- Email: [email protected]
Quantum coherence and formation of quantized states in semiconductors
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Physica E 21 (2004) 1126 – 1130 www.elsevier.com/locate/physe
Quantum coherence and formation of quantized states in semiconductors Masato Morifuji∗ , Katsuhiko Kato Department of Electronic Engineering, Osaka University, Suita, 2-1 Yamada-oka, Osaka 565-0871, Japan
Abstract We investigate the formation process of eigenstates in electromagnetic +elds. Electronic eigenfunctions in crystalline solids in the electromagnetic +elds are localized wavepackets associated with the discrete energy levels. The properties of such states (e.g. Landau levels in magnetic +elds and Stark-ladders in electric +elds) have been well revealed. However, it is obscure how transformation from the extended Bloch functions to the localized states occurs. By using the path-integral theory, we describe the transformation process of wave functions into quantized eigenstates. It is shown that self-interference, which may be interpreted as an afterimage of electronic motion, plays an important role in the transformation. ? 2003 Elsevier B.V. All rights reserved. PACS: 71.15−m; 73.23.Ad Keywords: Coherent dynamics; Quantized state; Path-integral theory
1. Introduction Electronic eigenstates in crystalline semiconductors are extended Bloch function dispersing energy into bands. On the other hand, when an electromagnetic +eld is applied, eigenstates are localized wavepackets with discrete energy levels. In magnetic +elds, such states are called the Landau levels. Stark-ladders are formed in static electric +elds, as well. Properties of these localized states have been well studied. However, as far as we know, it is still unclear how transition from the extended Bloch states to the localized states occurs. The purpose of this paper is to investigate the transformation process from Bloch states to localized states ∗
Corresponding author. Fax: +81-6-6879-7753. E-mail address: [email protected] (M. Morifuji).
so as to clarify the mechanism of quantization of electronic states.
2. Coherent dynamics of electron and formation of quantized state In order to show how eigenstates are formed from the free electronic states, we +rst study one-dimensional motion in harmonic potential V (x) = m!2 x2 =2 [1]. By using the path-integration theory, time evolution of an electron is described as ∞ (x ; t ) = K(x t ; x0 0) (x0 ; 0) d x0 ; (1) −∞
where K(x t ; x0 0) is the Feynman kernel for the harmonic potential [2–4].
1386-9477/$ - see front matter ? 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2003.11.192
M. Morifuji, K. Kato / Physica E 21 (2004) 1126 – 1130
E = 5.5hω
1127
t = 2π/ω
E = 3/2hω
Re[ψ(x ′,t ′)] [a.u.]
Re[χE(x′, t)] [a.u.]
2.0π/ω
1.5π/ω
1.0π/ω
t = 0.5π/ω / t=0 -5
0
5
(mω / h)1/2x Fig. 1. Time-evolution of a wave function in a harmonic potential.
As the initial state at time t = 0, we consider a function 1=4 2 e−( =2)x0 +ik0 x0 ; (2) (x0 ; 0) = where k0 is a wave number and −1=2 is the extension length of this state. This is a wavepacket put at the origin with velocity ˝k0 =m. The energy of this state is E = ˝2 k02 =2m, which is retained during the process of time evolution as is in the classical harmonic oscillator. We note that the has been introduced so as to show the oscillation in the real space clearly. In order to describe the transformation from the band states to the localized eigenstates, must be small enough so that the state is well regarded as a free electronic state with a continuous energy spectrum. Fig. 1 shows one period of time evolution of the wave function. We see that (x ; t ) is oscillating with change of the wavelength as well as the position. We stress here that the state is not quantized at this stage, that is, continuous value of energy E = ˝2 k02 =2m is possible for such an oscillator. Here we make a superposition of the oscillating states [4,5] as t 1 E (x ; t) = √ eiEt =˝ (x ; t ) dt : (3) t 0
-4
-2
0 (mω/h)1/2x′
2
4
Fig. 2. Formation process of eigen wave function. For the oscillator with energy 3=2˝!; E (x ; t) approaches the eigenstate with n = 1.
√ The factor 1= t has been added so that E (x ; t) is normalized. One can regard Eq. (3) as a Fourier transformation from time representation to the energy representation. In our interpretation, however, this may be called an afterimage eFect. In other words, E (x ; t) is a superposition of electron over the history. The function E (x ; t) has a physical meaning as a time-integrated probability amplitude, i.e., the probability of +nding an electron at energy E during the period 0∼t. In this sense, this is a wave function changing gradually from a band state to an eigenstate in an applied potential. The exponential factor in Eq. (3) has a role to cancel the time-dependent phase factor of (x ; t ), i.e., e−iEt =˝ . This cancellation is important to obtain a rational result. As we are considering a superposition of states belonging to diFerent times, if additional constant potential is applied, relative phase difference between the states changes. This results in completely diFerent wave functions; such a situation is irrational and undesirable. Therefore, from the viewpoint of the gauge invariance of wave functions, the time-dependent phase factor should be excluded from the superposition as shown in Eq. (3). In Fig. 2, we show E (x ; t) calculated at t = 0:5 =!∼2 =! for an oscillator with energy E=3=2˝!. This energy corresponds to the eigen energy of the
1128
M. Morifuji, K. Kato / Physica E 21 (2004) 1126 – 1130
E = 1.7hω
20π/ω
6π/ω <χE|χE> [a.u.]
Re[χE(x′, t)] [a.u.]
7π/ω
5π/ω 4π/ω t = 3π/ω
-4
12π/ω
6π/ω
-2
0
2
4
(mω/h)1/2x′ Fig. 3. Decay process of an oF-energy state. Since the energy of the oscillator is 1:7˝!; E (x ; t) decays as time increases.
n = 1 state of a harmonic oscillator. As shown in Fig. 1, the electron moves rightward at +rst, and at this stage the peak at (m!=˝)1=2 x 1 grows. As the time increases, the electron turns around to the left, and the dip in the left grows. As shown in Fig. 2, E (x ; t) gradually approaches the eigen wave function of the n = 1 state. Fig. 3 shows calculated E (x ; t) for E = 1:7˝!. This energy is not the eigen energy. In this case, the destructive self-interference results in the amplitude of E (x ; t) approaching zero with time. Fig. 4 shows the norm of E (x ; t) plotted as a function of energy. The curves are calculated for t = 2 =!∼20 =!. We see that the electronic states for any value of energy are possible for a small value of t. With increasing t, quantization gradually occurs; peaks at E = ˝!(n + 12 ) become sharper. These results indicate that the formation process of eigenstates is described by the function E (x ; t). At t = 0, the eigenstates are the extended band states given by Eq. (1). By applying the potential, the electronic motion shown in Fig. 1 occurs. As a result of self-interference, E (x ; t) changes gradually from a band state to a localized eigenstate for a particular value of E. Therefore, the function E (x ; t) describes the transformation from Bloch states (eigenstates for V = 0) to localized eigenstates in the electromagnetic +elds.
t = 2π/ω
E/ hω Fig. 4. Norm of the state E |E plotted as a function of energy. As the time increases, quantized states are formed and the peaks at E = ˝!(n + 12 ) grow.
From these results, we see that the self-interference plays an essential role in the formation of quantized states. For particular energies, an afterimage of electronic motion results in constructive interference; the amplitude becomes larger and larger with time. On the other hand for other energies, the amplitudes become gradually small due to the destructive self-interference. Therefore, discrete energy levels occurs. 3. Band electrons in high electric elds This theory enables us to describe both the Landau states in magnetic +elds and Stark-ladders in electric +elds in a uni+ed way [1]. As an example of the theory we investigate coherence of electrons in uniform electric +elds. In a static electric +eld F, an electron is accelerated as k(t) = k0 + eFt=˝. At the edge of the Brillouin zone, the electron is Bragg reIected due to the crystalline potential. As a result of this motion, a ladder-like energy spectrum associated with the localized wave functions realizes. This is known as a Stark-ladder [6–8]. Time-evolution of a Bloch function is given by (4) r|k(t) = d 3 r0 r|e−iHt=˝ |r0 r0 |k0 ;
M. Morifuji, K. Kato / Physica E 21 (2004) 1126 – 1130
and d 3 k |k k | = 1
= e−i=˝
d −t=2 k()
k0 +eFt=˝ (r);
(7)
k |e−iHt=˝ |k0 t=2
d −t=2 k()
(k − k0 − eFt=˝):
(8)
By superposing the time-evolving states given by Eq. (7) along the history as in Eq. (3), we have a wave function which expresses the formation of a Stark-ladder state. When the electric +eld is applied along the z-direction, we have the wave function Ek (r) =
2
0
de
2T t=T
-2
where k(t) = k0 + eFt=˝. To obtain Eq. (7), we have used that expression of the Green’s function in an electric +eld [9,10]
=e−i=˝
3T
(6)
into Eq. (4), we have r|k(t) = d 3 k r|k k |e−iHt=˝ |k0 t=2
4T
ρ(E, t) [a.u.]
where r|k0 ≡ k0 (r) is a Bloch function at the time t = 0. H is a Hamiltonian including the electric and crystalline potentials. In principle, the applied and crystal potentials should be treated on an equal footing. By applying the eFective mass approximation, the eFects from the crystalline potential are included in a cosine-like dispersion curve. By using the completeness relations d 3 r0 |r0 r0 | = 1 (5)
1129
−i=˝
t =2
−t =2
[kz () −E]d
k(t ) (r); (9)
which changes gradually from a Bloch function to a Stark-ladder. In Eq. (9), is the system size, k(t) = k0 + eFt, and = eFt . = eFt is change of wave vector during the period t. When is much smaller than Brillouin zone width, we can set as 2 =, and Ek (r) is regarded as a Bloch function. A band-like energy spectrum realizes in this case. With increasing , if n ≡ E=eFa is not an integer, the amplitudes interfere destructively with one another and cancel out. Whereas when E=eFa is an integer, Ek (r) becomes a rung of a Stark-ladder located at the nth position for
0 E/ eFa
2
Fig. 5. Time-dependent density of states in an electric +eld. The curves are calculated for the time t = T ∼ 4T , where T = 2 ˝=eFa is a period of Bloch oscillation.
2 =L. Therefore, a ladder-like spectrum associated with localized state is realized. To show how quantization occurs and the energy spectrum changes due to interference, we calculate a quantity (E; t) = Ek |Ek : (10) k
This quantity corresponds to a time-dependent density of states with weight of norm of Ek . Fig. 5 shows (E; t) plotted as a function of energy for various values of t. We have noted that the lateral degrees of freedom are neglected so as to show the discrete levels clearly in Fig. 5. The curves are calculated for t =T ∼4T , where T =2 ˝=eFa is a period of Bloch oscillation. When t is small, the density of states is continuous. With increasing time, interference accompanied with electronic motion occurs, and the quantized density of states realizes. Such temporal behavior of energy spectrum is similar to that shown in Fig. 4 but without the zero point energy. Qualitatively, such eFects of quantum coherence on electronic states are well expected. However, as far as we know, no quantitative description of self-interference has been done. The electronic coherence and formation of quantized states in the electric +elds are important for high +eld transport in recent
1130
M. Morifuji, K. Kato / Physica E 21 (2004) 1126 – 1130
nano-devices. Therefore, the quantitative treatment on quantum coherence as shown in this paper is necessary for precise description of the electrons in electromagnetic +elds. As an application of the present theory, we have shown that the quantum transport can be described by using the half-localized wavepacket states [8]. Sometimes we describe the transport by using Bloch functions, and sometimes we may regard transport as hopping motion of Stark-ladder states; the coherence is not properly considered in these treatments. On the other hand, the use of wavepacket states results in quantum eFects on description of the transport phenomena. Finally, we note that the use of wavepacket states is related to the Green’s function theory for transport. As we have shown in this paper, the half-localized states Ek (r) are written in terms of a Green’s function in an electric +eld. In order to improve the transport theories in high +elds, Green’s functions and Wigner distributions are often employed [11,12]. We note that there are some attempts to describe transport phenomena by using such a half-localized Stark-ladder state as well as Green’s functions [8,13]. 4. Conclusion In conclusion, we have shown that the formation process of quantized states in electromagnetic +elds is described in terms of self-interference. By superposing the extended Bloch functions through the history in which an electron has undergone, we derived a wave function for the formation process. For particular energies, self-interference results in the
enhancement of probability amplitude. Whereas for other energies, amplitudes are attenuated due to destructive interference. As a result, discrete energy levels occur. This theory enables us to describe both Stark-ladders in the electric +eld and Landau states in magnetic +elds in an uni+ed way. As an example of this theory we have shown how a Stark-ladder is formed in an electric +eld. Acknowledgements We acknowledge the +nancial support from the Handai Frontier Research Center. References [1] M. Morifuji, K. Kato, Phys. Rev. B 68 (2003) 035108. [2] R.P. Feynman, Rev. Mod. Phys. 20 (1948) 367. [3] R.P. Feynman, A.R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, New York, 1965. [4] L.S. Schulman, Techniques and Applications of Path Integration, Wiley, New York, 1996. [5] M.C. Gutzwiller, J. Math. Phys. 8 (1967) 1979. [6] M. Morifuji, S.K. Wah, C. Hamaguchi, Solid State Electron. 42 (1998) 1505. [7] M. Morifuji, C. Hamaguchi, Phys. Rev. B 58 (1998) 12842. [8] M. Morifuji, M. Ono, J. Phys. Soc. Jpn. 72 (2003) 229. [9] H. Haug, A.-P. Jauho, Quantum Kinetics in Transport and Optics of Semiconductors, Springer, Berlin, 1998. [10] J.H. Davies, J.W. Wilkins, Phys. Rev. B 38 (1998) 1667. [11] A. Wacker, A.-P. Jauho, Phys. Rev. Lett. 80 (1998) 369. [12] P. Bordone, A. Bertoni, R. Brunetti, C. Jacoboni, Math. Comput. Simat. 62 (2003) 307. [13] S. Rott, N. Linder, G.H. DOohler, Phys. Rev. B 65 (2002) 195 301.
Physica E 21 (2004) 1126 – 1130 www.elsevier.com/locate/physe
Quantum coherence and formation of quantized states in semiconductors Masato Morifuji∗ , Katsuhiko Kato Department of Electronic Engineering, Osaka University, Suita, 2-1 Yamada-oka, Osaka 565-0871, Japan
Abstract We investigate the formation process of eigenstates in electromagnetic +elds. Electronic eigenfunctions in crystalline solids in the electromagnetic +elds are localized wavepackets associated with the discrete energy levels. The properties of such states (e.g. Landau levels in magnetic +elds and Stark-ladders in electric +elds) have been well revealed. However, it is obscure how transformation from the extended Bloch functions to the localized states occurs. By using the path-integral theory, we describe the transformation process of wave functions into quantized eigenstates. It is shown that self-interference, which may be interpreted as an afterimage of electronic motion, plays an important role in the transformation. ? 2003 Elsevier B.V. All rights reserved. PACS: 71.15−m; 73.23.Ad Keywords: Coherent dynamics; Quantized state; Path-integral theory
1. Introduction Electronic eigenstates in crystalline semiconductors are extended Bloch function dispersing energy into bands. On the other hand, when an electromagnetic +eld is applied, eigenstates are localized wavepackets with discrete energy levels. In magnetic +elds, such states are called the Landau levels. Stark-ladders are formed in static electric +elds, as well. Properties of these localized states have been well studied. However, as far as we know, it is still unclear how transition from the extended Bloch states to the localized states occurs. The purpose of this paper is to investigate the transformation process from Bloch states to localized states ∗
Corresponding author. Fax: +81-6-6879-7753. E-mail address: [email protected] (M. Morifuji).
so as to clarify the mechanism of quantization of electronic states.
2. Coherent dynamics of electron and formation of quantized state In order to show how eigenstates are formed from the free electronic states, we +rst study one-dimensional motion in harmonic potential V (x) = m!2 x2 =2 [1]. By using the path-integration theory, time evolution of an electron is described as ∞ (x ; t ) = K(x t ; x0 0) (x0 ; 0) d x0 ; (1) −∞
where K(x t ; x0 0) is the Feynman kernel for the harmonic potential [2–4].
1386-9477/$ - see front matter ? 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2003.11.192
M. Morifuji, K. Kato / Physica E 21 (2004) 1126 – 1130
E = 5.5hω
1127
t = 2π/ω
E = 3/2hω
Re[ψ(x ′,t ′)] [a.u.]
Re[χE(x′, t)] [a.u.]
2.0π/ω
1.5π/ω
1.0π/ω
t = 0.5π/ω / t=0 -5
0
5
(mω / h)1/2x Fig. 1. Time-evolution of a wave function in a harmonic potential.
As the initial state at time t = 0, we consider a function 1=4 2 e−( =2)x0 +ik0 x0 ; (2) (x0 ; 0) = where k0 is a wave number and −1=2 is the extension length of this state. This is a wavepacket put at the origin with velocity ˝k0 =m. The energy of this state is E = ˝2 k02 =2m, which is retained during the process of time evolution as is in the classical harmonic oscillator. We note that the has been introduced so as to show the oscillation in the real space clearly. In order to describe the transformation from the band states to the localized eigenstates, must be small enough so that the state is well regarded as a free electronic state with a continuous energy spectrum. Fig. 1 shows one period of time evolution of the wave function. We see that (x ; t ) is oscillating with change of the wavelength as well as the position. We stress here that the state is not quantized at this stage, that is, continuous value of energy E = ˝2 k02 =2m is possible for such an oscillator. Here we make a superposition of the oscillating states [4,5] as t 1 E (x ; t) = √ eiEt =˝ (x ; t ) dt : (3) t 0
-4
-2
0 (mω/h)1/2x′
2
4
Fig. 2. Formation process of eigen wave function. For the oscillator with energy 3=2˝!; E (x ; t) approaches the eigenstate with n = 1.
√ The factor 1= t has been added so that E (x ; t) is normalized. One can regard Eq. (3) as a Fourier transformation from time representation to the energy representation. In our interpretation, however, this may be called an afterimage eFect. In other words, E (x ; t) is a superposition of electron over the history. The function E (x ; t) has a physical meaning as a time-integrated probability amplitude, i.e., the probability of +nding an electron at energy E during the period 0∼t. In this sense, this is a wave function changing gradually from a band state to an eigenstate in an applied potential. The exponential factor in Eq. (3) has a role to cancel the time-dependent phase factor of (x ; t ), i.e., e−iEt =˝ . This cancellation is important to obtain a rational result. As we are considering a superposition of states belonging to diFerent times, if additional constant potential is applied, relative phase difference between the states changes. This results in completely diFerent wave functions; such a situation is irrational and undesirable. Therefore, from the viewpoint of the gauge invariance of wave functions, the time-dependent phase factor should be excluded from the superposition as shown in Eq. (3). In Fig. 2, we show E (x ; t) calculated at t = 0:5 =!∼2 =! for an oscillator with energy E=3=2˝!. This energy corresponds to the eigen energy of the
1128
M. Morifuji, K. Kato / Physica E 21 (2004) 1126 – 1130
E = 1.7hω
20π/ω
6π/ω <χE|χE> [a.u.]
Re[χE(x′, t)] [a.u.]
7π/ω
5π/ω 4π/ω t = 3π/ω
-4
12π/ω
6π/ω
-2
0
2
4
(mω/h)1/2x′ Fig. 3. Decay process of an oF-energy state. Since the energy of the oscillator is 1:7˝!; E (x ; t) decays as time increases.
n = 1 state of a harmonic oscillator. As shown in Fig. 1, the electron moves rightward at +rst, and at this stage the peak at (m!=˝)1=2 x 1 grows. As the time increases, the electron turns around to the left, and the dip in the left grows. As shown in Fig. 2, E (x ; t) gradually approaches the eigen wave function of the n = 1 state. Fig. 3 shows calculated E (x ; t) for E = 1:7˝!. This energy is not the eigen energy. In this case, the destructive self-interference results in the amplitude of E (x ; t) approaching zero with time. Fig. 4 shows the norm of E (x ; t) plotted as a function of energy. The curves are calculated for t = 2 =!∼20 =!. We see that the electronic states for any value of energy are possible for a small value of t. With increasing t, quantization gradually occurs; peaks at E = ˝!(n + 12 ) become sharper. These results indicate that the formation process of eigenstates is described by the function E (x ; t). At t = 0, the eigenstates are the extended band states given by Eq. (1). By applying the potential, the electronic motion shown in Fig. 1 occurs. As a result of self-interference, E (x ; t) changes gradually from a band state to a localized eigenstate for a particular value of E. Therefore, the function E (x ; t) describes the transformation from Bloch states (eigenstates for V = 0) to localized eigenstates in the electromagnetic +elds.
t = 2π/ω
E/ hω Fig. 4. Norm of the state E |E plotted as a function of energy. As the time increases, quantized states are formed and the peaks at E = ˝!(n + 12 ) grow.
From these results, we see that the self-interference plays an essential role in the formation of quantized states. For particular energies, an afterimage of electronic motion results in constructive interference; the amplitude becomes larger and larger with time. On the other hand for other energies, the amplitudes become gradually small due to the destructive self-interference. Therefore, discrete energy levels occurs. 3. Band electrons in high electric elds This theory enables us to describe both the Landau states in magnetic +elds and Stark-ladders in electric +elds in a uni+ed way [1]. As an example of the theory we investigate coherence of electrons in uniform electric +elds. In a static electric +eld F, an electron is accelerated as k(t) = k0 + eFt=˝. At the edge of the Brillouin zone, the electron is Bragg reIected due to the crystalline potential. As a result of this motion, a ladder-like energy spectrum associated with the localized wave functions realizes. This is known as a Stark-ladder [6–8]. Time-evolution of a Bloch function is given by (4) r|k(t) = d 3 r0 r|e−iHt=˝ |r0 r0 |k0 ;
M. Morifuji, K. Kato / Physica E 21 (2004) 1126 – 1130
and d 3 k |k k | = 1
= e−i=˝
d −t=2 k()
k0 +eFt=˝ (r);
(7)
k |e−iHt=˝ |k0 t=2
d −t=2 k()
(k − k0 − eFt=˝):
(8)
By superposing the time-evolving states given by Eq. (7) along the history as in Eq. (3), we have a wave function which expresses the formation of a Stark-ladder state. When the electric +eld is applied along the z-direction, we have the wave function Ek (r) =
2
0
de
2T t=T
-2
where k(t) = k0 + eFt=˝. To obtain Eq. (7), we have used that expression of the Green’s function in an electric +eld [9,10]
=e−i=˝
3T
(6)
into Eq. (4), we have r|k(t) = d 3 k r|k k |e−iHt=˝ |k0 t=2
4T
ρ(E, t) [a.u.]
where r|k0 ≡ k0 (r) is a Bloch function at the time t = 0. H is a Hamiltonian including the electric and crystalline potentials. In principle, the applied and crystal potentials should be treated on an equal footing. By applying the eFective mass approximation, the eFects from the crystalline potential are included in a cosine-like dispersion curve. By using the completeness relations d 3 r0 |r0 r0 | = 1 (5)
1129
−i=˝
t =2
−t =2
[kz () −E]d
k(t ) (r); (9)
which changes gradually from a Bloch function to a Stark-ladder. In Eq. (9), is the system size, k(t) = k0 + eFt, and = eFt . = eFt is change of wave vector during the period t. When is much smaller than Brillouin zone width, we can set as 2 =, and Ek (r) is regarded as a Bloch function. A band-like energy spectrum realizes in this case. With increasing , if n ≡ E=eFa is not an integer, the amplitudes interfere destructively with one another and cancel out. Whereas when E=eFa is an integer, Ek (r) becomes a rung of a Stark-ladder located at the nth position for
0 E/ eFa
2
Fig. 5. Time-dependent density of states in an electric +eld. The curves are calculated for the time t = T ∼ 4T , where T = 2 ˝=eFa is a period of Bloch oscillation.
2 =L. Therefore, a ladder-like spectrum associated with localized state is realized. To show how quantization occurs and the energy spectrum changes due to interference, we calculate a quantity (E; t) = Ek |Ek : (10) k
This quantity corresponds to a time-dependent density of states with weight of norm of Ek . Fig. 5 shows (E; t) plotted as a function of energy for various values of t. We have noted that the lateral degrees of freedom are neglected so as to show the discrete levels clearly in Fig. 5. The curves are calculated for t =T ∼4T , where T =2 ˝=eFa is a period of Bloch oscillation. When t is small, the density of states is continuous. With increasing time, interference accompanied with electronic motion occurs, and the quantized density of states realizes. Such temporal behavior of energy spectrum is similar to that shown in Fig. 4 but without the zero point energy. Qualitatively, such eFects of quantum coherence on electronic states are well expected. However, as far as we know, no quantitative description of self-interference has been done. The electronic coherence and formation of quantized states in the electric +elds are important for high +eld transport in recent
1130
M. Morifuji, K. Kato / Physica E 21 (2004) 1126 – 1130
nano-devices. Therefore, the quantitative treatment on quantum coherence as shown in this paper is necessary for precise description of the electrons in electromagnetic +elds. As an application of the present theory, we have shown that the quantum transport can be described by using the half-localized wavepacket states [8]. Sometimes we describe the transport by using Bloch functions, and sometimes we may regard transport as hopping motion of Stark-ladder states; the coherence is not properly considered in these treatments. On the other hand, the use of wavepacket states results in quantum eFects on description of the transport phenomena. Finally, we note that the use of wavepacket states is related to the Green’s function theory for transport. As we have shown in this paper, the half-localized states Ek (r) are written in terms of a Green’s function in an electric +eld. In order to improve the transport theories in high +elds, Green’s functions and Wigner distributions are often employed [11,12]. We note that there are some attempts to describe transport phenomena by using such a half-localized Stark-ladder state as well as Green’s functions [8,13]. 4. Conclusion In conclusion, we have shown that the formation process of quantized states in electromagnetic +elds is described in terms of self-interference. By superposing the extended Bloch functions through the history in which an electron has undergone, we derived a wave function for the formation process. For particular energies, self-interference results in the
enhancement of probability amplitude. Whereas for other energies, amplitudes are attenuated due to destructive interference. As a result, discrete energy levels occur. This theory enables us to describe both Stark-ladders in the electric +eld and Landau states in magnetic +elds in an uni+ed way. As an example of this theory we have shown how a Stark-ladder is formed in an electric +eld. Acknowledgements We acknowledge the +nancial support from the Handai Frontier Research Center. References [1] M. Morifuji, K. Kato, Phys. Rev. B 68 (2003) 035108. [2] R.P. Feynman, Rev. Mod. Phys. 20 (1948) 367. [3] R.P. Feynman, A.R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, New York, 1965. [4] L.S. Schulman, Techniques and Applications of Path Integration, Wiley, New York, 1996. [5] M.C. Gutzwiller, J. Math. Phys. 8 (1967) 1979. [6] M. Morifuji, S.K. Wah, C. Hamaguchi, Solid State Electron. 42 (1998) 1505. [7] M. Morifuji, C. Hamaguchi, Phys. Rev. B 58 (1998) 12842. [8] M. Morifuji, M. Ono, J. Phys. Soc. Jpn. 72 (2003) 229. [9] H. Haug, A.-P. Jauho, Quantum Kinetics in Transport and Optics of Semiconductors, Springer, Berlin, 1998. [10] J.H. Davies, J.W. Wilkins, Phys. Rev. B 38 (1998) 1667. [11] A. Wacker, A.-P. Jauho, Phys. Rev. Lett. 80 (1998) 369. [12] P. Bordone, A. Bertoni, R. Brunetti, C. Jacoboni, Math. Comput. Simat. 62 (2003) 307. [13] S. Rott, N. Linder, G.H. DOohler, Phys. Rev. B 65 (2002) 195 301.