Magnetooptical Properties of Rydberg Excitons - Center-of-Mass Quantization Approach
Sylwia Zielińska-Raczyńska, David Ziemkiewicz, Gerard Czajkowski
aa r X i v : . [ c ond - m a t . m e s - h a ll ] O c t Magnetooptical Properties of Rydberg Excitons - Center-of-Mass QuantizationApproach
Sylwia Zieli´nska-Raczy´nska, David Ziemkiewicz, and Gerard Czajkowski
Institute of Mathematics and Physics, UTP University of Science and Technology,Al. Prof. S. Kaliskiego 7, PL 85-789 Bydgoszcz (Poland)
We show how to compute the magnetooptical functions (absorption, reflection, and transmis-sion) when Rydberg Exciton-Polaritons appear, including the effect of the coherence between theelectron-hole pair and the electromagnetic field, and the polaritonic effect. Using the Real DensityMatrix Approach the analytical expressions for magnetooptical functions are obtained and numer-ical calculations for Cu I. INTRODUCTION
The concept of excitons has been formulated more than 80 years ago by Y. Frenkel [1], who predictedtheir existence in molecular crystals. A few years later, G. Wannier [2] and N. de Mott [3] described theseelectron-hole bound states for inorganic semiconductors. In 1952 E. Gross and N. Karriiev [4] discovered suchWannier-Mott excitons experimentally in a copper oxide semiconductor. After that time excitons remainan important topic of experimental and theoretical research, since they play a dominant role in the opticalproperties of semiconductors (molecular crystals etc.,). The excitons have been studied in great details invarious types of semiconductor nanostructures and in bulk crystals. Since there is a large number of papers,monographs, review articles devoted to excitons, we refer to only a small collection of them [5]-[10]. In thepast decades the main effort of the researchers was focussed on excitons in nanostructures [11]-[21], but veryrecently, new attention has been drawn back to the subject of excitons in bulk crystals by an experimentalobservation of the so-called yellow exciton series in Cu O up to a large principal quantum number of n =25 [22]. Such excitons in copper oxide, in analogy to atomic physics, have been named Rydberg excitons.By virtue of their special properties Rydberg excitons are of fascination in solid and optical physics. Theseobjects whose size scales as the square of the Rydberg principal quantum number n , are ideally suited forfundamental quantum interrogations, as well as detailed classical analysis. Several theoretical approaches tocalculate optical properties of Rydberg excitons have been presented [23]-[35]. Recently quantum coherenceof Rydberg excitons in this system has been investigated [32] which opens new avenue for their furtherimplementation in quantum information processing.When external constant fields (electric or/and magnetic) are applied, the Rydberg excitons, especiallythose with high principal number n , show effects which are not observable in exciton systems with a fewnumber of excitonic states. This effects range from a large Stark shift, overlapping of states, creation ofhigher order excitons (F, H, etc) to quantum chaos and new type statistics for exciton states [31]. Althoughthe Stark and other electrooptic effects on Rydberg excitons in Cu O have been measured and analized [24],[25], [33], there are only few results available regarding the magnetooptic properties of Rydberg excitons[31] where the excitonic spectra of coprous oxide subjected to an external magnetic field up to 7 T has beenmeasured and the complex splitting pattern of crossing and overlapping levels has been demonstrated.Highly excited Rydberg excitons in Cu O crystal provide a well-accessible venue for combined theoreticaland experimental studies of magnetic field effects on the systems. From a different perspective magnetic fieldsmay offer a promising possibility for a controlled manipulation of Rydberg excitons, which would be otherwisedifficult to trap by standard optical techniques, developed for ground states. Due to the fact that free-spaceRydberg polaritons have recently drawn intense interest as tools for quantum information processing one canexpected Rydberg excitons in solid may become highly required object for creating high-fidelity photonicquantum materials [36]. Magnetic fields can strongly affect the Rydberg-Rydberg interactions by breakingthe Zeeman degeneracy that produces Foster zeros.In the presented paper we will focus on magnetooptical properties of Rydberg excitons in Cu O motivatedby the results presented in [31]. As in our previous papers [26], [33], we will use the method based onthe Real Density Matrix Approach (RDMA). Our main purpose is to obtain the analytical expressions forthe magnetooptical functions of semiconductor crystals (reflectivity, transmissivity, absorption, and bulkmagnetosusceptibility), including a high number of Rydberg excitons, taking into account the effect ofanisotropic dispersion and the coherence of the electron and hole with the radiation field, as well to calculatethe positions of excitonic resonances in the situation when degeneracies of exciton states with different orbitaland spin angular momentum are lifted by magnetic field. Presented approach, owing to application of thefull form of Hamiltonian for excitons in an external magnetic field, allows one to get so-called positive shiftsof resonances (connected with a linear dependence on the field strength and quadratic exciton diamagneticshifts. Due to the specific structure of Cu O crystal, particulary to a small radius of Wannier excitons, it isjustified to assume infinite confinement potentials at the crystal surface. All these factors result in complexpattern of spectra which, especially for higher order of excitons, becomes even more intricate for states withhigher principal number n .In very recent paper Schweiner et al [35] has pointed out that an external magnetic field influences the sys-tem, reducing its cubic symmetry and leads to the complex splitting of excitonic lines in absorption spectra.They have solved numerically Schr¨odinger equation and then calculated oscillator strengths including thecomplete valence band structure into their considerations. The most results for Rydberg excitons was con-centrated on the energy values of the excitonic states (for example, [34]). We, in turn, extend such approachincluding the polaritonic effects. This will be done by means of the so-called center-of-mass quantizationapproach [37]-[47]. Thanks to this approach one can include into account the influence of the internal struc-ture of the electromagnetic wave propagating in the crystal. It is important because in the experiments onthe Rydberg excitons in Cu O ([22], [24], [31]) the crystal size in the propagation direction exceeds largelythe wavelength. Finally, we will examine the influence of the effective mass anisotropy on the magnetoopticproperties.The paper is organized as follows. In Sec. II we recall the basic equations of the RDMA and formulatethe equations for the case when the constant magnetic field is applied. In Sec. III we describe an iterationprocedure, which will be applied to solve a system of coupled integro-differential equations and finally obtainthe magneto-optic functions. The second iteration step, from which the magneto-optic functions, includingthe polariton effects, will be calculated, is given in Sec. IV. The formulas, derived in this Section, are thanapplied in Sec V. to calculate the magneto-optic functions for a Cu O crystal, considered in Ref. [31]. Finally,in Sec. VI we draw conclusions of the model studied in this paper. The derivations of useful matrix elementsand calculations with a lot of technical details are established in Appendices.
II. DENSITY MATRIX FORMULATION
Having in mind the above mentioned experiments on Rydberg excitons, we will compute the linear responseof a semiconductor slab to a plain electromagnetic wave, whose electric field vector has a component of theform E i ( z, t ) = E in exp(i k z − i ωt ) , k = ωc , (1)attaining the boundary surface of the semiconductor located at the plane z = 0. The second boundary islocated at the plane z = L . In the case of the examined Cu O crystals the extension will be of the order 30 µ m.The electromagnetic wave is then reflected, transmitted and partially absorbed. The wave propagatingin the medium has the form of polaritons, defined as joint field-medium excitations. Polaritons are mixedmodes of the electromagnetic wave and discrete excitations of the crystal (excitons). Below we assumethe separation of the relative electron-hole motion with well defined quantum levels and the center-of-mass(COM) motion which interacts with the radiation field and produces the mixed modes.In the RDMA all this processes are described by a set of the so-called constitutive equations for thecoherent amplitudes Y ν ( r e , r h ) of the electron-hole pair of coordinates r h (hole) and r e (electron). In thecase of Cu O ν means P, F, H, . . . excitons. The equations have been described, for example, in Refs. [26],[33] and have the form (se also [7], [9])˙ Y ( R , r ) + (i / ¯ h ) H eh Y ( R , r ) + (1 / ¯ h ) Γ Y ( R , r ) = (i / ¯ h ) M ( r ) E ( R ) , (2)where R jest is the excitonic center-of-mass coordinate, r = r e − r h the relative coordinate, M ( r ) thesmeared-out transition dipole density, E ( R ) is the electric field vector of the wave propagating in the crystal,and Γ stands for the dissipation processes. The smeared-out transition dipole density M ( r ) is related to thebilocality of the amplitude Y and describes the quantum coherence between the macroscopic electromagneticfield and the interband transitions. The two-band Hamiltonian H eh includes the electron- and hole kineticenergy terms, the electron-hole interaction potential and the confinement potentials. When constant fields,magnetic and electric, are applied, the Hamiltonian has the form H = E g + 12 m e (cid:18) p e − e r e × B (cid:19) + 12 m hz (cid:18) p h + e r h × B (cid:19) z + 12 m h k (cid:18) p h + e r h × B (cid:19) k + e F · ( r e − r h ) + V conf ( r e , r h ) − e πǫ ǫ b | r e − r h | , (3) B is the magnetic field vector, F the electric field vector, V conf are the surface potentials for electrons andholes, m hz , m h k are the components of the hole effective mass tensor, and the electron mass is assumedto be isotropic. Separating the exciton center-of-mass and relative motion, and considering the case when B k z , F = 0, we transform the Hamiltonian (3) into the form H = H + P z M z + P k M k + 18 µ k ω c ρ + e µ ′k B L z − eM k P k · (cid:0) r k × B (cid:1) + V conf ( R , r ) , (4)where ω c = eB/µ k is the cyclotron frequency, the reduced mass µ ′k is defined as1 µ ′k = 1 m e − m h k , (5)and H is the two-band Hamiltonian for the relative electron-hole motion, as used in the papers [26], [33].The operator L z is the z -component of the angular momentum operator. We must solve the constitutiveequations with the above Hamiltonian to obtain the polarization and finally the polariton modes.The coherent amplitude Y define the excitonic counterpart of the polarization P exc ( R ) = 2 Z d r M ∗ ( r ) Y ( R , r ) , (6)which is than used in the Maxwell field equation c ∇ R E − ǫ b ¨ E ( R ) = 1 ǫ ¨ P exc ( R ) , (7)with the use of the bulk dielectric tensor ǫ b and the vacuum dielectric constant ǫ . In the present paper wesolve the equations (2)-(7) with the aim to compute the magnetoooptical functions (reflectivity, transmission,and absorption) for the case of Cu O.In semiconductors like, for example, GaAs, when only a few lowest excitonic states are excited, it ispossible to solve the polariton dispersion relation and to determine the amplitudes of the polariton waves.Analogous methods cannot be applied in the case of Rydberg excitons, where, as, for example, in Cu O,even 25 excitonic states are observed. There is a question what approach is appropriate for such case. Oneof the possibilities is to use the so-called exciton center-of-mass (COM) quantization. In this approach itis assumed, that no electron- or hole separately are confined within the semiconductor crystal, but theircenter-of-mass [37]-[47]. This approach is justified for small-radius Wannier excitons, as is the case of Cu O(about 1 nm), and certainly not appropriate for semiconductors with large-radius excitons, like GaAs (about15 nm). In the COM approach mostly infinite confinement potentials at the crystal surfaces z = 0 , L areassumed, therefore the eigenfunctions and eigenvalues of the COM motion have the form w N ( Z ) = r L ∗ sin (cid:18) N πL ∗ Z (cid:19) ,W N = ¯ h M z N π L ∗ = N S, (8) S = µ k M z (cid:18) πa ∗ L ∗ (cid:19) R ∗ , where M z is the total excitonic mass in the z − direction, a ∗ is the excitonic radius, R ∗ the excitonic Rydberg,and L ∗ the effective crystal size in the z − direction.Having the confinement functions, we look for a solution Y ( Z, r ) = X Nnℓm c Nnℓm R nℓm ( r ) Y ℓm ( θ, φ ) w N ( Z ) . (9)where R nℓm are the radial functions of an anisotropic Schr¨odinger equation [33] R nℓm ( r ) = (cid:18) η ℓm na ∗ (cid:19) / ℓ + 1)! s ( n + ℓ )!2 n ( n − ℓ − × (cid:18) η ℓm rna ∗ (cid:19) ℓ e − η ℓm r/na ∗ M (cid:18) − n + ℓ + 1 , ℓ + 2 , η ℓm rna ∗ (cid:19) (10)= (cid:18) η ℓm na ∗ (cid:19) / s ( n − ℓ − n ( n + ℓ )! (cid:18) η ℓm rna ∗ (cid:19) ℓ L ℓ +1 n − ℓ − (cid:18) η ℓm rna ∗ (cid:19) e − η ℓm r/na ∗ ,η ℓm = Z dΩ | Y ℓm | sin θ + ( µ k /µ z ) cos θ , and E nℓm the corresponding eigenvalues E nℓm = − η ℓm n R ∗ , (11) M ( a, b, z ) being the confluent hypergeometric function in the notation of [48], and L pn ( x ) are the Laguerrepolynomials. III. ITERATION PROCEDURE
The electric field E ( Z ) of the wave propagating in the crystal, acting as a source in Eq. (2), must satisfythe Maxwell equation (7) which, for the wave propagating in the Z direction and with the harmonic timedependence fulfils the propagation equationd E d Z + k b E ( Z ) = − ω c ǫ P exc ( Z ) , k b = √ ǫ b ωc , (12)and P exc ( Z ) is the excitonic part of the crystal polarization P exc ( Z ) = 2 Z M ∗ ( r ) Y ( Z, r )d r. (13)The function Y ( Z, r ), with regard to (2),(9), and for the Faraday configuration, satisfies the equation X Nnℓm E g − ¯ hω + W N + E nℓm + m µ k µ ′k γR ∗ − i Γ + R ∗ a ∗ γ r sin θ ! c Nnℓm R nℓ ( r ) Y ℓm ( θ, φ ) w N ( Z )= M ( r ) E ( Z ) , (14)where γ = ¯ hω c / R ∗ is the dimensionless strength of the magnetic field. Even under applying the COMquantization, we are left with a system of two coupled integro-differential equations for the functions Y and E . Having the field E ( Z ) we can determine the optical functions, reflectivity, transmissivity, and absorption,by the relations R = (cid:12)(cid:12)(cid:12)(cid:12) E (0) E in − (cid:12)(cid:12)(cid:12)(cid:12) , T = (cid:12)(cid:12)(cid:12)(cid:12) E ( L ∗ ) E in (cid:12)(cid:12)(cid:12)(cid:12) ,A = 1 − R − T, (15)where E in is the amplitude of the normally incident wave. The solution of the equations (12)-(13), whichyield the electric field and the optical functions, can be obtained by several methods. However, the methods,which were used for GaAs layers [50], or Quantum Dots ([51], [52] and references therein), cannot be appliedfor the considered case of Rydberg excitons, as we discussed in our previous paper [26], [33]. Below wepropose an iteration procedure. The first step in this procedure is the solution of the system of equations(14), where on the r.h.s. we put, instead of the full solution E ( Z ), its (known) homogeneous part E hom ,satisfying the equation d E d Z + k b E ( Z ) = 0 , (16)and the appropriate boundary conditions. It has the form E hom ( Z ) = E in k f ( L − Z )( k b + k ) W ,f ( z ) = e − i k b Z + k b − k k b + k e i k b Z , (17) W = e − i k b L − (cid:18) k b − k k b + k (cid:19) e i k b L . Inserting the above expression on the r.h.s. of the equations (14), the following set of equations will beobtained, from which the coefficients c Nnℓm can be determined X Nnℓm (cid:18) W Nnℓm + R ∗ a ∗ γ r sin θ (cid:19) c Nnℓm R nℓm ( r ) Y ℓm ( θ, φ ) w N ( Z )= M ( r ) E hom ( Z ) , (18) W Nnℓm = E g − ¯ hω + W N + E nℓm + m µ k µ ′k γR ∗ − i Γ. The expression for the dipole density, which should be used in (14), has the form: for P excitons, ℓ = 1 [26] M (1) ( r ) = e r M r + r r r e − r/r = e r M ( r ) = i M r + r r r r π Y , − − Y , ) e − r/r + j M r + r r r r π Y , − + Y , ) e − r/r + k M r + r r r r π Y e − r/r , (19)and for F excitons, ℓ = 3, in normalized (with respect to r ) form M (3) ( r ) = i M r r "r π Y − Y − ) − r π Y − Y − ) e − r/r + j M r r ( − i "r π Y + Y − ) − r π Y + Y − ) e − r/r (20)+ k M r r (cid:20) r π Y e − r/r (cid:21) . Having in mind the experiments by Aßmann et al [31], we consider the field B as perpendicular to thecrystal surface, ( k z ), and the wave propagating in the z direction and characterized by the electric field E .The wave is assumed linearly polarized, E = ( E x , , E x = E hom ( Z ). Thus we takethe x components of the densities M defined in (19) and (20) M (1) x ( r ) = M r + r r r r π Y − − Y ) e − r/r , (21) M (3) x ( r ) = M r r "r π Y − Y − ) − r π Y − Y − ) e − r/r , (22)with the coherence radius r (see [26] for the discussion about r ). Using the above formulas, together withthe expression (17) for homogenous field E hom , we obtain, in the first step of iteration, a system of equations: X nℓm (cid:20) E g − ¯ hω + W N + E nℓm + m µ k µ ′k γR ∗ − i Γ + R ∗ a ∗ γ r sin θ (cid:21) c Nnℓm R nℓm ( r ) Y ℓm ( θ, φ ) w N ( Z ) = M x E ( Z ) , (23)By appropriate integration and making use of the orthonormality of the eigenfunctions, the equations forthe expansion coefficients obtain the form W N n ℓ m c N n m ℓ + R ∗ a ∗ γ X nℓm h R n ℓ m | r | R nℓm ih Y ℓ m | sin θ | Y ℓm i c N nℓm = h w N | E x ( Z ) ih R n ℓ m Y ℓ m | M x i . (24)Introducing the notation V ( nn ) ℓℓ mm = R ∗ a ∗ γ h R n ℓ m | r | R nℓm ih Y ℓ m | sin θ | Y ℓm i , (25)we put the equations (24) into the form W N n ℓ m c n m ℓ + X nℓm V ( nn ) ℓℓ mm c N nℓm = h w N | E x ih R n ℓ m Y ℓ m | M x i . (26)As in the case of previously discussed electro-optic properties [33], we take into account only the states n = n .This assumption is justified by the fact, that the diamagnetic shift, related to these matrix elements, is muchsmaller than the Zeeman splitting. Now we obtain the equations W Nnℓm c Nnℓm + V ( n ) ℓℓm c Nnℓm + V ( n ) ℓℓ +2 m c Nnℓ +2 m + V ( n ) ℓℓ − m c Nnℓ − m = h w N | E x ih R nℓ m Y ℓm | M x i , (27) V ( n ) ℓℓm = R ∗ γ ( ℓ + ℓ + m − ℓ − ℓ + 3) (cid:18) nη ℓm (cid:19) [5 n + 1 − ℓ ( ℓ + 1)] ,V ( n ) ℓℓ +2 m = − R ∗ γ s ( ℓ + 2 − m )( ℓ + 1 − m )( ℓ + m + 1)( ℓ + m + 2)(2 ℓ + 1)(2 ℓ + 3) (2 ℓ + 5) ∞ Z d ρρ R nℓ R nℓ +2 ,V ( n ) ℓℓ − m = − R ∗ γ s ( ℓ − m )( ℓ − m − ℓ + m )( ℓ + m − ℓ − ℓ − (2 ℓ + 1) ∞ Z d ρρ R nℓm R nℓ − m . The effect of overlapping of different states is included in the resonant denominators. The detailed form forthe coefficients c Nnℓm , h R n ℓ m Y ℓ m | M x i , h E x | w N i , and the derivation of the matrix elements V ( n ) ℓℓm is givenin Appendices A and B. The equations (27) are the basic equations in the presented paper, which will beused in the numerical calculations of the optical functions. IV. SECOND ITERATION STEP - MAGNETO-OPTIC FUNCTIONS
In the first iteration step we have computed the coefficients c Nnℓm . Having them, we determine theamplitude Y ( Z, r ) from Eq. (9) and the excitonic polarization from the Eq. (13), obtaining P exc ( Z ) = 2 Z M ∗ ( r ) Y ( Z, r )d r = 2 Z M ∗ ( r ) X Nnℓm c Nnℓm R nℓm Y ℓm w N ( Z )d r = X N P N w N ( Z ) ,P N = 2 X nℓm c Nnℓm h M ∗ | R nℓm Y ℓm i = ǫ ǫ b ∆ ( P ) LT h w N | E hom ( Z ) i X nℓm χ Nnℓm (28)= ǫ ǫ b ∆ ( P ) LT I N E hom (0) X nℓm χ Nnℓm , where I N are defined in Eq. (D6), and χ Nnℓm in Eqn. (F3, F4). The so obtained polarization will be used asa source in the Maxwell equation (12). As is known, the solution of a nonhomogeneous differential equationis composed of two parts, which are the solution of a homogeneous equation and of the nonhomogeneousone: E ( Z ) = E hom ( Z ) + E nhom ( Z ) , (29)where the homogeneous part satisfies the equation (16). The nonhomogeneous part will be obtained bymeans of the appropriate Green function, satisfying the equationd d Z G E ( Z, Z ′ ) + k b G E ( Z, Z ′ ) = − δ ( Z − Z ′ ) , (30)and having the form (for example, [9]) G E ( Z, Z ′ ) = i2 k b W (cid:18) e − i k b Z < + k b − k k b + k e i k b Z < (cid:19) × (cid:18) k b − k k b + k e i k b L − i k b Z > + e − i k b L +i k b Z > (cid:19) , (31)where Z < = min( Z, Z ′ ) , Z > = max( Z, Z ′ ). Using the above Green’s function we obtain the nonhomogeneouspart in the form E nhom ( Z ) = k ǫ L Z G E ( Z, Z ′ ) P exc ( Z ′ )d Z ′ . (32)The equations (29), (31), and (32) give the total electric field of the wave propagating in the crystal, fromwhich the reflectivity R and transmissivity T are obtained. They have the form R = (cid:12)(cid:12)(cid:12)(cid:12) E (0) E in − (cid:12)(cid:12)(cid:12)(cid:12) (33)= R FP (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:18) k b L (cid:19) (1 − r ∞ )(1 − r ∞ e iΘ ) (1 − r ∞ e iΘ ) r ∞ (1 − e iΘ ) X N (cid:18) I N L (cid:19) X nℓm ∆ ( P ) LT χ Nnℓm (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ,T = (cid:12)(cid:12)(cid:12)(cid:12) E ( L ) E in (cid:12)(cid:12)(cid:12)(cid:12) = T FP (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:18) E E L (cid:19) (1 − r ∞ e iΘ )(1 − r ∞ e iΘ ) X N (cid:18) I N L (cid:19) X nℓm ∆ ( P ) LT χ Nnℓm (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (34)where Θ = 2 k b L , r ∞ is defined by r ∞ = k − k b k + k b , (35) I n /L are given in Eq. (D7), and R FP , T FP are the well-known formulas for the Fabry-Perot normal incidencereflectivity and transmissivity of a lossless dielectric slab of thickness L (see, e.g., [53]), R FP = F sin (Θ / F sin (Θ / , T FP = 11 + F sin (Θ / ,F = 4 r ∞ (1 − r ∞ ) . (36)The derivation of the formulas (33) and (34) is given in Appendix E. V. RESULTS
We have performed numerical calculations of magneto-optical functions (absorption, reflectivity, and trans-missivity) for the Cu O crystal having in mind the experiments by Aßmann et al [31]. Considering thespecific properties of Cu O and the crystal dimension large compared to the exciton Bohr radius, we ob-serve that the COM quantized energies W N (8) are small compared to the remaining components of theexcitonic energies W Nnℓm (18). Therefore, in the first approximation, we can neglect them in the formulasdefining the coefficients c Nnℓm (Appendix A) and in expressions χ Nnℓm (F3,F4), and calculate the bulkmagneto-susceptibility χ ( ω ) = ǫ b ∆ ( P ) LT [ χ N + χ N − + χ N + χ N − + χ Nn ± + ˜ χ Nn ± + ˜ χ Nn ± . . . ] , (37)Using the formula α = ( ω/c )Im √ ǫ b + χ we have calculated the magneto-absorption, taking into accountthe lowest n = 2 −
10 excitonic states. The parameters we used are the energies E nℓm , the gap energy E g , the L-T energy ∆ ( P ) LT , and the dissipation parameter Γ . We have used the electron and hole effectivemasses: m e = 1 . , m h k = 0 . , µ k = 0 . , µ z = 0 .
672 (the masses in free electron mass) m , from themcalculated the reduced mass µ ′k = − .
25, which gives µ k /µ ′k ≈ − . . Since the LT splitting for P excitonsis not known, we have established a relation between the known ∆ LT for S excitons and the quantity ∆ ( P ) LT for P excitons. First, using the bulk dispersion c k ω = ǫ b + 2 ǫ Z d rM ∗ Y, (38)for k = 0 , n = 2, we establish the relation between the splitting and the dipole matrix element ǫ b ∆ ( P ) LT = 2 ǫ | I + I | (cid:20) W + 1 W − (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) B =0 , | M | = 4 ǫ ǫ b a ∗ ∆ ( P ) LT π ( r /a ∗ ) η , (39)where I , I are defined in Eq.(B3,B4). Using an analogous expression for for S excitons ([54]) | M | = πǫ ǫ b a ∗ ∆ ( S ) LT η one can determine ∆ ( P ) LT as function of ∆ ( S ) LT ∆ ( P ) LT = π (cid:16) r a ∗ (cid:17) η η ∆ ( S ) LT . (40)The energies W Nnℓm were obtained from the relations (11), and (18) (without W N ) with the effective Rydbergenergy R ∗ . We have used the values E g = 2172 meV , R ∗ = 86 .
981 meV, ∆ ( S ) LT = 10 µ eV which is commonvalue in available literature, µ k /µ z = 0 . Γ = 0 . n ≥ P and F excitons, for example, the overlapping of states, are observed. This is shown inFig. 3 for the n = 4 exciton state, where the dependence of the absorption on the applied field strength isshown. The Zeeman splitting is clearly visible and the lines are shifted towards higher energy with increasingfield strength. Moreover, some absorption lines become visible only for sufficiently strong magnetic field. Ourcalculation scheme can be extended to higher principal quantum number; the absorption spectrum for n=4-25 excitonic states is shown on the Fig. 4. In Fig. 5 we show that our method allows to identify the excitonicstates. In the RDMA approach we consider the role of the coherence of the radiation field with excitons,which enters via the smeared dipole density M ( r ) and its parameters M , r . The influence of the coherenceradius r is illustrated in Fig. 6. One can observe that an increase of r causes an increase of absorption,which is due to the related increase of the L-T splitting and oscillator strength, see Eq. (40). The COMquantization approximation, used in our calculations, allowed to calculate the reflection and transmissionspectra, taking into account both the microscopic electronic excitations (excitons) and their interaction withthe radiation field, resulting in creation of polariton modes. We have obtained analytical expressions for thereflection coefficient R and the transmissivity T , and their shapes are shown in Figs. 7-10. The calculationhas been performed up to N=350 and for excitonic states with principal number n=4-7. One can observethe overlapping of the Fabry-Perot modes, typical for a dielectric slab, with excitonic resonances. Notably,the Fabry-Perot interference has the dominating effect, strongly affecting the reflection and transmissioncoefficient over the whole energy range. Conversely, the variation of reflection coefficient due to the excitonicresonances has a comparably small amplitude, but is quickly varying with energy. In the Fig. 7 inset, onecan see that these two effects are readily separable. VI. CONCLUSIONS
The main results of our paper can be summarized as follows. We have proposed a procedure basedon the RDMA approach that allows to obtain analytical expressions for the magneto-optical functions ofsemiconductor crystals including high number Rydberg excitons. Our results have general character becausearbitrary exciton angular momentum number and arbitrary applied field strength are included. We havechosen the example of cuprous oxide, inspired by the recent experiment by Aßmann et al [31]. We havecalculated the magneto-optical functions (susceptibility, absorption, reflection, and transmission), obtaininga fairly good agreement between the calculated and the experimentally observed spectra.As each method using iterative procedure presented approach is a kind of approximation. Although wehave solved the problem of excitons in semiconductor when an external magnetic field is applied, but we donot include the interaction between states with different n into the Hamiltonian we have used. It shouldemphasized that presented approach is analytical until the last step in which the numerical code is used.The choice of dipole density model and therefore the oscillator strengths, which is intricate function of freeparameters has an impact on accuracy of our calculations and might be the source of discrepancy between theexperimental results. Our results confirm the fundamental peculiarity of magneto-optical effects: shifting,splitting and, as a result for higher excitonic states, mixing of spectral lines. In particular, we obtained thesplitting of P and F excitons, with increasing number of peaks corresponding to the increasing state number.All these interesting features of excitons with high n number which are examined and discussed on the basison our theory might possibly provide deep insight into the nature of Rydberg excitons in solids and provoketheir application to design all-optical flexible switchers and future implementation in quantum informationprocessing. Appendix A: Expansion coefficients c Nnℓm
Using the eigenfunctions (10) and transition dipole densities (21), (22) we calculate the expansion coeffi-cients c Nnℓm . For n = 2 , P excitons: W N n ± c N n ± + V ( n )11 ± c N n ± = h w N | E x ih R Y ± | M (1) x i ,c N n ± = h w N | E x ih R n Y ± | M (1) x i W N n ± + V ( n )11 ± . (A1)For n ≥ P and F . For n = 4 we have three equations W N ± c N ± + V (4)11 ± c N ± + V (4)13 ± c N ± = h w N | E x ih R Y ± | M (1) x i ,W N ± c N ± + V (4)33 ± c N ± + V (4)31 ± c N ± = h w N | E x ih R Y ± | M (3) x i , (A2) W N ± c N ± + V (4)33 ± c N ± = h w N | E x ih R Y ± | M (3) x i . a c N ± + a c N ± = b ,a c N ± + a c N ± = b ,a = W N ± + V (4)11 ± , (A3) a = V (4)13 ± = a ,a = W N ± + V (4)33 ± ,b = h w N | E x ih R Y ± | M (1) x i ,b = h w N | E x ih R Y ± | M (3) x i , with solutions c N ± = h w N | E x i a h R Y ± | M (1) x i − a h R Y ± | M (3) x i ∆ ,c N ± = h w N | E x i a h R Y ± | M (3) x i − a h R Y ± | M (1) x i ∆ , (A4)∆ = a a − a . The third of equations (A2) has the solution in the form c N ± = h w N | E x ih R Y ± | M (3) x i W N ± + V (4)33 ± . (A5)For n = 5 we obtain quite analogous equations W N ± c N ± + V (5)11 ± c N ± + V (5)13 ± c N ± = h w N | E x ih R Y ± | M (1) x i ,W N ± c N ± + V (5)33 ± c N ± + V (5)31 ± c N ± = h w N | E x ih R Y ± | M (3) x i , (A6) W N ± c N ± + V (5)33 ± c N ± = h w N | E x ih R Y ± | M (3) x i . The third of the above equations yields c N ± = h w N | E x ih R Y ± | M (3) x i W N ± + V (5)33 ± , (A7)whereas the first two give the coefficients c N ± , c N ± c N ± = h w N | E x i a h R Y ± | M (1) x i − a h R Y ± | M (3) x i a a − a ,c N ± = h w N | E x i a h R Y ± | M (3) x i − a h R Y ± | M (1) x i a a − a , (A8)where a = W N ± + V (5)11 ± ,a = V (5)13 ± = a ,a = W N ± + V (5)33 ± . c N n ± = h w N | E x ih R n Y ± | M (3) x i W N n ± + V ( n )33 ± ,c N n ± = h w N | E x i a h R n Y ± | M (1) x i − a h R n Y ± | M (3) x i a a − a , (A9) c N n ± = h w N | E x i a h R n Y ± | M (3) x i − a h R n Y ± | M (1) x i a a − a ,a = W N n ± + V ( n )11 ± ,a = V ( n )13 ± = a ,a = W N n ± + V ( n )33 ± , ∆ = a a − a c N n ± = h w N | E x i ∆ ( a " ∓ r π M (cid:16) r a ∗ (cid:17) η / a ∗ / r n − n − a " ∓ . · r πM (cid:16) r a ∗ (cid:17) η / a ∗ / r ( n − n − n − n = h w N | E x i ∆ (cid:16) r a ∗ (cid:17) r πM a ∗ / r n − n ( ± a i η / ± a · . · (cid:16) r a ∗ (cid:17) η / r ( n − n − n ) = ± h w N | E x i (cid:16) W N n ± + V ( n )111 (cid:17) (cid:16) W N n ± + V ( n )331 (cid:17) − (cid:16) V ( n )131 (cid:17) (cid:16) r a ∗ (cid:17) r πM a ∗ / r n − n × ( i η / (cid:16) W N n ± + V ( n )331 (cid:17) + 0 . · V ( n )131 (cid:16) r a ∗ (cid:17) η / r ( n − n − n ) c N n ± = h w N | E x i a h R n Y ± | M (3) x i − a h R n Y ± | M (1) x i a a − a = h w N | E x i ∆ ( a " ∓ . · r πM (cid:16) r a ∗ (cid:17) η / a ∗ / r ( n − n − n − n − a " ∓ r π M (cid:16) r a ∗ (cid:17) η / a ∗ / r n − n = ∓ h w N | E x i ∆ (cid:16) r a ∗ (cid:17) r πM a ∗ / r n − n ( a · . · (cid:16) r a ∗ (cid:17) η / r ( n − n − n + a i η / ) = ∓ h w N | E x i (cid:16) W N n ± + V ( n )111 (cid:17) (cid:16) W N n ± + V ( n )331 (cid:17) − (cid:16) V ( n )131 (cid:17) (cid:16) r a ∗ (cid:17) r πM a ∗ / r n − n × (h W N n ± + V ( n )111 i · . · (cid:16) r a ∗ (cid:17) η / r ( n − n − n + V ( n )131 · i η / o Appendix B: Determination of the quantities h R n Y ± | M (1) x i , h R n Y ± | M (3) x i , h R n Y ± | M (3) x i Below we calculate the quantities h R n Y ± | M (1) x i , h R n Y ± | M (3) x i , h R n Y ± | M (3) x i , which enter in theabove derived formulas for the coefficients c Nnℓm . Using the definition (21, 22) one obtains h R n Y ± | M (1) x x i = ∓ r π M r ∞ Z r d r r + r r e − r/r R n ( r ) . (B1)Inserting on the r.h.s. the expression for the radial function R n (see Eq. (10)) we get h R n Y ± | M (1) x x i = ∓ r π M r ∞ Z d r ( r + r ) e − λr (cid:18) η na ∗ (cid:19) / s ( n + 1)!2 n ( n − × (cid:18) η rna ∗ (cid:19) M (cid:18) − n + 2 , , η rna ∗ (cid:19) , (B2)where λ = 1 r + η na ∗ = na ∗ + η r na ∗ r . The r.h.s. of Eq. (B2) consists of two parts: I = r π M r (cid:18) η na ∗ (cid:19) / (cid:18) η na ∗ (cid:19) s ( n + 1)!2 n ( n − ∞ Z d r r e − λr × M (cid:18) − n + 2 , , η rna ∗ (cid:19) (B3)= r π M r (cid:18) η na ∗ (cid:19) / s ( n + 1)!2 n ( n − λ − F (cid:18) − n + 2 , , , η r na ∗ + η r (cid:19) ,I = r π M r r (cid:18) η na ∗ (cid:19) / s ( n + 1)!2 n ( n − × ∞ Z r d r e − λr M (cid:18) − n + 2 , , η na ∗ · r (cid:19) = r π M r r λ − (cid:18) η na ∗ (cid:19) / s ( n + 1)!2 n ( n − F (cid:18) − n + 2 , , , η r na ∗ + η r (cid:19) ,F ( α, β, γ, z ) being the hypergeometric series (for example, [49]) F = 1 + αβγ z
1! + α ( α + 1) β ( β + 1) γ ( γ + 1) z
2! + . . . . (B5)3Performing the summation we obtain I + I = r π M r (cid:18) η na ∗ (cid:19) / s ( n + 1)!2 n ( n − λ − F (cid:18) − n + 2 , , , η r na ∗ + η r (cid:19) + r π M r r λ − (cid:18) η na ∗ (cid:19) / s ( n + 1)!2 n ( n − F (cid:18) − n + 2 , , , η r na ∗ + η r (cid:19) ≈ r π M r (cid:18) η na ∗ (cid:19) / s ( n + 1)!2 n ( n − n a ∗ r (3 na ∗ + η r )( na ∗ + η r ) = r π M r (cid:18) η na ∗ (cid:19) / s ( n + 1)!2 n ( n − , where the assumption r < a ∗ has been used. Finally h R n Y ± | M (1) x i = ∓ r π M i (cid:16) r a ∗ (cid:17) η / a ∗ / r n − n . (B6)The following expression will be useful in calculation of oscillator strengths | I + I | = 23 π | M | (cid:16) r a ∗ (cid:17) η a ∗ n − n . (B7)In the next step we calculate the quantity h R n Y ± | M (3) x x i . Using the definitions (20) and (10) one obtains h R n Y ± | M (3) x x i = ∓ r π M r ∞ Z d r e − r/r R n ( r )= ∓ r π M r (cid:18) η na ∗ (cid:19) / ∞ Z d r r e − λr s ( n + 3)!2 n ( n − × M (cid:18) − n + 4 , , η na ∗ · r (cid:19) with λ = 1 r + η na ∗ = na ∗ + η r na ∗ r . Performing the integration we arrive at the formulas h R n Y ± | M (3) x i = ∓ r π M r (cid:18) η na ∗ (cid:19) / s ( n + 3)!2 n ( n − λ − F (cid:18) − n + 4 , , , η na ∗ λ (cid:19) = ∓ r π M r (cid:18) η na ∗ (cid:19) / s ( n + 3)!2 n ( n − n a ∗ r ( na ∗ + η r ) F (cid:18) − n + 4 , , , η na ∗ λ (cid:19) ≈ ∓ . · r πM (cid:16) r a ∗ (cid:17) η / a ∗ / r ( n − n − n − n , where again the assumption r < a ∗ has been used. The formula (cid:12)(cid:12)(cid:12) h R n Y ± | M (3) x i (cid:12)(cid:12)(cid:12) ≈ π · | M | a ∗ (cid:18) (cid:19) (2 η ) (cid:16) r a ∗ (cid:17) ( n − n − n − n = 2 . · − π | M | a ∗ ( η ) (cid:16) r a ∗ (cid:17) ( n − n − n − n Y , ± harmonics, have the form h R n Y ± | M (3) x i = ∓ r π M r (cid:18) η na ∗ (cid:19) / s ( n + 3)!2 n ( n − λ − F (cid:18) − n + 4 , , , η na ∗ λ (cid:19)(cid:12)(cid:12)(cid:12) h R n Y ± | M (3) x i (cid:12)(cid:12)(cid:12) ≈ π · | M | a ∗ (cid:18) (cid:19) (2 η ) (cid:16) r a ∗ (cid:17) ( n − n − n − n (B8)= 3 . · − π | M | a ∗ ( η ) (cid:16) r a ∗ (cid:17) ( n − n − n − n . Appendix C: Derivation of the matrix elements V ( nn ) ℓℓ mm In order to calculate the matrix elements V ( nn ) ℓℓ mm , as defined in Eq. (25), we start with the integralcontaining the angular dependence I ℓℓ mm = h Y ℓ m | sin θ | Y ℓm i = Z dΩ Y ∗ ℓ m (1 − cos θ ) Y ℓm = δ ℓℓ δ mm − Z dΩ Y ℓ m cos θ Y ℓm . (C1)Making use of the definition of the spherical harmonic functions Y ℓm ( θ, φ ) = s (2 ℓ + 1)( ℓ − m )!4 π ( ℓ + m )! P mℓ (cos θ ) e i mφ , (C2)in terms of the associated Legendre polynomials P mℓ , the second of the integrals on the r.h.s. of Eq. (C1)can be put into the form I = Z dΩ Y ∗ ℓ m cos θ Y ℓm (C3)= δ mm π s (2 ℓ + 1)( ℓ − m )!4 π ( ℓ + m )! (2 ℓ + 1)( ℓ − m )!4 π ( ℓ + m )! +1 Z − d x P mℓ ( x ) x P m ℓ ( x ) . Making use of the recurrence relation ([49]) xP mℓ ( x ) = 12 ℓ + 1 (cid:2) ( ℓ − m + 1) P mℓ +1 ( x ) + ( ℓ + m ) P mℓ − ( x ) (cid:3) , (C4)we arrive at the integral +1 Z − d x P mℓ ( x ) x P m ℓ ( x ) = +1 Z − d x (cid:26) ℓ + 1 (cid:2) ( ℓ − m + 1) P mℓ +1 ( x ) + ( ℓ + m ) P mℓ − ( x ) (cid:3) × ℓ + 1 (cid:2) ( ℓ − m + 1) P mℓ +1 ( x ) + ( ℓ + m ) P mℓ − ( x ) (cid:3)(cid:27) . (C5)Performing the multiplication and integration, using the orthogonality relation +1 Z − P mℓ ( x ) P mℓ ( x )d x = 22 ℓ + 1 ( ℓ + m )!( ℓ − m )! δ ℓℓ , +1 Z − d x P mℓ ( x ) x P m ℓ ( x ) = 2(2 ℓ + 2 ℓ − − m )( ℓ + m )!(2 ℓ − ℓ + 1)(2 ℓ + 3)( ℓ − m )! δ ℓℓ + 2( ℓ − m + 1)( ℓ + m + 2)( ℓ + m + 1)!(2 ℓ + 1)(2 ℓ + 3)(2 ℓ + 5)( ℓ + 1 − m )! δ ℓ +1 ,ℓ − (C6)+ 2( ℓ − m − ℓ + m )( ℓ + m − ℓ − ℓ − ℓ + 1)( ℓ − − m )! δ ℓ +1 ,ℓ − . The above result inserting into the Eq. (C3) gives for the angular integration I = δ mm π s (2 ℓ + 1)( ℓ − m )!4 π ( ℓ + m )! (2 ℓ + 1)( ℓ − m )!4 π ( ℓ + m )! × (cid:20) ℓ + 2 ℓ − − m )( ℓ + m )!(2 ℓ − ℓ + 1)(2 ℓ + 3)( ℓ − m )! δ ℓℓ + 2( ℓ − m + 1)( ℓ + m + 2)( ℓ + m + 1)!(2 ℓ + 1)(2 ℓ + 3)(2 ℓ + 5)( ℓ + 1 − m )! δ ℓ +1 ,ℓ − (C7)+ 2( ℓ − m − ℓ + m )( ℓ + m − ℓ − ℓ − ℓ + 1)( ℓ − − m )! δ ℓ +1 ,ℓ − (cid:21) . Using the above results we obtain the diagonal matrix element V nn ℓℓ mm when n = n V nℓ ℓ m m = R ∗ γ δ ℓℓ mm (cid:18) − ℓ + 2 ℓ − − m (2 ℓ − ℓ + 3) (cid:19) ∞ Z d ρρ R nℓ R n ℓ (C8)= δ ℓℓ mm R ∗ γ ℓ + ℓ + m − ℓ − ℓ + 3) ∞ Z d ρ ρ [ R nℓ ( ρ )] = δ ℓℓ mm R ∗ γ ( ℓ + ℓ + m − ℓ − ℓ + 3) (cid:18) nη ℓm (cid:19) [5 n + 1 − ℓ ( ℓ + 1)]where we used the formula (for example [55] where η ℓm = 1) h ρ i = 12 [5 n + 1 − ℓ ( ℓ + 1)] . (C9)In a similar way the the off-diagonal elements can be computed with the results displayed in Eq. (27).The integrals containing the radial eigenfunctions R nℓm can be expressed by the integrals over Laguerrepolynomials I nℓsm = ∞ Z d ρρ R nℓm R nsm = ∞ Z d ρρ (cid:18) η ℓm n (cid:19) / s ( n − ℓ − n ( n + ℓ )! (cid:18) η ℓm ρn (cid:19) ℓ × L ℓ +1 n − ℓ − (cid:18) η ℓm ρn (cid:19) e − η ℓm ρ/n × (cid:18) η ℓm n (cid:19) / s ( n − s − n ( n + s )! (cid:18) η ℓm ρn (cid:19) s L s +1 n − s − (cid:18) η ℓm ρn (cid:19) e − η ℓm ρ/n = (cid:18) n η ℓm (cid:19) s ( n − ℓ − n ( n + ℓ )! s ( n − s − n ( n + s )! ∞ Z x ℓ + s +4 L ℓ +1 n − ℓ − ( x ) L s +1 n − s − ( x ) e − x d x n = 4 , ℓ = 1 , s = 3 , m = 1, one has I = ∞ Z d ρρ R R = (cid:18) η (cid:19) r · · r · · ∞ Z x L ( x ) L ( x ) e − x d x = (cid:18) η (cid:19) r · r ∞ Z x (cid:2) x − x + 20 (cid:3) e − x d x = (cid:18) η (cid:19) r −
90 + 20) = 20 (cid:18) η (cid:19) r ≈ (cid:18) η (cid:19) . Appendix D: Calculation of the coefficients h w N | E i The homogeneous solution of the field equation (17) can be put into the form E hom ( Z ) = Ae i k b Z + Be − i k b Z , (D1)where A = 2 k ( k + k b ) W e − i k b L E in ,B = 2 k ( k b − k )( k + k b ) W e i k b L E in . (D2)With these expressions one obtains for h w N | E ih w N | E i = AI + BI ∗ , (D3)with the notation I = h w N | e i k b Z i = r L L Z sin N πL Z e i k b Z d Z. (D4)With the use of the relations sin x = e i x − e − i x , e ± i Nπ = cos N π, (D5)the integral I becomes I = I N = √ L (1 − cos N π exp [i k b L ]) N π ( N π ) − ( k b L ) . (D6)The quantity I N /L which appears in the expressions for the optical functions, has the form I N L = 2 N π h ( k b L ) − N π i (1 − cos N π exp [i k b L ]) . (D7)The expression k b L can be transformed in the following way: k b L = ¯ hω ¯ hc √ ǫ b L = EE L ,E L = ¯ hc √ ǫ b L = 0 . · − eV s · · m/s √ ǫ b L = 1 . · meV · µ m √ ǫ b L ( µ m) . (D8)7For the Cu O data ǫ b = 7 . E L = 72 . L meV , (D9)where L is expressed in µ m. For L = 30 µ m we have E L = 2 .
41 meV. Since I N = I ∗ N , it follows from (D3) h w N | E hom ( Z ) i = I N ( A + B ) = I N E hom (0) , (D10)where E hom (0) = 2 k ( k + k b ) W E in (cid:18) e − i k b L + k b − k k b + k e i k b L (cid:19) (D11)= E in r ∞ − r ∞ exp(iΘ) (cid:0) − r ∞ e iΘ (cid:1) . Appendix E: Optical functions
Below we derive the formulas (33) and (34). They will be obtained from Eq. (15) by using the total electricfield E ( Z ) = E hom ( Z ) + E nhom ( Z ). In particular, for the reflection coefficient one obtains R = | r | , (E1)with r = r + r exc ,r = r ∞ (cid:0) − e iΘ (cid:1) − r ∞ e iΘ , (E2) r exc = k ǫ E in L Z G E (0 , Z ) P exc ( Z )d Z. Since G E (0 , Z ) = i(1 − r ∞ )2 k b (1 − r ∞ e iΘ ) (cid:0) e i k b Z − r ∞ e iΘ − i k b Z (cid:1) , (E3)we obtain r exc = k ǫ E in L Z G E (0 , Z ) P eks ( Z )d Z = k ǫ E in i(1 − r ∞ )2 k b (1 − r ∞ e iΘ ) X N L Z (cid:0) e i k b Z − r ∞ e iΘ − i k b Z (cid:1) P N w N ( Z )d Z (E4)= k ǫ E in i(1 − r ∞ ) k b (1 − r ∞ e iΘ ) X N (cid:2) h w N | e i k b Z i − r ∞ e iΘ h w N | e − i k b Z i (cid:3) P N = k ǫ E in i(1 − r ∞ ) k b (1 − r ∞ e iΘ ) (cid:0) − r ∞ e iΘ (cid:1) X N I N P N , from which, with respect to Eqn. (28), we get r exc = k E hom (0)2 E in i(1 − r ∞ ) k b (1 − r ∞ e iΘ ) (cid:0) − r ∞ e iΘ (cid:1) ǫ b X N I N X nℓm ∆ ( P ) LT χ Nnℓm . (E5)8Inserting in the above equation the expression (D11), we obtain r exc = ǫ b k k b i(1 − r ∞ )(1 − r ∞ e iΘ ) (cid:0) − r ∞ e iΘ (cid:1) X N I N X nℓm ∆ ( P ) LT χ Nnℓm = i2 ( k b L )(1 − r ∞ )(1 − r ∞ e iΘ ) (cid:0) − r ∞ e iΘ (cid:1) X N (cid:18) I N L (cid:19) X nℓm ∆ ( P ) LT χ Nnℓm . (E6)Now the total complex reflection coefficient r obtains the form r = r + r exc = r " (cid:18) E E L (cid:19) (1 − r ∞ )(1 − r ∞ e iΘ ) (1 − r ∞ e iΘ ) r ∞ (1 − e iΘ ) × X N (cid:18) I N L (cid:19) X nℓm ∆ ( P ) LT χ Nnℓm , (E7)which, using the Eq. (E1) immediately gives the result (33).Analogically, we determine the transmissivity T = (cid:12)(cid:12)(cid:12)(cid:12) E ( L ) E in (cid:12)(cid:12)(cid:12)(cid:12) , (E8)resulting from the equation T = | t | , (E9)where t = t + t exc ,t = 1 − r ∞ − r ∞ e iΘ e iΘ / , (E10) t exc = k ǫ E in L Z G E ( L, Z ) P exc ( Z )d Z. Since G E ( L, Z ) = i2 k b W (cid:18) e − i k b Z + k b − k k b + k e i k b Z (cid:19) × (cid:18) k b − k k b + k e i k b L − i k b L + e − i k b L +i k b L (cid:19) (E11)= i e iΘ / (1 − r ∞ )2 k b (1 − r ∞ e iΘ ) (cid:0) e − i k b Z − r ∞ e i k b Z (cid:1) , we have t exc = k E in L Z G E ( L, Z ) P exc ( Z )d Z = ( k b L )2 i e iΘ / (1 − r ∞ )(1 − r ∞ e iΘ ) (1 + r ∞ ) (cid:0) − r ∞ e iΘ (cid:1) X N (cid:18) I N L (cid:19) X nℓm ∆ ( P ) LT χ Nnℓm , and t = t + t exc = t " t ( k b L )2 i e iΘ / (1 − r ∞ )(1 − r ∞ e iΘ ) (1 + r ∞ ) (cid:0) − r ∞ e iΘ (cid:1) X N (cid:18) I N L (cid:19) X nℓm ∆ ( P ) LT χ Nnℓm = t " (cid:18) E E L (cid:19) (1 − r ∞ e iΘ )(1 − r ∞ e iΘ ) X N (cid:18) I N L (cid:19) X nℓm ∆ ( P ) LT χ Nnℓm . (E12)9Inserting the above result into Eq. (E9), we obtain the transmissivity (34). Appendix F: Derivation of the quantities χ Nnℓm
Using the dispersion relation c k ω = ǫ b + 2 ǫ Z d rM ∗ Y, ǫ Z d rM ∗ Y = 2 ǫ Z d rM (1) ∗ x X nℓm c Nnℓm R nℓm Y ℓm (F1)+ 2 ǫ Z d rM (3) ∗ x X nℓm c Nnℓm R nℓm Y ℓm we define the quantities χ Nnℓm and ˜ χ Nnℓm :2 ǫ Z d rM (1) ∗ x c N n ± R n Y ± = ǫ b ∆ ( P ) LT ˜ χ N n ± h E x | w N i ǫ Z d rM (3) ∗ x c N n ± R n Y ± = ǫ b ∆ ( P ) LT ˜ χ N n ± h E x | w N i (F2)2 ǫ Z d rM (3) ∗ x c N n ± R n Y ± = ǫ b ∆ ( P ) LT ˜ χ N n ± h E x | w N i . The notation ˜ χ denotes that the formula contains contributions from both excitons P and F. Using the for-mulas (21,22), and the expressions h R n Y ± | M (1) x i , h R n Y ± | M (3) x i , h R n Y ± | M (3) x i derived in AppendixB, we obtain obtaining the following formulas: for n=2,3 χ Nn ± = f n W n ± + V ( n )11 ± f n = 163 n − n . (F3)For n ≥ χ Nn ± = f n W Nn ± + V ( n )33 ± ,f n = 2 . · − η η (cid:16) r a ∗ (cid:17) ( n − n − n − n , ˜ χ Nn ± = 163 n − n (cid:16) W Nn ± + V ( n )111 (cid:17) (cid:16) W Nn ± + V ( n )331 (cid:17) − (cid:16) V ( n )131 (cid:17) × ((cid:16) W Nn ± + V ( n )331 (cid:17) − i · . · V ( n )131 (cid:16) r a ∗ (cid:17) η / η / r ( n − n − n ) , ˜ χ Nn ± = 0 . (cid:16) r a ∗ (cid:17) η / η n − n r ( n − n − n (F4) × (cid:16) W Nn ± + V ( n )111 (cid:17) (cid:16) W Nn ± + V ( n )331 (cid:17) − (cid:16) V ( n )131 (cid:17) × (h W Nn ± + V ( n )111 i · . · (cid:16) r a ∗ (cid:17) η / r ( n − n − n + i · V ( n )131 · η / ) . [1] Ya. J. Frenkel, Phys. Rev. , 17 ; , 1276 (1931).[2] G. H. Wannier, Phys. Rev. , 191 (1937).[3] N. F. Mott, Trans. Faraday Soc. , 500 (1938).[4] E. F. Gross and N. A. Karriiev, Dokl. Akad. Nauk SSSR , 471 (1952).[5] R. S. Knox, Theory of Excitons (Academic Press, New York, 1963).[6] V. M. Agranovich and V. L. Ginzburg,
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Figure 1: Changes in absorption spectrum due to the applied magnetic field for n=4 exciton. −3 −2 −1 E [meV] α −2 −2 −2 −2 Figure 2: The bulk magnetoabsorption of Cu O crystal calculated from the imaginary part of the bulk susceptibility(37) for B =2T. Insets show the detailed spectra around the excitonic lines corresponding to n=4, 5, 6, 7. E [meV] B [ T ] Figure 3: Absorption spectrum in the energetic region of n = 4 excitonic state as a function of the applied magneticfield strength. E [meV] B [ T ] Figure 4: Te same as in Fig 3, in the energetic region of n = 4 −
25 excitonic states. −3 −2 −1 E [meV] α χ χ χ χ χ χ χ χ Figure 5: The bulk magnetoabsorption of Cu O crystal, individual excitonic resonances are identified. −2 −1 E [meV] α r =0.2a * r =1.0a * Figure 6: The influence of the choice of the coherence radius r on the magnetoabsorption spectra of Cu O crystal. −2 −1 E [meV] R With excitonsWithout excitons
Figure 7: The magneto-reflection coefficient of Cu O crystal of thickness 30 µ m, when the magnetic field 2T is applied.The dashed curve (labelled without excitons) corresponds to Fabry-Perot reflection. Inset - reflection spectrum forn=5, 6, 7 exciton. −2 −1 E [meV] R With excitonsWithout excitons
Figure 8: The same as in Fig. 7, for crystal thickness 100 µ m. Inset - reflection spectrum for n=7 exciton. E [meV] T With excitonsWithout excitons
Figure 9: The transmissivity coefficient of Cu O crystal of thickness 30 µ m, when the magnetic field 2T is applied. Thedashed curve (labelled without excitons) corresponds to Fabry-Perot transmissivity. Inset - transmission spectrumfor n=7 exciton. E [meV] T With excitonsWithout excitons
Figure 10: The same as in Fig. 9, for crystal thickness 100 µµ