- Email: [email protected]
Laser theory for semiconductor quantum dots in microcavities
Superlattices and Microstructures 43 (2008) 470–473 www.elsevier.com/locate/superlattices
Laser theory for semiconductor quantum dots in microcavities Christopher Gies ∗ , Jan Wiersig, Frank Jahnke Institute for Theoretical Physics, University of Bremen, P.O. Box 330 440, 28334 Bremen, Germany Available online 10 September 2007
Abstract Quantum dots (QDs) used as active material in microresonators are currently of strong topical interest due to breakthroughs in growth and device structuring. From the theory side, however, atomic models are still used to analyse the emission from these semiconductor systems, despite known differences between QDs and atoms. We introduce a semiconductor laser theory based on a microscopic approach with the goal of better describing the characteristic behaviour of QD-based laser devices and to show differences from predictions based on atomic models. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Photon statistics; Microcavity lasers; Quantum-dot lasers; Microscopic theory
1. Overview Achievements in the fabrication of quantum dots and in the structuring of microcavities containing quantum dots (QDs) as the active material have led to novel applications that are based on controlled modifications of the emission properties, like lasers with reduced threshold, more efficient LEDs, or the generation of single or entangled photons. This development has kindled strong interest in the emission properties of QDs [1–3]. The combination of the threedimensional carrier and photon confinement of QDs in microcavities provides an ideal basis for the design of tailored laser devices. ∗ Corresponding author.
E-mail address: [email protected] (C. Gies, J. Wiersig, F. Jahnke). URL: http://www.itp.uni-bremen.de (C. Gies, J. Wiersig, F. Jahnke). c 2007 Elsevier Ltd. All rights reserved. 0749-6036/$ - see front matter doi:10.1016/j.spmi.2007.06.026
C. Gies et al. / Superlattices and Microstructures 43 (2008) 470–473
471
Due to the discrete level spectrum of QDs, their theoretical description is typically based on atomic models. Despite undeniable similarities, the inherent semiconductor properties of the QDs become obvious in several experiments, the frequently observed non-exponential and excitationintensity-dependent decay of the photoluminescence being one example [4]. The input/output curve of conventional atomic lasers is well described by rate equation models [5]. In order to calculate the photon statistics of the emitted light, master equations are used [6]. However, the application of these models to semiconductor QDs requires the assumption that carriers in the dot behave like carriers in an atom, which is not true in the general case, particularly when considering effects like a modified source term of spontaneous emission and Pauli blocking [7]. We use a microscopic approach to developing a laser theory which takes the semiconductor nature of QDs into account. Our theory includes many-body effects and provides a consistent scheme for the treatment of Coulomb- and light–matter-interaction-induced correlations between carriers, and between carriers and photons. Results for the input/output curves and the photon statistics are presented. 2. Model We use a semiconductor Hamiltonian for the interacting carrier–photon system, from which dynamical equations are derived via Heisenberg’s equation-of-motion technique. Carriers are treated in second quantization, and the light field is also quantized. The contributing parts are, from left to right, the free carrier spectrum, Coulomb interaction, free electromagnetic field, and the light–matter interaction; details are found in [8]: 0 H = Hcarr + HCoul + Hph + HLM .
(1)
A consequence of the equation-of-motion approach is the occurrence of an infinite hierarchy of coupled dynamical equations, arising due to the two-particle parts HCoul and HLM of the Hamiltonian. A truncation of this hierarchy for numerical calculation is unavoidable. At the same time, validity of the truncation method at a certain level is indispensable for the usability of the theory. The essential idea of what has become known as the cluster expansion method [9,10] is to reformulate all equations of motion for operator expectation values in such a way that equations of motion for the correlation functions are obtained. Correlation functions of nth order only contain processes which involve exactly n particles. The hierarchy of correlation functions is then truncated rather than the hierarchy of expectation values itself. This allows the consistent inclusion of n-particle effects. Using the above described formalism, we numerically determine Ď the dynamics of the photon population in the laser mode hbĎ bi and the carrier population hcν cν i Ď and 1 − hvν vν i for electrons and holes, respectively. Here, the operators bĎ and b are bosonic creation and annihilation operators for photons in the laser mode, and cν and vν are the fermionic equivalent for conduction and valence band carriers in the single-particle state ν. The photon autocorrelation function is of particular interest, as its value at zero time delay reveals the coherent or thermal nature of the emitted light. In our formalism [8] it is obtained from g (2) (τ = 0) =
hn 2 i − hni hbĎ bĎ bbi = hni2 hbĎ bi2
(2)
472
C. Gies et al. / Superlattices and Microstructures 43 (2008) 470–473
Fig. 1. Calculated output curve (left) and autocorrelation function g (2) (τ = 0) (right) for various values of β. The total spontaneous emission time is 100 ps, the cavity lifetime is 13 ps. Relaxation times of 5 ps (2.5 ps) are used for electrons (holes).
by calculating the time evolution of hbĎ bĎ bbi. In the dynamic equations further correlation functions occur, for which coupled equations are determined up to the order of four-particle processes. 3. Quantum-dot lasing In Fig. 1 results for the input/output characteristics and the photon statistics for an ensemble of 200 InGaAs QDs in GaAs/AlAs micropillars are presented. The number of resonant QDs is scaled with β −1 . These parameters are slightly modified from those used in [8], as we consider here more QDs in a cavity of lower quality. The results look similar but differ in detail in the sub-threshold behaviour. We choose to vary β as a parameter. Note that between different micropillars, other parameters like the spontaneous emission rate vary, too. The value β = 1 corresponds to complete coupling of the emitter to the laser mode. In this case, the input/output curve shows no kink. Since microcavity lasers can operate close to this regime, knowledge of the photon correlations is important for identifying the regime of truly coherent laser emission. Our results for the autocorrelation function demonstrate that for typical parameters of present systems there is always a clear transition from thermal to coherent emission even for β = 1, allowing for an identification of the coherent regime. In atomic systems, the jump of the intensity curve from below to above threshold is given by 1/β provided that nonradiative losses can be neglected. This feature is here spoiled by the fact that the spontaneous emission is not solely determined by the electron density, and because the system does not operate at full inversion. At high pump powers, saturation effects due to Pauli blocking are visible. 4. Comparison to atomic theory Due to the hierarchy problem that we discussed above, the applicability of our formalism is crucially dependent on the possibility of truncating the arising coupled equations of motion at a certain level. For the calculation of the photon statistics, which is determined by the average over four photon operators in Eq. (2), dynamic equations for at least up to four-particle effects must be explicitly calculated. In order to verify that this order is actually sufficient, we present the following method, which provides a direct measure for quantitatively verifying the truncation.
C. Gies et al. / Superlattices and Microstructures 43 (2008) 470–473
473
Fig. 2. Autocorrelation function (top) and input/output curve (bottom) for a fully inverted two-level system. Comparison between the master equation (symbols) and the two-level version of the semiconductor theory (solid lines). Refer to [8] for further details.
The difference between a QD and a two-level atom is that in an atom, only one electron per spin direction can be present. As a result, successive annihilation of an electron in two different states of the same atom is impossible. We can incorporate this idea into our theory and deduce an effective atomic laser theory, which is still based on the truncation of the hierarchy. For an atomic system the output intensity and the photon statistics can be calculated with carrier–photon correlations taken up to infinite order via a master equation approach [6]. We can then directly compare the truncated atomic theory to the exact version. Results are shown in Fig. 2, where symbols correspond to the master equation, and the solid lines are obtained from our truncated model in the limit of independent two-level atoms. For the input/output curves, the agreement is excellent. Looking at the photon statistics, a small deviation of about 5% in the value, but not in the position of the transition, is found for the case of maximum spontaneous emission coupling. Considering the full theory, where the atomic limit is not taken, this small deviation must be compared to the strong modifications due to semiconductor effects. In conclusion, we have studied the emission properties of QD–microcavity systems and discussed the differences from simpler atomic rate equations. An important verification of the validity of the truncation of carrier–photon correlations underlines the applicability of the cluster expansion technique, which we use to truncate the semiconductor theory. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
G.S. Solomon, M. Pelton, Y. Yamamoto, Phys. Rev. Lett. 86 (2001) 3903. H.-G. Park, et al., Science 305 (2004) 1444. S.M. Ulrich, et al., Phys. Rev. Lett. 98 (2007) 043906. M. Schwab, et al., Phys. Rev. B 74 (2006) 045323. H. Yokoyama, S.D. Brorson, J. Appl. Phys. 66 (1989) 4801. P.R. Rice, H.J. Carmichael, Phys. Rev. A 50 (1994) 4318. N. Baer, C. Gies, J. Wiersig, F. Jahnke, Eur. Phys. J. B 50 (2006) 411. C. Gies, J. Wiersig, M. Lorke, F. Jahnke, Phys. Rev. A 75 (2007) 013803. J. Fricke, Ann. Phys. 252 (1996) 479. W. Hoyer, M. Kira, S.W. Koch, Phys. Rev. B 67 (2003) 155113.
Laser theory for semiconductor quantum dots in microcavities Christopher Gies ∗ , Jan Wiersig, Frank Jahnke Institute for Theoretical Physics, University of Bremen, P.O. Box 330 440, 28334 Bremen, Germany Available online 10 September 2007
Abstract Quantum dots (QDs) used as active material in microresonators are currently of strong topical interest due to breakthroughs in growth and device structuring. From the theory side, however, atomic models are still used to analyse the emission from these semiconductor systems, despite known differences between QDs and atoms. We introduce a semiconductor laser theory based on a microscopic approach with the goal of better describing the characteristic behaviour of QD-based laser devices and to show differences from predictions based on atomic models. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Photon statistics; Microcavity lasers; Quantum-dot lasers; Microscopic theory
1. Overview Achievements in the fabrication of quantum dots and in the structuring of microcavities containing quantum dots (QDs) as the active material have led to novel applications that are based on controlled modifications of the emission properties, like lasers with reduced threshold, more efficient LEDs, or the generation of single or entangled photons. This development has kindled strong interest in the emission properties of QDs [1–3]. The combination of the threedimensional carrier and photon confinement of QDs in microcavities provides an ideal basis for the design of tailored laser devices. ∗ Corresponding author.
E-mail address: [email protected] (C. Gies, J. Wiersig, F. Jahnke). URL: http://www.itp.uni-bremen.de (C. Gies, J. Wiersig, F. Jahnke). c 2007 Elsevier Ltd. All rights reserved. 0749-6036/$ - see front matter doi:10.1016/j.spmi.2007.06.026
C. Gies et al. / Superlattices and Microstructures 43 (2008) 470–473
471
Due to the discrete level spectrum of QDs, their theoretical description is typically based on atomic models. Despite undeniable similarities, the inherent semiconductor properties of the QDs become obvious in several experiments, the frequently observed non-exponential and excitationintensity-dependent decay of the photoluminescence being one example [4]. The input/output curve of conventional atomic lasers is well described by rate equation models [5]. In order to calculate the photon statistics of the emitted light, master equations are used [6]. However, the application of these models to semiconductor QDs requires the assumption that carriers in the dot behave like carriers in an atom, which is not true in the general case, particularly when considering effects like a modified source term of spontaneous emission and Pauli blocking [7]. We use a microscopic approach to developing a laser theory which takes the semiconductor nature of QDs into account. Our theory includes many-body effects and provides a consistent scheme for the treatment of Coulomb- and light–matter-interaction-induced correlations between carriers, and between carriers and photons. Results for the input/output curves and the photon statistics are presented. 2. Model We use a semiconductor Hamiltonian for the interacting carrier–photon system, from which dynamical equations are derived via Heisenberg’s equation-of-motion technique. Carriers are treated in second quantization, and the light field is also quantized. The contributing parts are, from left to right, the free carrier spectrum, Coulomb interaction, free electromagnetic field, and the light–matter interaction; details are found in [8]: 0 H = Hcarr + HCoul + Hph + HLM .
(1)
A consequence of the equation-of-motion approach is the occurrence of an infinite hierarchy of coupled dynamical equations, arising due to the two-particle parts HCoul and HLM of the Hamiltonian. A truncation of this hierarchy for numerical calculation is unavoidable. At the same time, validity of the truncation method at a certain level is indispensable for the usability of the theory. The essential idea of what has become known as the cluster expansion method [9,10] is to reformulate all equations of motion for operator expectation values in such a way that equations of motion for the correlation functions are obtained. Correlation functions of nth order only contain processes which involve exactly n particles. The hierarchy of correlation functions is then truncated rather than the hierarchy of expectation values itself. This allows the consistent inclusion of n-particle effects. Using the above described formalism, we numerically determine Ď the dynamics of the photon population in the laser mode hbĎ bi and the carrier population hcν cν i Ď and 1 − hvν vν i for electrons and holes, respectively. Here, the operators bĎ and b are bosonic creation and annihilation operators for photons in the laser mode, and cν and vν are the fermionic equivalent for conduction and valence band carriers in the single-particle state ν. The photon autocorrelation function is of particular interest, as its value at zero time delay reveals the coherent or thermal nature of the emitted light. In our formalism [8] it is obtained from g (2) (τ = 0) =
hn 2 i − hni hbĎ bĎ bbi = hni2 hbĎ bi2
(2)
472
C. Gies et al. / Superlattices and Microstructures 43 (2008) 470–473
Fig. 1. Calculated output curve (left) and autocorrelation function g (2) (τ = 0) (right) for various values of β. The total spontaneous emission time is 100 ps, the cavity lifetime is 13 ps. Relaxation times of 5 ps (2.5 ps) are used for electrons (holes).
by calculating the time evolution of hbĎ bĎ bbi. In the dynamic equations further correlation functions occur, for which coupled equations are determined up to the order of four-particle processes. 3. Quantum-dot lasing In Fig. 1 results for the input/output characteristics and the photon statistics for an ensemble of 200 InGaAs QDs in GaAs/AlAs micropillars are presented. The number of resonant QDs is scaled with β −1 . These parameters are slightly modified from those used in [8], as we consider here more QDs in a cavity of lower quality. The results look similar but differ in detail in the sub-threshold behaviour. We choose to vary β as a parameter. Note that between different micropillars, other parameters like the spontaneous emission rate vary, too. The value β = 1 corresponds to complete coupling of the emitter to the laser mode. In this case, the input/output curve shows no kink. Since microcavity lasers can operate close to this regime, knowledge of the photon correlations is important for identifying the regime of truly coherent laser emission. Our results for the autocorrelation function demonstrate that for typical parameters of present systems there is always a clear transition from thermal to coherent emission even for β = 1, allowing for an identification of the coherent regime. In atomic systems, the jump of the intensity curve from below to above threshold is given by 1/β provided that nonradiative losses can be neglected. This feature is here spoiled by the fact that the spontaneous emission is not solely determined by the electron density, and because the system does not operate at full inversion. At high pump powers, saturation effects due to Pauli blocking are visible. 4. Comparison to atomic theory Due to the hierarchy problem that we discussed above, the applicability of our formalism is crucially dependent on the possibility of truncating the arising coupled equations of motion at a certain level. For the calculation of the photon statistics, which is determined by the average over four photon operators in Eq. (2), dynamic equations for at least up to four-particle effects must be explicitly calculated. In order to verify that this order is actually sufficient, we present the following method, which provides a direct measure for quantitatively verifying the truncation.
C. Gies et al. / Superlattices and Microstructures 43 (2008) 470–473
473
Fig. 2. Autocorrelation function (top) and input/output curve (bottom) for a fully inverted two-level system. Comparison between the master equation (symbols) and the two-level version of the semiconductor theory (solid lines). Refer to [8] for further details.
The difference between a QD and a two-level atom is that in an atom, only one electron per spin direction can be present. As a result, successive annihilation of an electron in two different states of the same atom is impossible. We can incorporate this idea into our theory and deduce an effective atomic laser theory, which is still based on the truncation of the hierarchy. For an atomic system the output intensity and the photon statistics can be calculated with carrier–photon correlations taken up to infinite order via a master equation approach [6]. We can then directly compare the truncated atomic theory to the exact version. Results are shown in Fig. 2, where symbols correspond to the master equation, and the solid lines are obtained from our truncated model in the limit of independent two-level atoms. For the input/output curves, the agreement is excellent. Looking at the photon statistics, a small deviation of about 5% in the value, but not in the position of the transition, is found for the case of maximum spontaneous emission coupling. Considering the full theory, where the atomic limit is not taken, this small deviation must be compared to the strong modifications due to semiconductor effects. In conclusion, we have studied the emission properties of QD–microcavity systems and discussed the differences from simpler atomic rate equations. An important verification of the validity of the truncation of carrier–photon correlations underlines the applicability of the cluster expansion technique, which we use to truncate the semiconductor theory. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
G.S. Solomon, M. Pelton, Y. Yamamoto, Phys. Rev. Lett. 86 (2001) 3903. H.-G. Park, et al., Science 305 (2004) 1444. S.M. Ulrich, et al., Phys. Rev. Lett. 98 (2007) 043906. M. Schwab, et al., Phys. Rev. B 74 (2006) 045323. H. Yokoyama, S.D. Brorson, J. Appl. Phys. 66 (1989) 4801. P.R. Rice, H.J. Carmichael, Phys. Rev. A 50 (1994) 4318. N. Baer, C. Gies, J. Wiersig, F. Jahnke, Eur. Phys. J. B 50 (2006) 411. C. Gies, J. Wiersig, M. Lorke, F. Jahnke, Phys. Rev. A 75 (2007) 013803. J. Fricke, Ann. Phys. 252 (1996) 479. W. Hoyer, M. Kira, S.W. Koch, Phys. Rev. B 67 (2003) 155113.