The propagation of a coherent optical linear wave in an ensemble of
semiconductor quantum dots is considered. It is shown that a distribution of
transition dipole moments of the quantum dots changes significantly the
polarization and Beer's absorption length of the ensemble of quantum dots.
Explicit analytical expressions for these quantities are presented

Adamashvili, G. T.

Maradudin, A. A.

English

arXiv

Adamashvili, GT

Maradudin, AA

eScholarship - University of California

UC IrvineUC Irvine Previously Published WorksTitleBeer's law in semiconductor quantum dotsPermalinkhttps://escholarship.org/uc/item/74c8t0vdAuthorsAdamashvili, GTMaradudin, AAPublication Date2010-03-02 Peer reviewedeScholarship.org Powered by the California Digital LibraryUniversity of CaliforniaarXiv:1003.0638v1  [physics.optics]  2 Mar 2010Beer’s law in semiconductor quantum dotsG. T. Adamashvili∗Max-Planck-Institut für Physik Komplexer Systeme Nöthnitzer Str. 38 D-01187 Dresden, Germanyemail:gadama@parliament.geA. A. MaradudinDepartment of Physics and Astronomyand Institute for Surface and Interface ScienceUniversity of California, Irvine, CA, 92697 USA.email: aamaradu@uci.eduThe propagation of a coherent optical linear wave in an ensemble of semiconductor quantumdots is considered. It is shown that a distribution of transition dipole moments of the quantumdots changes significantly the polarization and Beer’s absorption length of the ensemble of quantumdots. Explicit analytical expressions for these quantities are presented.PACS numbers: 78.67. HcI. INTRODUCTIONSemiconductor quantum dots (SQD), also referred to as zero-dimensional systems, are nanostructures thatallow confinement of charge carriers in all three spatial directions, which results in atomic-like discrete energyspectra and strongly enhanced carrier lifetimes. Such features make quantum dots similar to atoms in manyrespects (artificial atoms)[1]. Observation of optical coherence effects in ensembles of quantum dots is usuallyspoiled by the inhomogeneous line broadening due to dot-size fluctuations, with a full width at half maximumof typically more than several tens of meV. Real quantum dots often have a base length in the range 50-400 Å.Beside the frequency, the quantum dot-size fluctuations influence also the transition dipole moments of SQD.Borri et al.[2] have reported measurements of optical Rabi oscillations in the excitonic ground-state transitionof an inhomogeneously broadened InGaAs quantum dot ensemble. They found that a distribution with a 20percent standard deviation of transition dipole moments results in a strong damping of the oscillations versuspulse area. In the experiments reported in [2] it was also found that the period of the Rabi oscillations is changed.These results show quantitatively how uniformity in dot size and shape is important for any application basedon a coherent light-quantum dot ensemble interaction.In theoretical investigations of optical coherence effects in ensembles of SQD it is usually assumed thattransition dipole moments of the quantum dots are independent of the size of the dots [3-8]. It is obvious thatfor an adequate description of coherent optical phenomena it is necessary to take into account a distribution oftransition dipole moments in the ensemble of quantum dots.The purpose of the present work is to investigate theoretically the propagation and absorption of an opticallinear wave when a distribution of transition dipole moments in the ensemble of SQD is taken into account.II. THE OPTICAL BLOCH EQUATIONS IN SEMICONDUCTOR QUANTUM DOTSWe consider the propagation of a circularly polarized optical plane wave pulse in an ensemble of SQD witha pulse width T << T1,2, frequency ω >> T−1, wave vector ~k, and the electric field ~E, which gives rise to theexcitonic ground-state transitions (labeled 0-X).We use the method of slowly varying envelopes, and define a real envelope Ê by~E(z, t) = (√2)Ê(z, t)[~x cos(ωt− kz) + ~y sin(ωt− kz)],which defines an optical pulse propagating along the positive z axis, and ~x and ~y are unit vectors in the directionsof the x and y axes. We shall find it useful to define the complex polarization vectors~e+ =1√2(~x + i~y), ~e− =1√2(~x− i~y),in terms of which the electric field has the form~E(z, t) = Ê(z, t)(~e−ei(ωt−kz) + ~e+e−i(ωt−kz)), (1)∗Permanent address: Georgian Technical University, Kostava str. 77, Tbilisi, 0179, Georgia.2where Ê(z, t) is the slowly varying complex envelope of the electric field and e±i(kz−ωt) contains the rapidlyvarying phase of the carrier wave. We will assume that|∂Ê∂t| << ω|Ê|, |∂Ê∂z| << k|Ê|. (2)For the description of SQD in a circularly polarized optical pulse we use the two-level model (0-X transitions)that can be described by the states |1 > and |2 > with energies E1 = 0 and E2 = ~ω0, respectively, where |1 >is the ground state. The Hamiltonian and wave function of this system are:H = H0 + V̂ ,|Ψ >=∑n=1,2cn(t)exp(−i~Ent)|n >,where H0 = ~ω0|2 >< 2| is the Hamiltonian of the two-level SQD with the frequency of excitation ω21 = ω0,V̂ = −µ̂ ~E = −µ(σ̂−E+ + σ̂+E−)is the light-SQD interaction Hamiltonian, ~ is Planck’s constant,µ̂ = ~µ12σ̂+ + ~µ∗12σ̂−is the SQD’s dipole moment operator,~µ12 =µ√2(~x− i~y)is the electric dipole matrix element for the corresponding transition,µ = |~µ12|, ~µ21 = ~µ∗12,E± = Êe±i(ωt−kz),σ̂± =12(σ̂x ± iσ̂y),and the {σ̂i} are the Pauli matrices, which satisfy [σ̂x, σ̂y ] = 2iσ̂z, and commutation relations resulting fromcyclic permutations of the subscripts.The probability amplitudes c1 and c2 are determined by the Schrödinger equationsi~∂c1(t)∂t= −µ E+c2(t)e−iω0ti~∂c2(t)∂t= −µ E−c1(t)eiω0t. (3)The average values of the Pauli operators σ̂i for the state |Ψ >= c1 |1 > + c2 |2 >, are si =< σ̂i >=< Ψ|σ̂i|Ψ >, (where i = 1, 2, 3) and have the form [9]:sx = c∗1(t)c2(t)e−iω0t + c1(t)c∗2(t)eiω0t,sy = ic∗1(t)c2(t)e−iω0t − ic1(t)c∗2(t)eiω0t,sz = c∗2(t)c2(t)− c∗1(t)c1(t).To introduce explicitly the appropriate rotation matrix for the functions si in the rotating frame with com-ponents u, v, w, from the Schrödinger equations (3) we obtain the Bloch equations:u̇ = −∆v,v̇ = i∆u+ κwÊ,3ẇ = −κvÊ, (4)whereκ =2µ~, ∆ = ω0 − ω,while u and −v are the components, in units of the transition moment µ, of the SDQ’s dipole moment in-phaseand in-quadrature with the field ~E. In other words, v is the absorptive component of the SQD dipole moment,while u is the dispersive component, and 12~ω0sz is the expectation of the atom’s unperturbed energy.In addition to (4), we need a description of the pulse propagation in the medium. The wave equation for theelectric field ~E(z, t) of the optical pulse in the medium is given by∂2 ~E∂z2− εc2∂2 ~E∂t2=4πc2∂2 ~P∂t2, (5)where c is the speed of light in vacuum, ε = η2 is the dielectric permittivity, η is the refractive index of themedium, and ~P is the polarization of the ensemble of SQD.For the determination of the polarization of the ensemble of SQD we have to take into account that the dipolemoment of the SQD depends on the size of the quantum dot. The polarization of the ensemble of SQD is equalto~P =n0∑i=1< ~µi >,where n0 is the uniform dot density, and < ~µi > is the expectation value of the i-th dipole moment.We can rewrite this expression in another form:~P =∑j=1nj < ~µj >, (6)where nj is the number of dipoles with matrix elements µj in the interval ∆µj .We introduce the function h̃(µi) such that the quantity h̃(µi)∆µi is the the fraction of dipoles with dipolematrix element µi within ∆µi. Obviously its normalization is∫ ∞0h̃(µ)dµ = 1,or in another form ∫ ∞µminh̃(µ− µmin)dµ = 1,where µmin is the minimum value of the dipole matrix element in the ensemble of SQD. To take into accountthat when µ < µmin the number of dipoles is equal to zero, h̃(µ < µmin) = 0, we can rewrite the last equationin the form ∫ ∞−∞h̃(µ− µmin)dµ = 1.Because the quantity µ0 − µmin is constant, we can introduce the functionh(µ− µ0) = h̃(µ− µmin),and for this function the normalization condition has the form∫ ∞−∞h(µ− µ0)dµ = 1,where~µ0 =∑n0i=1 ~µin0is the main dot dipole matrix element.By using the function h(µ− µ0), we obtain that the quantity nj ≈ n0 h(µj − µ0) ∆µj . Substituting the lastexpression into Eq. (6) we obtain the polarization of the ensemble of SQD in the form~P ≈ n0ñ0∑j=1h(µj − µ0) ∆µj < ~µj > .4After transition to a limit we obtain the exact value of the polarization, namely~P = n0∫ ∞−∞h(µ− µ0) < ~µ > dµ. (7)Expressing the value of < µ > in Eq. (7) in terms of the slowly varying amplitudes u and v, we obtain~P =n0√2∫ ∞−∞µ h(µ− µ0) {u [~x cos(ωt− kz) + ~y sin(ωt− kz)] + v [−~x sin(ωt− kz) + ~y cos(ωt− kz)]} dµ.The fluctuations in the size of the SQD also lead to the inhomogeneous broadening of the spectral line. Upontaking into account the inhomogeneous broadening of the spectral line, the polarization of the ensemble of SQDis equal to~P (z, t) =n0√2∫ ∞−∞∫ ∞−∞µ h(µ− µ0) g(∆){u(z, t; ∆, µ)[~x cos(ωt− kz) + ~y sin(ωt− kz)]+v(z, t; ∆, µ)[−~x sin(ωt− kz) + ~y cos(ωt− kz)]} dµ d∆, (8)where g(∆) is the inhomogeneous broadening function. This equation generalizes the polarization of the en-semble of SQD to the case of a distribution of transition dipole moments. In the special case, when all dipolemoments are the same, the distribution function is h(µ − µ0) = δ(µ − µ0), and we obtain the polarization inthe usual form [9]:~Ph(z, t) =n0µ0√2∫ ∞−∞g(∆){u(z, t; ∆)[~x cos(ωt− kz) + ~y sin(ωt− kz)]+v(z, t; ∆)[−~x sin(ωt− kz) + ~y cos(ωt− kz)]} d∆. (9)By substituting equations (1) and (8) into the wave equation (5), and using the assumption of slowly varyingamplitudes (2), we obtain the in-phase and in-quadrature wave equations in the following forms:(k2 − εc2ω2)Ê =2πn0ω2c2∫ ∞−∞∫ ∞−∞µ h(µ− µ0) g(∆) u dµ d∆, (10)∂Ê∂z+ωεkc2∂Ê∂t=πn0ω2c2k∫ ∞−∞∫ ∞−∞µ h(µ− µ0) g(∆) v dµ d∆. (11)Equations (10) and (11) are general equations for the slowly varying amplitudes by means of which we canconsider a quite wide class of coherent optical phenomena (for instance: Rabi oscillations, photon echo, self-induced transparency, and others) in an ensemble of SQD in the presence of a distribution of transition dipolemoments in the ensemble. Equations (10) and (11) generalize the Maxwell equations that have been consideredup to now for atomic systems and for SQD when fluctuations of the dipole moments are neglected [1]. We haveto note that an analytic solution of these equations are significantly difficult than the solution of the Maxwellequations in the special case, when all dipole moments are the same, because in Eqs.(10) and (11) the quantitiesv and u are the functions also of the variable µ. Nevertheless analytic solution of the equations (10) and (11)are possible in some special cases, for instance, for Beer’s law in SQD.III. BEER’S LAW IN SEMICONDUCTOR QUANTUM DOTSFrom Eq. (10) we can obtain the dispersion relation for the wave and from Eq. (11) we can determine theexplicit form of the envelope of the electric field strength Ê.Usually g(∆) is a very broad and smooth function, so that the integral∫∞−∞g(∆)ud∆ in Eq. (10) is verysmall and is usually neglected [9]. Consequently, the dispersion law has the formk2 =εc2ω2. (12)After integration of the in-quadrature wave equation (11) from t = −∞ up to the time t̄ that occurs afterthe pulse has passed the point of observation z we obtain∂ϑ(t̄, z)∂z=2π2n0ω2g(0)c2k∫ ∞−∞µh(µ− µ0)v(µ, t0, z, 0)dµ, (13)whereϑ(t, z) =∫ t−∞Ê(t′, z)dt′ (14)5is proportional to the area of the pulse, θ(t, z) = 2µ~ϑ(t, z), the time t0 marks the end of the pulse, and at t0and for all later times the electric field envelope Ê is zero.For an analytic solution of Eq. (13) we have to determine the explicit form of the quantity v as a function ofthe electric field and dipole moment. From the Bloch equations (4) for the absorptive part of the on-resonancedipole amplitude, we find thatv(µ, t0, z, 0) = − sin2µ~ϑ(t0, z). (15)We consider Beer’s law of absorption for the electric field of the pulse in a SQD. For this purpose we considerthe limit of weak electric fields, in which the ”area” θ(t0, z) is small, i.e. |θ(t0, z)| << 1, for all values of µ.Under this condition, and taking into account that for all t > t0, ϑ(t̄, z) = ϑ(t0, z), from Eq.(15) we obtainv(µ, t0, z, 0) ≈ −2µ~ϑ(t0, z). (16)On substituting Eq. (16) into Eq. (13), we obtain the relation∂ϑ(t̄, z)∂z= −α2ϑ(t̄, z), (17)whereα =4 π2n0ω2g(0)c2~ k∫ ∞−∞µ2 h(µ− µ0) dµ (18)is the reciprocal Beer’s absorption length, in which fluctuations of the dipole moments of the quantum dotshave been taken into account.IV. CONCLUSIONWe note that Beer’s law, Eq.(17), shows that the electric field of the wave decays exponentially with increasingpenetration into the medium. For pulses with an electric field of arbitrary shape, Beer’s law is usually writtenfor the dimensionless pulse area 2µc~ϑ, where the dipole moment µc is the same for all atoms or SQD.For SQD with a distribution of their dipole moments, it is impossible to write the area of a pulse in the usualdimensionless form, and we have to use Beer’s law in the form of Eq. (17), for the quantity ϑ defined by Eq.(14).In the limiting case when the fluctuation of the dipole moments is neglected, the distribution function h(µ−µ0)transforms to the Dirac δ-function, and the reciprocal Beer’s absorption length reduces to the usual form which,in atomic systems with identical dipole moments in solids or in gases, where η = 1 [9], isαh =4 π2n0ωg(0)c~ ηµ02. (19)On comparing Eqs.(9),(18) and (19) we see that a distribution of the dipole moments in an SQD ensemblesignificantly changes the polarization Eq.(8) and Maxwell’s wave equations for an ensemble of quantum dotsEqs.(10) and (11) and, as a result, in a special case also the Beer’s absorption length Eq.(18).[1] D. Bimberg, M. Grundmann, and N. N. Ledentsov, Quantum Dot Heterostructures (Wiley, Chichester, 1999).[2] P. Borri, W. Langbein, S. Schneider, U. Woggon, R.L. Sellin, D. Ouyang, and D. Bimberg , Phys. Rev. Lett. 87,157401 (2001).[3] G. Panzarini, U. Hohenester, and E. Molinari, Phys. Rev. B 65, 165322 (2002).[4] J. Forstner, C. Weber, J. Danckwerts, and A. Knorr, Phys.Rev. Lett. 91, 127401 (2003).[5] G. T. Adamashvili and A. Knorr, Opt. Lett., 31, 74 (2006).[6] G. T. Adamashvili, C. Weber, A. Knorr, and N. T. Adamashvili, Phys. Rev. A 75, 063808 (2007).[7] G. T. Adamashvili and A. Knorr, Phys. Lett. A, 367, 220 (2007).[8] G. T. Adamashvili, C. Weber, and A. Knorr, The European Physical Journal D 47, 113 (2008).[9] L. Allen and J. H. Eberly, Optical resonance and two-level atoms (Dover, New York, 1975).

Beer's law in semiconductor quantum dots

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