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Excitons in Bulk Semiconductors Seminar: Optical Properties of Semiconductors

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Excitons in Bulk Semiconductors Seminar: Optical Properties of Semiconductors Excitons in Bulk Semiconductors Seminar: Optical Properties of Semiconductors Erkan Emre Summersemester 2011 Contents 1 Introduction 2 2 The Interband Polarization 2 3 Wannier Equation 6 4 Excitons 7 5 Optical Spectra 8 1 2 The Interband Polarization 1 Introduction The e®ects of an electric ¯eld on a semiconductor are described by examining the resulting polar- isation. Polarisation describes the displacement of an electrically charged particle under such an external ¯eld and is expressed via the so called susceptibility P (!) P(!) = = Â(!)E(!): (1) L3 When a semiconductor is exposed to an electric ¯eld where the ¯eld frequency is equal to or larger than the bandgap energy of the semiconductor, an electron excitation from the valence band to the conduction band occurs so that an electron-hole-pair is created. This is a well known e®ect and re- lates to the bandstructure of semiconductors. The following work reveals that by taking the Coulomb interaction between the electron and the hole of a semiconductor into account one can create a new kind of quasi-particles, which will remind one of the hydrogen atom problem. The desired results are achieved by describing the polarisation in quantum mechanical expressions and applying the equation of motion in the Heisenberg picture to ¯nd its time dependance. By applying the analogous solutions of the hydrogen atom problem to the newly found equation of motion one can determine the describing wavefunctions which describe the polarisation. This result enables us to ¯nally derive the susceptibility and absorption to gain more insight into the physical interpretation. 2 The Interband Polarization We start with the classical polarisation of a particle with the charge e, which is given by 3 P = L n0er: (2) To ¯nd a quantum mechanical expression we apply the expectation value of the polarisation in operator notation and introduce the second quantization with ¯eld operators Ã^. The polarisation is given by the expectation value of the dipole moment er Z P(t) = d3r hÃ^y(r; t) er Ã^(r; t)i: (3) The expression above gives the average value of the polarisation. This is due to the equilibrium statistical operator ½0, which is explicitly included in the equivalent notation Z h i 3 y P(t) = d r tr ½0 Ã^ (r; t) er Ã^(r; t) ; (4) where ½0 describes the state of the system before it is exposed into any light. By changing the basis we describe the electron ¯eld operators Ã^ by using Bloch functions X ^ Ã(r; t) = a^¸;k(t) ø(k; r): (5) ¸;k ¸ describes the band, in which the corresponding particle is contained, while k describes the momen- tum state the particle represents. The annihilation operatora ^¸;k(t) therefore destroys a particle in band ¸ with momentum k. Equation (3) therefore yields: X Z 3 y ¤ 0 P(t) = d r ha^¸;k(t) ø(k; r) er a^¸0;k0 (t) ø0 (k ; r)i (6) ¸;¸0;k;k0 2 2 The Interband Polarization y and also includes the creation operatora ^¸;k(t). The wave functions can be excluded out of the mean value due to not being operators. Because the spacial dependancy is not present in the mean value the integral can be applied only to the dipole moment and the wave functions. X Z y 3 ¤ 0 P(t) = ha^¸;k(t)a ^¸0;k0 (t)i d r ø(k; r) er ø0 (k ; r) (7) ¸;¸0;k;k0 Introducing the dipole approximation where only identical k-states in di®erent bands ¸ 6= ¸0 are coupled one can describe the integral as a product of a delta function and the dipole moment d¸¸0 Z 3 ¤ 0 d r ø(k; r) er ø0 (k ; r) ' ±k;k0 d¸¸0 : (8) This approximation is valid because the photon momentum of the light ¯eld is small in comparison to k and k0. By applying this approximation the polarisation is simply expressed as X X y P(t) = ha¸;k(t) a¸0;k(t)i d¸¸0 = P¸;¸0;k(t) d¸¸0 ; (9) ¸;¸0;k ¸;¸0;k where P¸;¸0;k(t) is called the pair function. It is important to notice, that the polarisation P(t) is only implicitly time dependant, because the time dependant factors only show in the creation and annihi- lation operators. This will prove useful in calculating the equation of motion in the following sections. With this result one can describe the dynamics of the system by applying the Heisenberg equation of motion. To do this, however, the Hamilton operator is needed. We describe the whole Hamiltonian by splitting it into three parts. Kinetic Energy The kinetic energy of the particles is simply given by X y Hkin = E¸;ka¸;k a¸;k: (10) ¸;k By introducing the two band approximation we can simplify the expression. We only consider ¸ to be the valence band v or the conduction band c. X ³ ´ y y Hkin = Ec;kac;k ac;k + Ev;kav;k av;k (11) k As one can see the single particle energies include the corresponding e®ective masses and the bandgap energy between the two bands 2 2 Ec;k = ~²c;k = Eg + ~ k =2mc (12) 2 2 Ev;k = ~²v;k = ~ k =2mv: (13) Electron-electric ¯eld interaction The interaction between the electron and the electric ¯eld is given by Z 3 y HI = d r Ã^ (r)(¡er) E(r; t) Ã^(r) (14) where the electric ¯eld can be split into a product of a time dependent and a spacial dependent part 1 E(r; t) = E(t) (exp (i q ¢ r) + exp (¡i q ¢ r)) : (15) 2 3 2 The Interband Polarization We again apply the dipole and two band approximations. The dipole approximation demands for the momentum of the electric ¯eld to be much smaller than the momentum of the particles, which are similar to each other k ¼ k0 À q ¼ 0: (16) This approximation results in the vanishing of the integral over the spacial part of the electric ¯eld. One can also project the dipole moment dcv (with the two band approximation already applied) into the direction of the electric ¯eld E(t), so that one can drop the vector notation. By inserting the Bloch functions into equation 14 we get X ³ ´ y y ¤ HI '¡ E(t) ac;k av;k dcv + av;k ac;k dcv (17) k Coulomb interaction The Coulomb interaction is denoted by the exchange of momentum q between two particles k and k0. The Hamilton operator therefore yields X 1 y y H = V a a a 0 0 a : (18) C 2 q ¸;k+q ¸0;k0¡q ¸ ;k ¸;k k;k0;q6=0;¸;¸0 In the above expressions it is assumed that the number of electrons in each band is conserved. This is because we do not consider relaxation from the conduction band due to Coulomb interaction; the interaction is also not su±cient to excite electrons from the valence band due to the large energetic di®erences between the Coulomb interaction and the bandgap energy. The case for two bands is expressed as 1 X H = V (19) C 2 q k;k0;q6=0 ¡ y y y y y y ¢ £ ac;k+q ac;k0¡q ac;k0 ac;k + av;k+q av;k0¡q av;k0 av;k + 2 ac;k+q av;k0¡q av;k0 ac;k with the Coulomb potential 4¼e2 1 Vq = 3 2 : (20) ²0L q Equation of motion By summing up the three di®erent parts of the Hamilton operator, one can ¯nally express the equation of motion for expectation values µ ¶ µ ¶ dhP (t)i @P (t) i~ vc;k = ¡h[H;P (t)]i + i~h vc;k i (21) dt vc;k @t | {z } = 0 As previously mentioned the polarisation is not explicitly dependent on time, which results in the vanishing of the partial derivative. By inserting the polarisation and the full Hamilton operator, we achieve µ ¶ dhP (t)i X i~ vc;k = ~(² ¡ ² ) P (t) + [n (t) ¡ n (t)] d E(t) + V dt c;k v;k vc;k c;k v;k cv q k0;q6=0 µ y y y y £ hac;k0+q av;k¡q ac;k0 ac;ki + hav;k0+q av;k¡q av;k0 ac;ki (22) ¶ y y y y +hav;k ac;k0¡q ac;k0 ac;k¡qi + hav;k av;k0¡q av;k0 ac;k¡qi 4 2 The Interband Polarization with the particle number y n¸;k = ha¸;ka¸;ki: (23) As one can see, the equation consists of the parts, where the ¯rst one describes the kinetic energy, the second the electron-electric ¯eld interaction and the last one the Coulomb interaction. Our following task will be the simplifaction of the Coulomb term, to gain more insight into the formula. We approximate the 4-operator average values with products of two operator. This expansion would be exact if the wafefunction of the system were a Slater determinant. y y y y y y hai aj ak ali = hai alihaj aki ¡ hai akihaj ali (24) to the match of creation and annihilation operators and postulate, that two particles in states k and k0 must interact, therefore prohibiting the noninteraction via q 6= 0. With these assumptions the four-particle operator meanvalue can be described as the sum of the product of two two-particle operator meanvalues, where the delta function ±q;0 returns zero. y y hac;k0+q av;k¡q ac;k0 ac;ki ' y y y y hac;k0+q ac;k0 ihav;k¡q ac;ki ±q;0 + hac;k0+q ac;kihav;k¡q ac;k0 i ±k¡q;k0 (25) = Pvc;k0 nc;k ±k¡q;k0 : By applying the same pattern to the remaining three parts of the sum one can ¯nd that y y hav;k0+q av;k¡q av;k0 ac;ki ' Pvc;k nv;k0 ±k¡q;k0 y y hav;k ac;k0¡q ac;k0 ac;k¡qi ' ¡Pvc;k nc;k¡q ±k;k0 (26) y y hav;k av;k0¡q av;k0 ac;k¡qi ' ¡Pvc;k¡q nv;k ±k;k0 : We return to the expression for the Coulomb interaction and proceed in the simpli¯cation.
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