* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Excitons in parabolic quantum dots in 1 electric and magnetic fields
Maxwell's equations wikipedia , lookup
Electrostatics wikipedia , lookup
EPR paradox wikipedia , lookup
Renormalization wikipedia , lookup
Casimir effect wikipedia , lookup
Quantum field theory wikipedia , lookup
Magnetic monopole wikipedia , lookup
Lorentz force wikipedia , lookup
Quantum potential wikipedia , lookup
Hydrogen atom wikipedia , lookup
Introduction to gauge theory wikipedia , lookup
Electromagnet wikipedia , lookup
Mathematical formulation of the Standard Model wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Field (physics) wikipedia , lookup
Superconductivity wikipedia , lookup
History of quantum field theory wikipedia , lookup
Time in physics wikipedia , lookup
Condensed matter physics wikipedia , lookup
Electromagnetism wikipedia , lookup
Old quantum theory wikipedia , lookup
Canonical quantization wikipedia , lookup
Semicond. Sci. Technol. 9 (1994) 1775-1780. Printed in the 1I UK Excitons in parabolic quantum dots in electric and magnetic fields S Jaziritl: and R Bennaceurl 7 Departement dePhysique d e I’Ecole Normale Superieure de Bizerte, Jarzouna, 7021 Bizerte, Tunisia f Laboratoirede Physique de la Matiere Condensee, Faculte des Sciences deTunis, 1060 Le Belvedere Tunis, Tunisia Received 16 February 1994, in final form 8 June 1994, accepted for publication 13June1994 Abstract. We report a perturbative-variational calculation on the effect of magnetic and electric fields on excitons in parabolic quantum dots. The effects of electric and magnetic fields on 1 s exciton energy and oscillator strength are investigated. The competition between the confinementand correlation effects on the one hand, and the external field effects (electric and magnetic) on the other hand, is also discussed 1. Introduction The development of semiconductor nanostructures, such as quantum boxes and dots, by the use of different techniques, has attracted much theoretical and experimental attention [l-123. In undoped quantum dots the optical properties are dominated by excitonic effects. In this paper, we consider quantum dots exhibiting a nearly parabolic confinement for both electrons and holes [IO, It]. These quantum dots can be fabricated by focused laser-beam-induced interdiffusion of a GaAs/GaAlAs quantum well system. For such a confinement, for both electrons and holes, the centre of mass and relative motion of the exciton separate [7, 121. In this case, the motion of the exciton in the ground state is essentially governed by the relative Hamiltonian [7, 121. The application of external fields (magnetic [7j or electric [ S . 121) influences exciton properties. The difficulties encountered in theoretical calculations arise from the need to include four different potentials governing exciton motion: these are the confinement, Coulomb, magnetic and electric potentials. Excitons in quantum boxes have been studied by Bryant [2], and have revealed the competing effects of quantum confinement and electron-hole correlation induced by the Coulomb potential. The influence of a magnetic field on excitons in parabolic quantum dots has been analysed by Halonen et al [7]. This work shows that the erect of the magnetic field is decreased when the confinement potential is increased and vice versa. The quantum confined stark erect (QCSE), induced in excitons by an electrostatic field, has been studied in spherical [SI and in parabolic [ l 2 j quantum dots. In particular, it is shown in [12] that quantum dots with a low confinement potential are more affected by the electric potential and exhibit a larger Stark shift. On the contrary. quantum 0268-1242/94/101775+06$19.50@ 1994 IOP Publishing Lid dots with strong confinement have the smallest Stark shift. The increase of confinement potential reduces the effect of the electric field. Therefore, it is interesting to study these effects on excitons in GaAs quantum dots, which exhibit an important quantum-size potential (with the radius of quantum dots ranging from SO to 200 A). In the present paper we shall present theoretical results on the properties of excitons in a parabolic quantum dot in the presence of electric ( F ) and magnetic ( B ) fields, applied parallel to the Iaxis. In our calculation we use an analytical approximation by means of a perturbativevariational approach [9, 131, where the Coulomb interaction potential is approximated by a force interacting according to Hooke’s law [6, 121. 2. nne!hod of ca!cu!l!!nn Our calculation is based on the effective-mass approximation and ignores vaIence band structure effects. For simplicity we consider a GaAs quantum dot with parabolic confinement for the electrons and holes having the same quantization energy hi2 [6, 10-123. The total Hamiltonian for the exciton in a parabolic quantum dot in the presence of electric ( F ) and a magnetic ( B ) fields applied in the same direction (z axis), can be expressed as H = H, + Hh --e’E r and 1775 S Jaziri and R Bennaceur where H, (Hh) and m, (mh) are the single-particle Hamiltonian and the effective mass for the electron (hole) and E is the GaAs background dielectric constant. In order to solve this total Hamiltonian we use the relative coordinate I' = ( I ' ~- rh) and the corresponding momenta p with reduced mass p (=m,m,/(m, + mh)) and centre-ofmass coordinate R (=(merh + mhrh)/(m, + mh)) and the corresponding momenta P with the total mass M = m, inh. For the magnetic vector potential, we consider the symmetric gauge vector potentials for the electron and hole as [7] A , = @ x (re - r h ) and A h = - $ E x (r, - vh) where the magnetic field is parallel to the x direction. Therefore the total Hamiltonian is + H + 4.+ Hcoup = HR P2 HR = 2M +Z I M Q ~ R ~ - _e2- eFi to a frequency a,. The io shift reflects tendency to a breaking of the electron-hole pair by the field, while the frequency Q, accounts for the remaining Coulombic attraction between the spatially separated electron and hole. In the (x,y) plane the electron-hole pair energy will also be lowered by the Coulombic term. Without Coulombic attraction the electric field would shift the electron and hole mean positions by zo (zo = -eF/pQ2) without altering the oscillation frequency (but giving rise to a quadratic negative shift -e2F2/2pQ2). The application of the magnetic field HL can be described by diamagnetic and Zeeman terms. The latter term makes no contribution to the exciton ground state. In the (x,y) plane, the diamagnetic term added to the parabolic confinement potential enhances the exciton energy. To apply the previous physical reasoning analytically we add to and subtract from the relative electron-hole Hamiltonian H, a term V(r) in order to split H, into two parts. The first one Ho should be analytically solvable while the second one H,, expected to be small, will be treated by first-order perturbation calculus on the eigenstates of H,,. There we choose [6, 121 &I PI,,,, = V(r) = j&u(nzr2 ihe BI' x V R - hQ) (34 where E. is still an unknown parameter. In this case Me where pR = he/2pc is the Bohr magneton, gP = (i 2) - is the g factor and o = Be/pc is the cyclotron frequency. The operator ipBg,,B(a/atp) corresponds to the z component of the angular momentum operator. The last term in the Hamiltonian, HEovp, couples the centre-of-mass and relative motions. According to [7, 141, this crossed term leads to a negligible perturbation of the ground exciton energy, and it couples only excited states of different relative and centre-of-mass motions and is neglected in the following. We can see that the exciton properties in the ground state can be essentially determined by the relative Hamiltonian H,. The relative Hamiltonian describing the exciton motion is H, = Hl + H5 + H, The perturbed term H , restores the character of the Coulomb interaction. The zeroth-order wavefunction and eigenenergy of the ground state of Ho are determined exactly. This is possible because the unperturbed eigenfunctions are separable and analytically expressed as x(R,r, E. F,2) = Q(R)$(p,j.)O(z,lJ =Q (R)(i>(L)'" RI& R,& x exp(-pz/2R$) exp( -zZ/2Rf) (4) where where pL and p z are respectively the transverse and longitudinal electron-hole relative momenta. In the absence of the magnetic field, the combined effects of the Coulombic potential, the electric field term -eFz and the parabolic confinement along z qualitatively result in the build up of a nearly parabolic potential along z displaced by io from the original one and corresponding 1776 RO (1 + J.),'4' The first term in (4) is the exciton centre-of-mass wavefunction for a harmonic oscillator and the last terms correspond to the exciton relative motion: $ ( p , % ) is Excitons in parabolic quantum dots the two-dimensional exciton wavefunction and e(z, 1)the one-dimensional exciton wavefunction. We determine the best choice of 1. as the one which ensures the fastest convergence of the perturbation series. so that only the first few terms will be used. Under these circumstances we shall write the total energy as - i.hQ + AE(1) (6) where E , is the gap energy and AH.?.) is approximated by the first-order energy matrix correction calculated by the perturbative method. The unknown parameter A in equations (3)-(6) is determined by minimization of the total energy value by requiring that the first-order energy shift AE(E.) = ( Y j IHllYA) vanishes, where Y;. is the ground state of Ifo. We calculate the oscillator strength of the exciton in the ground state in the dot normalized to that of a free exciton in a bulk material with volume V = $ltRi. The normalized oscillator strength is given by [9] where E,, and ET(F,B ) are the total exciton energies in the bulk and quantum dot respectively and a3Dis the bulk effective Bohr radius. 3. Results and discussions Results are presented for excitons confined in GaAs parabolic quantum dots placed in electric and magnetic fields parallel to the z direction, with the parameter values: gap energy E, = 1.520 eV. dielectric constant e = 13.1. effective masses for the electron (e), heavy hole (hh),m, = 0.067m0,m,, = 0.377m0 (m, is the free-electron mass). Following the perturbative-variational method developed in [13], the validity of the calculation can be tested by the dependence of the energy on the value of 1.We find that the energy will be independent of 71 if the true wavefunction is used to calculate the groundstate energy. We also compare our results with those obtained from conventional variational calculation for the Hamiltonian defined in (1) by using a Gaussian trial wavefunction of the type Yver(r) = N exp( -apz/2) exp(-flz2/2) where N is the normalization factor and E , fl are the variational parameters. Our results and the variational results are very close except at large values of R , where exciton energies calculated from our model are slightly higher. However, our results and the conventional ones are in agreement for R, < 200 A. For R, = 150 A and F = 80 kV cm-I and B = 5 T the two methods give E, values which differ by less than 0.2 meV. Our approximation is in fact coherent with the physics underlying equations (3)-(5): the exciton motion is mostly dominated by the confinement potential, the Coulombic term essentially correlates the electron-hole reduced motion in wider quantum dots. Also our approach is justified by the relativemagnitude ofthe variational parameters E-''' and fl-''z compared respectively with Rs(A) and R&) values for R, = 100 A and for F = 80 kV cm-' and B = 5 T; the deviation of our calculated values from the variational results is 4%. It should be noted that the perturbative-variational method is much simpler than the variational method. The effects of confinement, Coulomb, magnetic and electric potentials can be written in terms of characteristic length as respectively. These effects on the properties of the excitons can be classified with regard to the two intrinsic lengths: the dimension of the parabolic quantum dot R, and the excitonic Bohr radius a,D (the 3D Bohr radius = 140 8, for GaAs). In this work we consider only quantum dots with radii between 50 A and 200 8,where the confinement effect is predominant. In this case the correlation effect on excitons introduced by the Coulombic potential is weak. In order to analyse the effects of magnetic and electric fields on the energy E, and the oscillator strength of the ground exciton state we define first a total (effective) magnetic field E,,, = where Bo = is an effective magnetic field associated with the 3D parabolic quantum dot confinement, and LOff= is the corresponding magnetic length. In the absence of an applied electric field the increase in E, results in a diamagnetic compression of the exciton wavefunction in the (x, y ) directions. Under an applied electric field, the distance between electrons and holes is increased and induces a decrease of the energy E,. This decline of E, results in the spatial separation of the exciton wavefunction along the iaxis. In figure 1 we show the variation of the exciton energy in the ground state as a function of the applied electric field F for different quantum dot radii and values of the magnetic field B: R , = 50 8, for B = 0 and B = 20 T; R, = 120 8,for B = 0 and B = 10T and B = 20T. Note that the application of the magnetic field reduces the lateral dimension of the quantum dot corresponding to L,, = 46 8,(for R , = 50 ,&and B = 20 T) and Le,, = 74 8, (for R, = 120 8, and B = 10 T). We can see that for narrow dots (R, < 1008,) the curvature of E T ( F ) is negative, due to the quadratic Stark shift. For large quantum dots ( R , > 100 8,) the curvature is negative at low F and decreases more rapidly at large F. The reason i's that for fields such that the interacting electron-hole pair is well separated in the z direction, i.e. lzol > R, (-8 x 10-8F (kV cm-') > l), for this range of fields and dot sizes the energy is approximately equal to -e2/&lzoi.This regime cannot be reached in the case of m, 1777 S Jaziri and R Bennaceur 1640 - __Ros50A,B=O cl:Ro=50A,B=20T 620 1600-1580- % E ___ Ro=l '1560- OAB 0 bRO=l2$.~=~OT 3 narrow dots, where the effect of the field F (F = 100 kV cm- ') is not large enough; thus only the F*-like variation appears in the E,(F) curves. The effects of the Stark shift of the exciton in the quantum dot leads to a decrease of the oscillator strength in figure 2. The decrease of the oscillator strength in wider quantum dots can he explained by the field-induced polarization of the electron-hole wavefunction. Figure 3 shows the effect of the magnetic field on the exciton energy in quantum dots of different sizes for different values of applied electric field. The increase in the energy with increasing magnetic field is related to the compression of the exciton wavefunction in the (x,y ) directions. In the absence of an electric field ( F = 0), the application of the magnetic field creates an additional confinemenf lowering the value of LFrr(the corresponding magnetic length). We have two limiting situations: a diamagnetic-like shift ET Bo(l B2/2B3 whenever Bo % B and a high-field limit E,(B) % B(l + B$2B2) when Bo @ B. For strongly confined systems (B 6 Bo) we have very smooth parabolic dependencewith the external field. For wider dots both limiting situations are present, although the variations with B are sensibly smoothed by the effect of geometrical confinement. Under the application of an electric field the electrons and holes begin to separate. As the electrons and holes are pushed apart by the electric field a reduction in exciton energy with increasing electric field is expected. Figure 4 shows the magnetic-field dependences of the normalized oscillator strength. The increase in the oscillator strength with + 1420 50 0 F (kV/cm) Figure 1. Dependence of the energy of a 1 s exciton in a GaAs parabolic quantum dot on the electric field for different values of magnetic field and dot radius. ' 1640 --b :Ra=50A.B=20T 1620 4 1 Ro=S.OA.F=O a:Ro=5OA.F=20kV/cm --- ' --- Ro=lZOkB=O c :Ro=l2OA.B=1 OT d :Ro=l2OA.B=2OT 154.0 Figure 2. Dependence of the normalized oscillator strength of a 1 s exciton in a GaAs parabolic quantum dot on the electric field for differentvalues of magnetic field and dot radius. 1778 Figure 3. Dependence of the energy of a 1 s exciton in a GaAs parabolic quantum dot on the magnetic field for different values of electric fieid and dot radius. Excitons i n parabolic quantum dots a 1 I Ro=SOA.F=O a:Ro=50A,F=ZOkV/cm . --- --LRo=120A,F=0 b R o = l Z O h . F = l Okv/cm c:Ro=lZO&F=ZOkV/cm Figure 4. Dependence of the normalized oscillator strength of a 1 s exciton in a GaAs parabolic quantum dot on the magnetic field for different values of electric fieid and dot radius. I 1660 I 50 150 100 Ro (1) 200 Figure 5. Dependence of the energy of a 1 s exciton in a GaAs parabolic quantum dot on the radius of the dot for different values of electric and magnetic fields. Figure 6. Dependence of the normalized oscillator strength of a 1 s exciton in a GaAs parabolic quantum dot on the radius of the dot for different values of electric and the magnetic fields. increasing magnetic field is related to the compression of the exciton in the ( x , y ) directions and hence to the increase of f i f , , proportional to RB2. Lastly, figure 5 illustrates the dependence of the exciton energy on the size (R,) of the dot. In general, the energy decreases with increasing dot size. For narrower dots, only the confinement influences the energy increase, even in the presence of an external electric or magnetic field. For intermediate dot radii (R, > 100 A) the application of an electric field changes the rate of decrease of ET(Ro)(figure 5(b), (e)) We interpret this feature as competition between the electric field effect and the confinement. At large R,, the energy ET(R,) decreases. The application of a magnetic field does not change the behaviour of the E,(Ro) curve but increases the energy value. These same trends can be used to explain the size-dependent decrease of the ground exciton oscillator strength in figure 6. In conclusion, using a variational-perturbative calculation, we have calculated the 1 s exciton properties: energy and oscillator strength in a parabolic quantum dot in the presence of parallel electric and magnetic fields. First, we have tested the validity of the approximation and compared our results with calculations obtained by the conventional variational method. The application of an electric field induces a spatial separation ofthe carriers. leading to a decrease in the energy and the oscillator strength. The application of a magnetic field leads to additional confinement, which induces an enhancement 1779 S Jaziri and A Bennaceur in the exciton energy a n d oscillator strength. In the case of narrower dots, we have found that the exciton properties are little affected by the electric a n d magnetic fields. The influence of the external fields is greater on exciton properties for wider quantum dots ( R , > 100 A). Acknowledgments The authors would like t o thank Gkrald Bastard a n d Robson Ferreira for their fruitful discussions and help. References [l] Petroff P M, Cibert J, Gossard A C, Dolan G J and Tu C W 1987 J . Vuc. Sci. Technol. B 5 1204 Cibert J, Petroff P M, Dolan G J, Pearton S J, Gossard A C and English J H 1986 Auol. Phvs. Lett. 49 1275 Temkin H. Dolan G J. Panish M B and Chu S N G 1987 Appl. Phys. Lett. 50 413 Kash K, Scherer A, Worlock J M, Craighead H G and Tdmargo M C 1986 Appl. Phys. Lett 49 1043 Arnot H E G, Watt M, Sotomayor Torres C M, Glew R, Cusco R, Bates J and Beaumont S P 1989 Superlatt. Microsrruet. 5 459 [21 Hu Y Z, Lindberg M and Koch S W 1990 Phys. Reo. B 42 1713 Bryant G. W 1988 Phys. Rev. B 37 8763 Bryant G W 1992 Excitons in Coilfined Systems ed A DAndrea et al (New York American Institute of Physics) Takagahara 1992 Suf$ Sci. 267 310 Kdyanuma Y 1991 Phys. Reu. B 44 13085 I .. ~~ 1780 E31 Kash K, Scherer A, Worlock J M, Craighead H G and Tamargo M C 1986 Appl. Phys. Lett. 49 1043 Reed M A, Bate R T, Bradshaw K, Duncan W M, Frensley W R, Lee J W and Shih H D 1986 J. Vac. Sci. Technol. B 4 358 [41 Kohn W 1961 Phys. Rev. 123 1242 Peters F M 1990 P h p ; Re!,, B 42 1486 Maksym P A and Chdkraborty T 1990 Phys. Rev. Lett. 65 108 Demsey J, Johnson N F, Brey L and Halperin H I 1990 Phys. Rev. B 42 11708 Brey L, Johnson N F and Halperin B I 1989 Phys. Rev. B 40 10647 [SI SikorskiChandMerktU 1989Phys.Rev.Lett.622164 [6] Weiming Que 199%Solid State Commun. 81 721 Johnson N F and Payne M C 1991 Phys. Rev. L e t t 67 1157 [7] Halonen V, Chakraborty T and Pietilainen P 1992 Phys. Reu. 45 5986 [XI Chiba Y and Ohnishi S 1988 Phys. Rev. B 38 12988 191 Jaziri S, Bastard G and Bennaceur R 1993 Semicond. Sci. Technol. 8 670 Cl01 Bockelmann, U, Brunner K and Abstreiter G 1993 Proc. 6th Int. Con$ on Modulated Semiconductor Struetires [lll Brunner K, Bockelmann U, Abstreiter G, Walther M, Bohm G. Trankle G and Weimann G 1992 Phys. Reo. Lett. 69 3216 1121 Jaziri S, Bastard G and Bennaceur R 1993 3rd CunJ on Optics of Excitons in Confined Systems: 1993 J . Physique IV 3 367 U31 Lee Y C, Mei W N and Liu K C 1982 J. Phys. C: Solid State Phys. 15 L469 Dai C M, Pei I H and Chuu D S 1990 Physica B 160 317 Huang W J, Chou W C, Chuu D S, Hdn C S and Mei W N 1988 Semicond. Sci. Technol. 3 202 [14] Jaziri S 1994 Solid State Commuii. 91 171