljo(Y/ikhal + j,(&khaljo(khal = 0 (21 where hkh= d2m E h 3,2, B = ml/mh is the relation between the light and heavy hole effective mass. This state is four-fold degenerate with respect to momentum projection M = 73/2. T1/2 and the corresponding wave functions are: \yM= 2 1 R1(r) 1 m+u=p,j LmUp (31 1=0.2 where u are the Bloch functions of the four-fold degenera:e valence band TB (~(=f1/2,+3/21, YI m are the normalized spherical functions, i kl ( 1 are the mnp 3j Wigner symbols, R. 2 are the radial functions which satisfied the set of equations [II. It is seen that these functions have mixed s-d type symmetry. In the limit ml/mh<<l the energy and the wave functions have simple analytical forms h2v; E RO=CI jo(V2r/al-j,(V,ll s/2 = - I 2mhd2 R2=Cj2( V2r/a 1 (41 where, V2= 5.76 is the first root of j, , C = 6.044/a3’* 121. 0749-6036/92/020167 +03 $02.00/O 0 1992 Academic Press Limited