- Email: [email protected]
Quantum kinetics of carrier capture processes into a quantum dot
Physica B 314 (2002) 455–458
Quantum kinetics of carrier capture processes into a quantum dot T. Kuhn*, M. Glanemann, V.M. Axt Institut fur Westfalische Wilhelms-Universitat, . Festkorpertheorie, . . . Wilhelm-Klemm-Str. 10, 48149 Munster, Germany .
Abstract The capture of electrons moving in a quantum wire into an embedded quantum dot due to the interaction with optical phonons is studied on a quantum kinetic level within the density matrix approach. We find even at low temperatures where only phonon emission is possible a non-monotonic time-dependence of the bound state occupations due to virtual transitions. In the case of several bound dot states the capture in general results in a coherent superposition of these states. r 2002 Elsevier Science B.V. All rights reserved. PACS: 72.10.Bg; 73.63.Kv; 73.63.Nm Keywords: Carrier capture; Quantum kinetics; Quantum transport; Quantum dot
Many experimental and theoretical investigations in the past few years have shown that on a femtosecond time scale the concept of an instantaneous transition between single-particle states with well-defined energies is no more adequate. Features like a time-dependent broadening of phonon replicas due to energy–time uncertainty [1,2] or phonon quantum beats due to correlations between electrons and phonons [3,4] can only be explained on a quantum kinetic level. In modern optical techniques like time-resolved near-field microscopy these ultrashort time scales are more and more combined with ultrashort length scales [5]. In this case, where spatial inhomogeneities occur on a nanometer scale also the concept of a *Corresponding author. Tel.: +49-251-83-36312; fax: +49251-83-33669. E-mail address: [email protected] (T. Kuhn).
local transition between well-defined momentum states looses its validity due to position–momentum uncertainty. In this regime, the fields of quantum kinetics, which has mainly addressed quantum features in the temporal dynamics, and quantum transport, where mainly quantum features in the spatial transport have been considered, merge. Transitions between states of different dimensionality are of relevance in a variety of modern semiconductor nanostructures. In quantum well lasers, carriers are trapped from the three-dimensional contact regions into the active twodimensional layers. In many experiments with self-assembled quantum dots, carriers are generated in the two-dimensional wetting layer and they are then trapped into the zero-dimensional dot states. In this paper, we will study the case where carriers are generated in a one-dimensional
0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 1 4 3 7 - 5
456
T. Kuhn et al. / Physica B 314 (2002) 455–458
quantum wire and are subsequently captured into a quantum dot embedded in the quantum wire by the emission of optical phonons. Such a wire–dot system has recently been realized by a zigzag quantum wire on a patterned substrate where dot states form at the corners of the wire [6]. Near-field optical measurements on this sample indeed indicate a transfer of carriers between the wire and the dot regions [6]. Our calculations are based on the density matrix approach to electron–phonon quantum kinetics which has been successfully applied in the past years to many aspects of the short-time dynamics in spatially homogeneous systems of different dimensions [1,4,7]. The basic variable from which most experimentally observable quantities can be obtained is the single-particle density matrix fk0 ;k ¼ /cwk0 ck S where cwk ðck Þ denotes the creation (annihilation) operator of an electron with momentum k: Here, we restrict ourselves to the case of a oneband model; the extension to the two-band case can be found in Ref. [1] for a homogeneous system where f is diagonal and in Ref. [8] for an inhomogeneous system. Of course, any other representation of the density matrix is equivalent; most convenient for the interpretation is often a Wigner P representation obtained according to Fk ðrÞ ¼ q eiqr fkð1=2Þq;kþð1=2Þq from which we obtain directly, e.g., the local carrier density P nðrÞ ¼ V1 k Fk ðrÞ where V is a normalization volume. Being an interacting many body system, the equation of motion for the single-particle density matrix in a coupled electron–phonon system is not closed. Instead it involves higher order density matrices, here the so-called phonon-assisted density matrices which in our case are conveniently defined as sk0 ;q;k ¼ i_1 gq /cwk0 bq ck S where bq is the annihilation operator of an optical phonon with . wave vector q and gq denotes the Frohlich coupling matrix element. This variable describes correlations between electrons and phonons and eventually the scattering processes. It is the starting point for an infinite hierarchy which for most purposes is truncated by factorizing density matrices involving four operators into the respective lower orders. The resulting equation of motion for s contains a source term with electron
and phonon distribution functions which has the structure of the usual Boltzmann scattering term. In addition, if the electrons move in a single particle potential V ðrÞ it involves the Fourier transform of this potential. For the case of a potential due to a homogeneous electric field, for example, this gives rise to an intracollisional field effect. It is this term which makes the description independent of the basis chosen for the electrons. However, if a Markov approximation is performed leading, e.g., to the Boltzmann equation, this base independence is lost. We apply the theory to the problem of the capture of an electronic wave packet traveling in a quantum wire of 100 nm2 cross section (only lowest subband considered) into localized states of a quantum dot embedded in this wire. The dot is modeled by a potential V ðzÞ ¼ V0 sechðazÞ: The parameters V0 and a determine the depth and the width of the dot, respectively, and, therefore, the number and energetic position of bound states. The wave packet is assumed to be generated locally, e.g. by means of a near-field microscope, in the vicinity of the dot with an excess energy of 18 meV and an energetic width of 7:5 meV: Calculations have been performed at a temperature of 4 K: Fig. 1 displays the motion of the wave packet in the presence of a shallow dot (V0 ¼ 18 meV; a ¼ 0:2 nm1 ) with one bound state. Part (a) shows results without phonon interaction. The wave packet spreads due to the dispersion and as long as it is in the region of the dot a dip appears because of the higher kinetic energy and the resulting higher group velocity of the front part of the wave packet above the dot. After the passage, it reshapes again and essentially no carrier density remains in the dot region. Due to the smooth potential shape there is very little reflection. The situation changes if the phonon interaction is taken into account [Fig. 1(b)]. Again a part of the wave packet traverses the dot; however, it is clearly seen that now a significant contribution remains in the dot region which indicates a capture into localized states. For a clear interpretation, we have projected the electronic density matrix on the bound state of the dot. This gives us the occupation of the bound state
T. Kuhn et al. / Physica B 314 (2002) 455–458
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Fig. 1. Spatial profile of the electronic wave packet moving along the wire at different times (upper part) and potential profile including the bound state energy level of the quantum dot (lower part). Part (a) refers to a calculation without carrierphonon interaction, in (b) carrier–phonon interaction is included.
10 -4
ρ 11
5
0 0
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time (fs) Fig. 2. Occupation of the bound state in the quantum dot as a function of time for the case shown in Fig. 1(b).
which is plotted as a function of time in Fig. 2. We see that indeed at about 200 fs; when the wave packet approaches the dot, the occupation increases. However, it does not exhibit a monotonic rise as would be expected in a semiclassical
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Fig. 3. Same as Fig. 1 but for a deeper quantum dot potential with three bound states.
treatment since at 4 K only phonon emission is possible. This is a consequence of virtual transitions which are possible on short time scales due to energy–time uncertainty and which are known to give rise to oscillatory occupations [1]. In Fig. 3, the dynamics of the wave packet is displayed for the case of a deeper quantum dot (V0 ¼ 40 meV; a ¼ 0:1 nm1 ) with three bound states again without (a) and with (b) carrier– phonon interaction. In contrast to the previous case, the capture into one of the bound states, here the second one, is now approximately in resonance because of the phonon energy of 36:4 meV and the initial energy of 18 meV: This results in a much higher trapped density. The most interesting feature, however, is the fact that now also the trapped wave packet exhibits a pronounced oscillatory dynamics. Again, we get a deeper insight by projecting the density matrix on the reduced subspace of bound states. The resulting diagonal elements, i.e., the occupations of the three states, are plotted in Fig. 4(a) while Fig. 4(b) shows the imaginary part of the off-diagonal elements describing coherences between the states. The real parts (not shown) exhibit a similar behavior. As can be expected, the occupation of
T. Kuhn et al. / Physica B 314 (2002) 455–458
458
density matrix elements (
10 -4 )
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ρ 11 ρ 22 ρ 33
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0 4 (b)
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Im ( ρ 12 ) Im ( ρ ) 13
Im ( ρ 23 )
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Fig. 4. Diagonal (a) and off-diagonal (b) matrix elements of the reduced density matrix in the subspace of the bound states in the quantum dot as functions of time for the wave packet shown in Fig. 3(b). The indices 1, 2, and 3 refer to the bound states at 30:6; 15:2; and 5:8 meV; respectively.
the second state which is in resonance with the phonon emission process is the biggest. However, also the other states are occupied both due to the finite width of the wave packet and energy–time uncertainty. The latter again gives rise to the nonmonotonic behavior which is more pronounced the closer the bound state is to the continuum. Part (b) shows that the off-diagonal elements are of the same order as the off-resonant diagonal elements and even not much smaller than the resonant one. Thus, the final state in the capture process is obviously a linear combination of the bound states. This linear combination depends on the dynamics and cannot easily be determined a priori. This shows the importance of a treatment which does not already select a specific basis. The
existence of coherences among bound states also implies that the capture of a wave packet should be accompanied by the emission of electromagnetic radiation in the THz range. In conclusion, we have presented an analysis of the short-time dynamics of carrier capture processes into localized states based on a quantum kinetic density matrix approach. The presence of virtual transitions at short times due to energy– time uncertainty results in a non-monotonic behavior of the occupations of bound states even at low temperatures where only phonon emission is possible. In the presence of several bound states, the capture results in a coherent superposition of bound states and, thus, a wave packet inside the quantum dot is formed which exhibits an oscillatory motion. Therefore, we conclude that the choice of the final state in a capture process is nontrivial. This makes an approach which is independent of the chosen basis indispensable. This work has been financially supported by the DFG within the framework of the Schwerpunktporgramm Quantenkoh.arenz in Halbleitern and a Habilitandenstipendium for V.M.A. as well as by the European Commission within the TMR network Ultrafast Quantum Optoelectronics.
References [1] [2] [3] [4] [5] [6] [7]
[8]
J. Schilp, T. Kuhn, G. Mahler, Phys. Rev. B 50 (1994) 5435. C. Furst, . et al., Phys. Rev. Lett. 78 (1997) 3733. L. B!anyai, et al., Phys. Rev. Lett. 75 (1995) 2188. V.M. Axt, M. Herbst, T. Kuhn, Superlat. Microstruct. 26 (1999) 117. V. Emiliani, et al., J. Phys. 11 (1999) 5889. C. Lienau, et al., Phys. Stat. Sol. (a) 178 (2000) 471. T. Kuhn, in: Theory of Transport Properties of Semicon. (Ed.), Chapman & Hall, ductor Nanostructures, E. Scholl London, 1998, p. 173. T. Kuhn, Adv. Solid State Physics 41 (2001) 125.
Quantum kinetics of carrier capture processes into a quantum dot T. Kuhn*, M. Glanemann, V.M. Axt Institut fur Westfalische Wilhelms-Universitat, . Festkorpertheorie, . . . Wilhelm-Klemm-Str. 10, 48149 Munster, Germany .
Abstract The capture of electrons moving in a quantum wire into an embedded quantum dot due to the interaction with optical phonons is studied on a quantum kinetic level within the density matrix approach. We find even at low temperatures where only phonon emission is possible a non-monotonic time-dependence of the bound state occupations due to virtual transitions. In the case of several bound dot states the capture in general results in a coherent superposition of these states. r 2002 Elsevier Science B.V. All rights reserved. PACS: 72.10.Bg; 73.63.Kv; 73.63.Nm Keywords: Carrier capture; Quantum kinetics; Quantum transport; Quantum dot
Many experimental and theoretical investigations in the past few years have shown that on a femtosecond time scale the concept of an instantaneous transition between single-particle states with well-defined energies is no more adequate. Features like a time-dependent broadening of phonon replicas due to energy–time uncertainty [1,2] or phonon quantum beats due to correlations between electrons and phonons [3,4] can only be explained on a quantum kinetic level. In modern optical techniques like time-resolved near-field microscopy these ultrashort time scales are more and more combined with ultrashort length scales [5]. In this case, where spatial inhomogeneities occur on a nanometer scale also the concept of a *Corresponding author. Tel.: +49-251-83-36312; fax: +49251-83-33669. E-mail address: [email protected] (T. Kuhn).
local transition between well-defined momentum states looses its validity due to position–momentum uncertainty. In this regime, the fields of quantum kinetics, which has mainly addressed quantum features in the temporal dynamics, and quantum transport, where mainly quantum features in the spatial transport have been considered, merge. Transitions between states of different dimensionality are of relevance in a variety of modern semiconductor nanostructures. In quantum well lasers, carriers are trapped from the three-dimensional contact regions into the active twodimensional layers. In many experiments with self-assembled quantum dots, carriers are generated in the two-dimensional wetting layer and they are then trapped into the zero-dimensional dot states. In this paper, we will study the case where carriers are generated in a one-dimensional
0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 1 4 3 7 - 5
456
T. Kuhn et al. / Physica B 314 (2002) 455–458
quantum wire and are subsequently captured into a quantum dot embedded in the quantum wire by the emission of optical phonons. Such a wire–dot system has recently been realized by a zigzag quantum wire on a patterned substrate where dot states form at the corners of the wire [6]. Near-field optical measurements on this sample indeed indicate a transfer of carriers between the wire and the dot regions [6]. Our calculations are based on the density matrix approach to electron–phonon quantum kinetics which has been successfully applied in the past years to many aspects of the short-time dynamics in spatially homogeneous systems of different dimensions [1,4,7]. The basic variable from which most experimentally observable quantities can be obtained is the single-particle density matrix fk0 ;k ¼ /cwk0 ck S where cwk ðck Þ denotes the creation (annihilation) operator of an electron with momentum k: Here, we restrict ourselves to the case of a oneband model; the extension to the two-band case can be found in Ref. [1] for a homogeneous system where f is diagonal and in Ref. [8] for an inhomogeneous system. Of course, any other representation of the density matrix is equivalent; most convenient for the interpretation is often a Wigner P representation obtained according to Fk ðrÞ ¼ q eiqr fkð1=2Þq;kþð1=2Þq from which we obtain directly, e.g., the local carrier density P nðrÞ ¼ V1 k Fk ðrÞ where V is a normalization volume. Being an interacting many body system, the equation of motion for the single-particle density matrix in a coupled electron–phonon system is not closed. Instead it involves higher order density matrices, here the so-called phonon-assisted density matrices which in our case are conveniently defined as sk0 ;q;k ¼ i_1 gq /cwk0 bq ck S where bq is the annihilation operator of an optical phonon with . wave vector q and gq denotes the Frohlich coupling matrix element. This variable describes correlations between electrons and phonons and eventually the scattering processes. It is the starting point for an infinite hierarchy which for most purposes is truncated by factorizing density matrices involving four operators into the respective lower orders. The resulting equation of motion for s contains a source term with electron
and phonon distribution functions which has the structure of the usual Boltzmann scattering term. In addition, if the electrons move in a single particle potential V ðrÞ it involves the Fourier transform of this potential. For the case of a potential due to a homogeneous electric field, for example, this gives rise to an intracollisional field effect. It is this term which makes the description independent of the basis chosen for the electrons. However, if a Markov approximation is performed leading, e.g., to the Boltzmann equation, this base independence is lost. We apply the theory to the problem of the capture of an electronic wave packet traveling in a quantum wire of 100 nm2 cross section (only lowest subband considered) into localized states of a quantum dot embedded in this wire. The dot is modeled by a potential V ðzÞ ¼ V0 sechðazÞ: The parameters V0 and a determine the depth and the width of the dot, respectively, and, therefore, the number and energetic position of bound states. The wave packet is assumed to be generated locally, e.g. by means of a near-field microscope, in the vicinity of the dot with an excess energy of 18 meV and an energetic width of 7:5 meV: Calculations have been performed at a temperature of 4 K: Fig. 1 displays the motion of the wave packet in the presence of a shallow dot (V0 ¼ 18 meV; a ¼ 0:2 nm1 ) with one bound state. Part (a) shows results without phonon interaction. The wave packet spreads due to the dispersion and as long as it is in the region of the dot a dip appears because of the higher kinetic energy and the resulting higher group velocity of the front part of the wave packet above the dot. After the passage, it reshapes again and essentially no carrier density remains in the dot region. Due to the smooth potential shape there is very little reflection. The situation changes if the phonon interaction is taken into account [Fig. 1(b)]. Again a part of the wave packet traverses the dot; however, it is clearly seen that now a significant contribution remains in the dot region which indicates a capture into localized states. For a clear interpretation, we have projected the electronic density matrix on the bound state of the dot. This gives us the occupation of the bound state
T. Kuhn et al. / Physica B 314 (2002) 455–458
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Fig. 1. Spatial profile of the electronic wave packet moving along the wire at different times (upper part) and potential profile including the bound state energy level of the quantum dot (lower part). Part (a) refers to a calculation without carrierphonon interaction, in (b) carrier–phonon interaction is included.
10 -4
ρ 11
5
0 0
100
200
300
400
500
time (fs) Fig. 2. Occupation of the bound state in the quantum dot as a function of time for the case shown in Fig. 1(b).
which is plotted as a function of time in Fig. 2. We see that indeed at about 200 fs; when the wave packet approaches the dot, the occupation increases. However, it does not exhibit a monotonic rise as would be expected in a semiclassical
pot. (meV)
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Fig. 3. Same as Fig. 1 but for a deeper quantum dot potential with three bound states.
treatment since at 4 K only phonon emission is possible. This is a consequence of virtual transitions which are possible on short time scales due to energy–time uncertainty and which are known to give rise to oscillatory occupations [1]. In Fig. 3, the dynamics of the wave packet is displayed for the case of a deeper quantum dot (V0 ¼ 40 meV; a ¼ 0:1 nm1 ) with three bound states again without (a) and with (b) carrier– phonon interaction. In contrast to the previous case, the capture into one of the bound states, here the second one, is now approximately in resonance because of the phonon energy of 36:4 meV and the initial energy of 18 meV: This results in a much higher trapped density. The most interesting feature, however, is the fact that now also the trapped wave packet exhibits a pronounced oscillatory dynamics. Again, we get a deeper insight by projecting the density matrix on the reduced subspace of bound states. The resulting diagonal elements, i.e., the occupations of the three states, are plotted in Fig. 4(a) while Fig. 4(b) shows the imaginary part of the off-diagonal elements describing coherences between the states. The real parts (not shown) exhibit a similar behavior. As can be expected, the occupation of
T. Kuhn et al. / Physica B 314 (2002) 455–458
458
density matrix elements (
10 -4 )
20
(a)
ρ 11 ρ 22 ρ 33
10
0 4 (b)
0
Im ( ρ 12 ) Im ( ρ ) 13
Im ( ρ 23 )
-4 0
100
200 300 time (fs)
400
500
Fig. 4. Diagonal (a) and off-diagonal (b) matrix elements of the reduced density matrix in the subspace of the bound states in the quantum dot as functions of time for the wave packet shown in Fig. 3(b). The indices 1, 2, and 3 refer to the bound states at 30:6; 15:2; and 5:8 meV; respectively.
the second state which is in resonance with the phonon emission process is the biggest. However, also the other states are occupied both due to the finite width of the wave packet and energy–time uncertainty. The latter again gives rise to the nonmonotonic behavior which is more pronounced the closer the bound state is to the continuum. Part (b) shows that the off-diagonal elements are of the same order as the off-resonant diagonal elements and even not much smaller than the resonant one. Thus, the final state in the capture process is obviously a linear combination of the bound states. This linear combination depends on the dynamics and cannot easily be determined a priori. This shows the importance of a treatment which does not already select a specific basis. The
existence of coherences among bound states also implies that the capture of a wave packet should be accompanied by the emission of electromagnetic radiation in the THz range. In conclusion, we have presented an analysis of the short-time dynamics of carrier capture processes into localized states based on a quantum kinetic density matrix approach. The presence of virtual transitions at short times due to energy– time uncertainty results in a non-monotonic behavior of the occupations of bound states even at low temperatures where only phonon emission is possible. In the presence of several bound states, the capture results in a coherent superposition of bound states and, thus, a wave packet inside the quantum dot is formed which exhibits an oscillatory motion. Therefore, we conclude that the choice of the final state in a capture process is nontrivial. This makes an approach which is independent of the chosen basis indispensable. This work has been financially supported by the DFG within the framework of the Schwerpunktporgramm Quantenkoh.arenz in Halbleitern and a Habilitandenstipendium for V.M.A. as well as by the European Commission within the TMR network Ultrafast Quantum Optoelectronics.
References [1] [2] [3] [4] [5] [6] [7]
[8]
J. Schilp, T. Kuhn, G. Mahler, Phys. Rev. B 50 (1994) 5435. C. Furst, . et al., Phys. Rev. Lett. 78 (1997) 3733. L. B!anyai, et al., Phys. Rev. Lett. 75 (1995) 2188. V.M. Axt, M. Herbst, T. Kuhn, Superlat. Microstruct. 26 (1999) 117. V. Emiliani, et al., J. Phys. 11 (1999) 5889. C. Lienau, et al., Phys. Stat. Sol. (a) 178 (2000) 471. T. Kuhn, in: Theory of Transport Properties of Semicon. (Ed.), Chapman & Hall, ductor Nanostructures, E. Scholl London, 1998, p. 173. T. Kuhn, Adv. Solid State Physics 41 (2001) 125.