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Movement and amplification of exciton condensed phase pulses and interaction between pulses in semiconductor quantum wells
Physics Letters A 376 (2012) 2804–2807
Contents lists available at SciVerse ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Movement and amplification of exciton condensed phase pulses and interaction between pulses in semiconductor quantum wells O.I. Dmytruk a , V.I. Sugakov b,∗ a b
Taras Shevchenko National University of Kyiv, 2, Prosp. Academician Glushkov, Kyiv 03022, Ukraine Institute for Nuclear Research, National Academy of Sciences of Ukraine, 47, Prosp. Nauky, Kyiv 03680, Ukraine
a r t i c l e
i n f o
Article history: Received 7 June 2012 Received in revised form 13 August 2012 Accepted 14 August 2012 Available online 20 August 2012 Communicated by V.M. Agranovich
a b s t r a c t Formation and movement of an exciton pulse in an inhomogeneous potential are studied. It is shown that the pulse does not blur and the maximum of the exciton density in the pulse remains constant during the exciton lifetime if the pulse is formed from the condensed phase. The path, traversed by the excitons, can be increased by imposing an additional laser pulse on the system. Thereby, such a system can be used for information transmission over the exciton condensed phase. © 2012 Elsevier B.V. All rights reserved.
Keywords: Indirect excitons Condensed phase Exciton pulse Amplification
1. Introduction Lately, several non-trivial effects in so-called indirect exciton systems in GaAsAl double quantum wells were observed. In the presence of an external electrical field perpendicular to the quantum well plane, an electron and a hole of an indirect exciton are separated in the space into different quantum wells that inhibits their mutual recombination and leads to the significant increase of their lifetime. Among the interesting facts, discovered for indirect excitons, we should mention periodical structures on the ring outside the laser spot that were observed in the emission spectrum of double quantum wells [1]. Same structures were observed under the perimeter of the window in the metal electrode [2]. To explain the phenomenon several models were proposed. They explained instability by Bose statistics of excitons [3], or by BEC [4,5] or by Mott transition [6]. In the papers [7–10] the formation of periodic structures was explained by the theory which is based on two approaches. 1. There is an exciton condensed phase caused by the attractive interaction between excitons. At not very far distances between quantum wells, when the exciton dipole moment is not too large, the exchange and van der Waals interactions exceed the dipole–dipole repulsion. An existence of attractive inter-
*
Corresponding author. E-mail addresses: [email protected] (O.I. Dmytruk), [email protected] (V.I. Sugakov). 0375-9601/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2012.08.023
action between excitons is confirmed by the calculations of biexcitons [11,12], and also under investigation of many exciton system [13]. 2. The finite value of the exciton lifetime plays an important role in the formation of spatial distribution of exciton condensed phases. Usually, the exciton lifetime exceeds significantly the time of an establishment of a local equilibrium state. By this reason the lifetime of excitons is suggested to be equal to infinity in theoretical solutions of many exciton problems. But, it is not correct in inhomogeneous systems with several phases. It is necessary to take into account the finite value of the exciton lifetime under the investigation of the mutual spatial distribution phases in several phase systems, because the time of the establishment of the equilibrium between phases is determined by slow diffusion processes and it is less than the exciton lifetime. The approach [7–10] has allowed to describe the spatial distribution of the exciton condensed phase and its dynamics under changing pumping and temperature, observed in [1,2]. Significant indirect excitons lifetime can be promising for using them for information transmission in semiconductors [14,15]. Authors of [16] demonstrated an experimental proof of the principle for an exciton optoelectronic transistor. It is interesting to study the exciton pulse motion, when excitons are in the condensed phase. The movement of periodical structures under homogeneous pumping is studied in the paper [17]. But it is important to study the movement of an exciton single pulse in external potentials for information transmission.
O.I. Dmytruk, V.I. Sugakov / Physics Letters A 376 (2012) 2804–2807
2805
The present Letter is addressed to the study of creation and properties of the exciton condensed phase pulses. 2. Condensed exciton phase pulse To describe an exciton density distribution we use the spinodal decay model generalized so that it takes into account the finite lifetime of indirect excitons and the presence of the pumping [7–10]. The considered system is non-equilibrium one. Usually the time of establishment of local equilibrium in exciton system is much less than the exciton lifetime. The assumption about local equilibrium is widely applied in the non-equilibrium process theory. In each small environment volume in the local equilibrium state the free energy is the same function of macroscopic variables as it is for an equilibrium system [18]. In the state of local equilibrium the system may be described by the free energy which depends on exciton density. The equation for the density of excitons [17] is as follows:
∂n n = − div j + G (r , t ) − , ∂t τ
(1)
Fig. 1. Spatial dependence of the exciton density at various values of the pumping: G = 0.005 < G 1 for the continuous line, G 1 < G = 0.008 < G 2 for the periodic line, and G = 0.009 > G 2 for the dashed line. τ = 100, b1 = −2.1.
where n is the exciton density, G (r , t ) is the pumping (the number of excitons created per time unit per space unit of a quantum μ is the exciton current well), τ is the exciton lifetime, j = − M ∇ density, M is the exciton mobility. For the exciton mobility M we use Einstein formula M = nD , where D is a diffusion coefficient, kT k is Boltzmann constant, T is the temperature. Expressed the chemical potential as μ = δδnF , chose the free energy in the Landau model:
dr
F [n] =
K 2
2 (∇ n) + f (n) + nV .
(2)
2
n) characterizes the energy of inhomogeneity, f (n) The term K2 (∇ is a free energy density, nV considers potential energy. We approximate the density of the free energy in the form:
f (n) = f 0 + kT n ln
n n0
a
b
c
2
3
4
− 1 + n2 + n3 + n4 ;
(3)
n n0
where the term kT n(ln − 1) dominates for low exciton densities and describes the diffusive motion while terms of the power series for the exciton density are important for large n. The three last terms are the most important in studying the system in the vicinity of the exciton concentration at which the condensed phase is formed. The phenomenological parameters a, b, c satisfy the conditions a > 0, b < 0, c > 0. At such conditions the free energy describes both the growth of the energy per one exciton with increasing density due to dipole–dipole repulsion between excitons and the presence of minimum of the chemical potential due to exciton–exciton attractive exchange interaction with further increasing density. Let us emphasize that Eq. (3) is not the expansion of the density of a free energy in the row n. Eq. (3) is an approximation of the free energy density under the assumption about the existence of a local equilibrium. Let introduce dimensionless variables for length, density, energy and time:
lu =
K a
,
nu =
a c
V u = anu ,
,
tu =
d1l2u D
.
(4)
Introducing the dimensionless variables and substituting (2) to (1) we get a nonlinear equation that determines the exciton density:
∂n −n + n + b1n2 + n3 + V + G − n , (5) n∇ = d1 n + ∇ ∂t τ where b1 = √b < 0, d1 = ac
kT Vu
pendent on the coordinates.
and V is an external potential, de-
Fig. 2. Spatial dependence of the exciton density after the end of the laser pulse acting in different time moments for below-threshold pumpings: G = 0.04, τ = 1000, b1 = −2.1.
Eq. (5) was solved numerically in the one-dimensional case. Such one-dimensional system may emerge, for example, in the presence of the slot in a metal electrode quantum well under the slot [17]. Initial and border conditions were chosen to be zero. In the case of the homogeneous external potential and homogeneous pumping the solutions of (5) have the spatial dependency shown in Fig. 1. In the case of low pumpings G < G 1 the homogeneous density distribution is formed that corresponds to the gas phase. In the case of pumpings higher than a particular threshold value G 1 the system becomes unstable on the formation of periodic structures. Regions of maximum correspond to the condensed phase of excitons, regions of minimum — to the gas phase. The periodic structure exists in the pumpings range G 1 < G < G 2 . In the case of high pumpings G > G 2 density distribution becomes homogeneous again and corresponds to the exciton condensed phase. G 1 and G 2 values depend on the exciton lifetime τ and interaction b1 . In the paper [17] the author studies the motion of a periodic distribution of excitonic structure in an external potential, which depends linearly on the coordinates, in the case of spatially uniform and constant in time pumping. However, it is important for information transmission to study the movement of the condensed exciton phase pulses in space. This Letter is addressed to the solution of this issue. Let us suggest that the system is exposed to rectangular in time and space laser impulse with amplitude greater than that at which
2806
O.I. Dmytruk, V.I. Sugakov / Physics Letters A 376 (2012) 2804–2807
Fig. 3. Spatial dependence of the exciton density after the end of the laser pulse acting in different moments of time, when the pumping exceeds some threshold value: G = 0.3, τ = 1000, b1 = −2.1.
Fig. 4. Dependence of exciton density maximum on time at various values of the pumping: squares — G = 0.2 > G 1 (the pulse is formed from the condensed phase), triangles — G = 0.08 < G 1 (the pulse is formed from the gas phase).
the condensed phase is formed, then the pumping has the following dependence on the coordinates and time
G ( z, t ) =
const, 0,
if z1 < z < z2 , t 1 < t < t 2 ; in all other cases.
(6)
We have fulfilled also calculations for the Gaussian pulse form. The results obtained with the Gaussian pulse form are similar to the results with the rectangular form given by formula (6). To create a single pulse the spatial width of the pumping pulse (z2 − z1 ) must be less than or the order of periodic structure period in the homogeneous excitement (Fig. 1). If the pulse width is greater than this value, then the exciton pulse with several maxima is created. At first, we study the case of a tight pumping pulse (z2 − z1 = 10) which creates an exciton pulse with a single maximum. Let us consider the dynamics of a pulse shape in the case of a homogeneous external potential (V = 0 in Eq. (5)). At low pumpings the exciton gas phase emerges, condensed phase is not formed, and the distribution density of excitons blurs diffusely over time (Fig. 2). If the value of the pumping pulse acting on the system is greater than a threshold value, than in the system the density distribution is created a pulse maximum of which corresponds to the density in the condensed phase. In the case of small pulse width with the above-threshold pumping a single island of exciton condensed phase is formed. The time of the pulse formation is much less than the exciton lifetime (≈ 0.1τ ). The shape of the exciton density distribution indicates the formation of the condensed phase, namely, density distribution varies in the space with the transition from the high-density region to the low-density one (Fig. 3). It is clearly observed in Fig. 4 that in case of the condensed state maximum value of the exciton density remains constant during the exciton lifetime while width of the gas phase pulse decreases. 3. Movement of exciton condensed phases pulses in an inhomogeneous field Let us consider the pulse motion in the potential that linearly depends on the coordinates. An experimental implementation of the linear potential is obtained in [19]. For such a case let us put V = δ z in Eq. (5). Calculated Eq. (5) numerically we obtained that for the above-threshold pumping the pulse is moving in the way, so its maximum is shifted with the velocity v ≈ − δkTD . This is illustrated in Fig. 5. In this case the pulse width decreases and its height remains almost constant while it is moving, and only af-
Fig. 5. Spatial dependence of the exciton density G = 0.2. Different curves are presented for different moments of time over uniform time intervals.
ter a value of the exciton lifetime condensed phase disappears and the pulse diffuses. Thus, the maximum density lasts in the condensed phase longer than in the case of the exciton motion in the gas phase. This is illustrated in Fig. 4, which represents the dependence of the exciton pulse maximum value on time for different values of the pumping (for below threshold value and for above threshold value). One can observe that at pumpings below the threshold the maximum value of a moving exciton pulse rapidly decreases over time, whereas for above-threshold pumpings the maximum value of density remains almost constant during the exciton lifetime or even bigger in the case of wide initial laser pulses. For the quantitative description, we chose the following values of parameters: T = 4 K, D = 100
cm2 , s
δ = 4.5
eV . cm
In this case ex-
citon pulse moves with the velocity v = − δkTD = 1.3 · 106
cm . s
At
exciton lifetime τ = 10−7 s the maximum of pulse remains constant during the path of 1.3 mm. 4. Interaction of exciton condensed phases pulses in an inhomogeneous field Let us consider the amplification possibility for the initial pulse by imposing another laser pulse (Fig. 6). Assume that at the moment of condensed phase evanescence in the first pulse and pulse is diffusing (Fig. 6, a), the system is exposed to another pulse (Fig. 6, b) such that its amplitude is lesser than the threshold
O.I. Dmytruk, V.I. Sugakov / Physics Letters A 376 (2012) 2804–2807
2807
then the signal being transmitted by the system in the condensed phase can be received on a greater distance (Fig. 7). 5. Discussion and conclusions In this Letter we studied the motion dynamics of the exciton condensed phase pulses using the phase transitions theory for the system of unstable particles considering the finite exciton lifetime and the presence of a pumping. The numerical simulation of the exciton density equation results to the following conclusions.
Fig. 6. Spatial dependence of the exciton density over uniform time intervals. (a) Only first laser pulse acting, (b) only second laser pulse acting, (c) the cumulative laser pulse acting.
1. Under the pumping pulse, limited in time and space, the exciton condensed phase pulse is formed if the intensity of laser pulse is greater than the threshold value. 2. In the condensed phase the maximum exciton density value remains constant during the exciton lifetime, while width of the pulse decreases. But in case of the gas phase the maximum value of the pulse fades. 3. In the presence of an external inhomogeneous potential exciton pulse moves in the space. In the case of linear potential pulse is moving with the velocity, proportional to the slope of this potential. 4. The path, traversed by the maximum of the exciton condensed phase pulse, might be increased by imposing an additional laser pulse on the system. The value of this pulse may be below the threshold value for condensed phase formation (due to the amplification of the first impulse). If a detector reacts to the signal amplitude, then the maximum in the distribution of the exciton density can be detected on larger distance. References
Fig. 7. Dependence of the exciton density on time in the point of observation z = 158 for three cases. First pulse is imposed in point z = 15, second pulse (less than threshold value) is imposed in point z = 115, continuous line describes cumulative density for two pulses.
one and the position by the imposing time matches the position of the first pulse maximum. As it is shown in Fig. 6, c, the condensed phase is formed again as a result, the maximum of which decreases slower than for the gas phase. It is also clear that on great distances the maximum value of the cumulative pulse is significantly higher than the maximum value of either of pulses separately. Thus the path traversed by a pulse can be increased having the excitons in the condensed phase. Dependence of exciton density on time in case of the first, second and cumulative exciton pulses in the point of space is presented in Fig. 7. Maximum exciton density in the cumulative pulse is bigger than sum of the exciton densities in two different pulses (n1,max + n2,max < nΣ ). If a detector reacts to the signal amplitude,
[1] L.V. Butov, A.C. Gossard, D.S. Chemla, Nature 418 (2002) 751. [2] A.V. Gorbunov, V.B. Timofeev, Pis’ma Zh. Eksp. Teor. Fiz. 83 (2006) 178; A.V. Gorbunov, V.B. Timofeev, Usp. Fiz. Nauk 176 (2006) 652. [3] L.S. Levitov, B.D. Simons, L.V. Butov, Phys. Rev. Lett. 94 (2005) 176404. [4] A.V. Paraskevov, T.V. Khabarova, Phys. Lett. A 151 (2007) 368. [5] R.B. Saptsov, Pis’ma Zh. Eksp. Teor. Fiz. 86 (2007) 779. [6] V.S. Babichenko, arXiv:0706.0994 [cond-mat], 2007. [7] V.I. Sugakov, Ukr. J. Phys. 49 (2004) 1117; V.I. Sugakov, Solid State Commun. 134 (2005) 63. [8] V.I. Sugakov, Fiz. Tverd. Tela 48 (2006) 1868; V.I. Sugakov, Fiz. Nizk. Temper. 32 (2006) 1449; V.I. Sugakov, Phys. Rev. B 76 (2007) 115303. [9] A.A. Chernyuk, V.I. Sugakov, Phys. Rev. B 74 (2006) 085303. [10] V.I. Sugakov, A.A. Chernyuk, Pis’ma Zh. Eksp. Teor. Fiz. 85 (2007) 699. [11] M.Y.I. Tan, N.D. Drummord, I. Needs, Phys. Rev. B 71 (2005) 033303. [12] Ch. Schindler, R. Zimmermann, Phys. Rev. B 78 (2008) 045313. [13] Yu.E. Lozovik, O.L. Berman, Pis’ma Zh. Eksp. Teor. Fiz. 64 (1996) 526. [14] A. Gartner, L. Prechtel, D. Schuh, A.W. Holleitner, J.P. Kotthaus, Phys. Rev. B 76 (2007) 085304. [15] A.G. Winbow, J.R. Leonard, M. Remeika, Y.Y. Kuznetsova, A.A. High, A.T. Hammack, L.V. Butov, J. Wilkes, A.A. Guenther, A.L. Ivanov, M. Hanson, A.C. Gossard, Phys. Rev. Lett. 106 (2011) 196806. [16] A.A. High, A.T. Hammack, L.V. Butov, A.C. Gossard, Optics Lett. 32 (2007) 2466. [17] V.I. Sugakov, J. Phys.: Condens. Matter 21 (2009) 275803. [18] P. Glansdorff, I. Prigagine, Thermodynamic Theory of Structure, Stability and Fluctuations, Wiley, New York, 1971. [19] J.R. Leonard, M. Remeika, M.K. Chu, Y.Y. Kuznetsova, A.A. High, L.V. Butov, J. Wilkes, M. Hanson, A.C. Gossard, Appl. Phys. Lett. 100 (2012) 231106.
Contents lists available at SciVerse ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Movement and amplification of exciton condensed phase pulses and interaction between pulses in semiconductor quantum wells O.I. Dmytruk a , V.I. Sugakov b,∗ a b
Taras Shevchenko National University of Kyiv, 2, Prosp. Academician Glushkov, Kyiv 03022, Ukraine Institute for Nuclear Research, National Academy of Sciences of Ukraine, 47, Prosp. Nauky, Kyiv 03680, Ukraine
a r t i c l e
i n f o
Article history: Received 7 June 2012 Received in revised form 13 August 2012 Accepted 14 August 2012 Available online 20 August 2012 Communicated by V.M. Agranovich
a b s t r a c t Formation and movement of an exciton pulse in an inhomogeneous potential are studied. It is shown that the pulse does not blur and the maximum of the exciton density in the pulse remains constant during the exciton lifetime if the pulse is formed from the condensed phase. The path, traversed by the excitons, can be increased by imposing an additional laser pulse on the system. Thereby, such a system can be used for information transmission over the exciton condensed phase. © 2012 Elsevier B.V. All rights reserved.
Keywords: Indirect excitons Condensed phase Exciton pulse Amplification
1. Introduction Lately, several non-trivial effects in so-called indirect exciton systems in GaAsAl double quantum wells were observed. In the presence of an external electrical field perpendicular to the quantum well plane, an electron and a hole of an indirect exciton are separated in the space into different quantum wells that inhibits their mutual recombination and leads to the significant increase of their lifetime. Among the interesting facts, discovered for indirect excitons, we should mention periodical structures on the ring outside the laser spot that were observed in the emission spectrum of double quantum wells [1]. Same structures were observed under the perimeter of the window in the metal electrode [2]. To explain the phenomenon several models were proposed. They explained instability by Bose statistics of excitons [3], or by BEC [4,5] or by Mott transition [6]. In the papers [7–10] the formation of periodic structures was explained by the theory which is based on two approaches. 1. There is an exciton condensed phase caused by the attractive interaction between excitons. At not very far distances between quantum wells, when the exciton dipole moment is not too large, the exchange and van der Waals interactions exceed the dipole–dipole repulsion. An existence of attractive inter-
*
Corresponding author. E-mail addresses: [email protected] (O.I. Dmytruk), [email protected] (V.I. Sugakov). 0375-9601/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2012.08.023
action between excitons is confirmed by the calculations of biexcitons [11,12], and also under investigation of many exciton system [13]. 2. The finite value of the exciton lifetime plays an important role in the formation of spatial distribution of exciton condensed phases. Usually, the exciton lifetime exceeds significantly the time of an establishment of a local equilibrium state. By this reason the lifetime of excitons is suggested to be equal to infinity in theoretical solutions of many exciton problems. But, it is not correct in inhomogeneous systems with several phases. It is necessary to take into account the finite value of the exciton lifetime under the investigation of the mutual spatial distribution phases in several phase systems, because the time of the establishment of the equilibrium between phases is determined by slow diffusion processes and it is less than the exciton lifetime. The approach [7–10] has allowed to describe the spatial distribution of the exciton condensed phase and its dynamics under changing pumping and temperature, observed in [1,2]. Significant indirect excitons lifetime can be promising for using them for information transmission in semiconductors [14,15]. Authors of [16] demonstrated an experimental proof of the principle for an exciton optoelectronic transistor. It is interesting to study the exciton pulse motion, when excitons are in the condensed phase. The movement of periodical structures under homogeneous pumping is studied in the paper [17]. But it is important to study the movement of an exciton single pulse in external potentials for information transmission.
O.I. Dmytruk, V.I. Sugakov / Physics Letters A 376 (2012) 2804–2807
2805
The present Letter is addressed to the study of creation and properties of the exciton condensed phase pulses. 2. Condensed exciton phase pulse To describe an exciton density distribution we use the spinodal decay model generalized so that it takes into account the finite lifetime of indirect excitons and the presence of the pumping [7–10]. The considered system is non-equilibrium one. Usually the time of establishment of local equilibrium in exciton system is much less than the exciton lifetime. The assumption about local equilibrium is widely applied in the non-equilibrium process theory. In each small environment volume in the local equilibrium state the free energy is the same function of macroscopic variables as it is for an equilibrium system [18]. In the state of local equilibrium the system may be described by the free energy which depends on exciton density. The equation for the density of excitons [17] is as follows:
∂n n = − div j + G (r , t ) − , ∂t τ
(1)
Fig. 1. Spatial dependence of the exciton density at various values of the pumping: G = 0.005 < G 1 for the continuous line, G 1 < G = 0.008 < G 2 for the periodic line, and G = 0.009 > G 2 for the dashed line. τ = 100, b1 = −2.1.
where n is the exciton density, G (r , t ) is the pumping (the number of excitons created per time unit per space unit of a quantum μ is the exciton current well), τ is the exciton lifetime, j = − M ∇ density, M is the exciton mobility. For the exciton mobility M we use Einstein formula M = nD , where D is a diffusion coefficient, kT k is Boltzmann constant, T is the temperature. Expressed the chemical potential as μ = δδnF , chose the free energy in the Landau model:
dr
F [n] =
K 2
2 (∇ n) + f (n) + nV .
(2)
2
n) characterizes the energy of inhomogeneity, f (n) The term K2 (∇ is a free energy density, nV considers potential energy. We approximate the density of the free energy in the form:
f (n) = f 0 + kT n ln
n n0
a
b
c
2
3
4
− 1 + n2 + n3 + n4 ;
(3)
n n0
where the term kT n(ln − 1) dominates for low exciton densities and describes the diffusive motion while terms of the power series for the exciton density are important for large n. The three last terms are the most important in studying the system in the vicinity of the exciton concentration at which the condensed phase is formed. The phenomenological parameters a, b, c satisfy the conditions a > 0, b < 0, c > 0. At such conditions the free energy describes both the growth of the energy per one exciton with increasing density due to dipole–dipole repulsion between excitons and the presence of minimum of the chemical potential due to exciton–exciton attractive exchange interaction with further increasing density. Let us emphasize that Eq. (3) is not the expansion of the density of a free energy in the row n. Eq. (3) is an approximation of the free energy density under the assumption about the existence of a local equilibrium. Let introduce dimensionless variables for length, density, energy and time:
lu =
K a
,
nu =
a c
V u = anu ,
,
tu =
d1l2u D
.
(4)
Introducing the dimensionless variables and substituting (2) to (1) we get a nonlinear equation that determines the exciton density:
∂n −n + n + b1n2 + n3 + V + G − n , (5) n∇ = d1 n + ∇ ∂t τ where b1 = √b < 0, d1 = ac
kT Vu
pendent on the coordinates.
and V is an external potential, de-
Fig. 2. Spatial dependence of the exciton density after the end of the laser pulse acting in different time moments for below-threshold pumpings: G = 0.04, τ = 1000, b1 = −2.1.
Eq. (5) was solved numerically in the one-dimensional case. Such one-dimensional system may emerge, for example, in the presence of the slot in a metal electrode quantum well under the slot [17]. Initial and border conditions were chosen to be zero. In the case of the homogeneous external potential and homogeneous pumping the solutions of (5) have the spatial dependency shown in Fig. 1. In the case of low pumpings G < G 1 the homogeneous density distribution is formed that corresponds to the gas phase. In the case of pumpings higher than a particular threshold value G 1 the system becomes unstable on the formation of periodic structures. Regions of maximum correspond to the condensed phase of excitons, regions of minimum — to the gas phase. The periodic structure exists in the pumpings range G 1 < G < G 2 . In the case of high pumpings G > G 2 density distribution becomes homogeneous again and corresponds to the exciton condensed phase. G 1 and G 2 values depend on the exciton lifetime τ and interaction b1 . In the paper [17] the author studies the motion of a periodic distribution of excitonic structure in an external potential, which depends linearly on the coordinates, in the case of spatially uniform and constant in time pumping. However, it is important for information transmission to study the movement of the condensed exciton phase pulses in space. This Letter is addressed to the solution of this issue. Let us suggest that the system is exposed to rectangular in time and space laser impulse with amplitude greater than that at which
2806
O.I. Dmytruk, V.I. Sugakov / Physics Letters A 376 (2012) 2804–2807
Fig. 3. Spatial dependence of the exciton density after the end of the laser pulse acting in different moments of time, when the pumping exceeds some threshold value: G = 0.3, τ = 1000, b1 = −2.1.
Fig. 4. Dependence of exciton density maximum on time at various values of the pumping: squares — G = 0.2 > G 1 (the pulse is formed from the condensed phase), triangles — G = 0.08 < G 1 (the pulse is formed from the gas phase).
the condensed phase is formed, then the pumping has the following dependence on the coordinates and time
G ( z, t ) =
const, 0,
if z1 < z < z2 , t 1 < t < t 2 ; in all other cases.
(6)
We have fulfilled also calculations for the Gaussian pulse form. The results obtained with the Gaussian pulse form are similar to the results with the rectangular form given by formula (6). To create a single pulse the spatial width of the pumping pulse (z2 − z1 ) must be less than or the order of periodic structure period in the homogeneous excitement (Fig. 1). If the pulse width is greater than this value, then the exciton pulse with several maxima is created. At first, we study the case of a tight pumping pulse (z2 − z1 = 10) which creates an exciton pulse with a single maximum. Let us consider the dynamics of a pulse shape in the case of a homogeneous external potential (V = 0 in Eq. (5)). At low pumpings the exciton gas phase emerges, condensed phase is not formed, and the distribution density of excitons blurs diffusely over time (Fig. 2). If the value of the pumping pulse acting on the system is greater than a threshold value, than in the system the density distribution is created a pulse maximum of which corresponds to the density in the condensed phase. In the case of small pulse width with the above-threshold pumping a single island of exciton condensed phase is formed. The time of the pulse formation is much less than the exciton lifetime (≈ 0.1τ ). The shape of the exciton density distribution indicates the formation of the condensed phase, namely, density distribution varies in the space with the transition from the high-density region to the low-density one (Fig. 3). It is clearly observed in Fig. 4 that in case of the condensed state maximum value of the exciton density remains constant during the exciton lifetime while width of the gas phase pulse decreases. 3. Movement of exciton condensed phases pulses in an inhomogeneous field Let us consider the pulse motion in the potential that linearly depends on the coordinates. An experimental implementation of the linear potential is obtained in [19]. For such a case let us put V = δ z in Eq. (5). Calculated Eq. (5) numerically we obtained that for the above-threshold pumping the pulse is moving in the way, so its maximum is shifted with the velocity v ≈ − δkTD . This is illustrated in Fig. 5. In this case the pulse width decreases and its height remains almost constant while it is moving, and only af-
Fig. 5. Spatial dependence of the exciton density G = 0.2. Different curves are presented for different moments of time over uniform time intervals.
ter a value of the exciton lifetime condensed phase disappears and the pulse diffuses. Thus, the maximum density lasts in the condensed phase longer than in the case of the exciton motion in the gas phase. This is illustrated in Fig. 4, which represents the dependence of the exciton pulse maximum value on time for different values of the pumping (for below threshold value and for above threshold value). One can observe that at pumpings below the threshold the maximum value of a moving exciton pulse rapidly decreases over time, whereas for above-threshold pumpings the maximum value of density remains almost constant during the exciton lifetime or even bigger in the case of wide initial laser pulses. For the quantitative description, we chose the following values of parameters: T = 4 K, D = 100
cm2 , s
δ = 4.5
eV . cm
In this case ex-
citon pulse moves with the velocity v = − δkTD = 1.3 · 106
cm . s
At
exciton lifetime τ = 10−7 s the maximum of pulse remains constant during the path of 1.3 mm. 4. Interaction of exciton condensed phases pulses in an inhomogeneous field Let us consider the amplification possibility for the initial pulse by imposing another laser pulse (Fig. 6). Assume that at the moment of condensed phase evanescence in the first pulse and pulse is diffusing (Fig. 6, a), the system is exposed to another pulse (Fig. 6, b) such that its amplitude is lesser than the threshold
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then the signal being transmitted by the system in the condensed phase can be received on a greater distance (Fig. 7). 5. Discussion and conclusions In this Letter we studied the motion dynamics of the exciton condensed phase pulses using the phase transitions theory for the system of unstable particles considering the finite exciton lifetime and the presence of a pumping. The numerical simulation of the exciton density equation results to the following conclusions.
Fig. 6. Spatial dependence of the exciton density over uniform time intervals. (a) Only first laser pulse acting, (b) only second laser pulse acting, (c) the cumulative laser pulse acting.
1. Under the pumping pulse, limited in time and space, the exciton condensed phase pulse is formed if the intensity of laser pulse is greater than the threshold value. 2. In the condensed phase the maximum exciton density value remains constant during the exciton lifetime, while width of the pulse decreases. But in case of the gas phase the maximum value of the pulse fades. 3. In the presence of an external inhomogeneous potential exciton pulse moves in the space. In the case of linear potential pulse is moving with the velocity, proportional to the slope of this potential. 4. The path, traversed by the maximum of the exciton condensed phase pulse, might be increased by imposing an additional laser pulse on the system. The value of this pulse may be below the threshold value for condensed phase formation (due to the amplification of the first impulse). If a detector reacts to the signal amplitude, then the maximum in the distribution of the exciton density can be detected on larger distance. References
Fig. 7. Dependence of the exciton density on time in the point of observation z = 158 for three cases. First pulse is imposed in point z = 15, second pulse (less than threshold value) is imposed in point z = 115, continuous line describes cumulative density for two pulses.
one and the position by the imposing time matches the position of the first pulse maximum. As it is shown in Fig. 6, c, the condensed phase is formed again as a result, the maximum of which decreases slower than for the gas phase. It is also clear that on great distances the maximum value of the cumulative pulse is significantly higher than the maximum value of either of pulses separately. Thus the path traversed by a pulse can be increased having the excitons in the condensed phase. Dependence of exciton density on time in case of the first, second and cumulative exciton pulses in the point of space is presented in Fig. 7. Maximum exciton density in the cumulative pulse is bigger than sum of the exciton densities in two different pulses (n1,max + n2,max < nΣ ). If a detector reacts to the signal amplitude,
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