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Temperature and size dependence of fractal MCF in semiconductor billiards
Microelectronic Engineering 51–52 (2000) 241–247 www.elsevier.nl / locate / mee
Temperature and size dependence of fractal MCF in semiconductor billiards a, a,e b a c c A.P. Micolich *, R.P. Taylor , J.P. Bird , R. Newbury , T.M. Fromhold , C.R. Tench , a d d H. Linke , Y. Aoyagi , T. Sugano a
School of Physics, University of New South Wales, Sydney, NSW, 2052, Australia Center for Solid State Research, Arizona State University, Tempe, AZ 85287 -6206, USA c School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2 RD, UK d Nanoelectronics Materials Laboratory, RIKEN, 2 -1 Hirosawa, Wako-shi, Saitama 351 -01, Japan e Physics Department, University of Oregon, Eugene, OR 97403 -1274, USA b
Abstract Fractal behaviour in the magneto-conductance fluctuations of mesoscopic systems has been predicted using semiclassical theory and observed experimentally in semiconductor billiards. Surface-gate billiards provide an ideal environment for investigations of fractal behaviour due to the ease with which parameters in the experimental system can be tuned. In this study we vary temperature and billiard size to investigate the classical and quantum limits of the semiclassical theory for fractal magneto-conductance fluctuations. 2000 Elsevier Science B.V. All rights reserved. Keywords: Fractal conductance fluctuations; Mesoscopic semiconductor billiards
Modern semiconductor growth and fabrication technologies have allowed extensive investigations of low-dimensional electron systems in recent decades [1]. Lately, considerable interest has focused on semiconductor billiards, where material-induced length scales, such as the electron mean free path and the inelastic scattering length, exceed the electron confinement length and, as a consequence, phase-coherent ballistic transport effects are observed [1–4]. Sub-micron billiards can be defined in the two-dimensional electron gas (2DEG) of an AlGaAs / GaAs heterostructure using patterned metal gates deposited on the heterostructure surface. Application of a negative bias to these surface-gates leads to electrostatic depletion of the 2DEG in the regions directly below the gates and the formation of a billiard whose geometry closely resembles that of the patterned surface-gates [5]. At low temperatures, reproducible fluctuations in billiard conductance, due to electron quantum interference effects, are observed as a function of magnetic field [2–4]. These magneto-conductance fluctuations (MCF) provide an important probe of electron transport through the billiard. *Corresponding author. Fax: 161-2-9385-6060. E-mail address: [email protected] (A.P. Micolich) 0167-9317 / 00 / $ – see front matter PII: S0167-9317( 99 )00484-0
2000 Elsevier Science B.V. All rights reserved.
242
A.P. Micolich et al. / Microelectronic Engineering 51 – 52 (2000) 241 – 247
Whilst traditionally it has been assumed that the confining potential of semiconductor billiards may be adequately approximated by an infinite square-well potential profile, recent studies of more realistic ‘soft-wall’ billiard potentials have received significant attention. These ‘soft-wall’ potentials have been shown to generate a mixed phase-space, containing regions of both chaotic and stable (i.e. non-chaotic) trajectories [6]. It has been speculated that the boundary between chaotic and non-chaotic regions generates a power-law distribution of the area A enclosed by electron trajectories [7], in contrast to the exponential distribution expected for a purely chaotic phase-space. A semiclassical theory [7], which monitors the phase accumulated by electron waves travelling along the classical trajectories, predicts that this power-law area distribution generates fractal MCF in billiard electron transport experiments. One of the key properties of fractal MCF is self-similarity. In the MCF of square billiards statistical self-similarity is observed, where the statistical properties of the fluctuations are the same at different magnetic field scales [8]. The consequence of fluctuations at different scales is that the data trace occupies more space than a smooth one-dimensional line but less space than a two-dimensional plane. The fractal dimension DF quantifies this space-filling property [9]. Hence fractal MCF satisfy 1 , DF , 2 [3]. Another important factor is the highest and lowest magnifications over which fractal behaviour can be observed. Whilst mathematically-generated fractals (e.g. the Mandelbrot Set) occur over an infinite range of magnifications, fractals in physical systems have an upper and lower limit beyond which fractal behaviour is no longer observed (e.g. a coastline where the upper limit is the length of the coastline and the lower limit the scale of the grains of sand comprising the coastline) [3,9,10]. The presence of fractal behaviour in the conductance fluctuations of mesoscopic systems is now well-established following studies of quasi-ballistic gold nanowires [11] and ballistic semiconductor billiards [3,4]. Unlike many natural systems, the advantage of studying fractals in semiconductor billiards is the ease with which we can tune various parameters in the experimental system and examine how this effects the fractal behaviour. In this study we focus on temperature and billiard size. These two parameters allow us to investigate the classical and quantum limits of the semiclassical theory for fractal conductance fluctuations [7]. Experiments were performed using billiards defined by the gate geometry shown in Fig. 1 (inset). Four different sized square billiards (w 3 w) were used in this study with lithographic widths w 5 0.4, 0.6, 1.0 and 2.0 mm. In all cases the billiards are significantly smaller than the electronic mean free path of the material ( . 5 mm), ensuring ballistic transport within each billiard. Typical carrier concentrations (hs ) and mobilities ( m ) for these billiards are 4–5 3 10 15 m 2 and 20–70 m 2 / V s respectively giving a Fermi wavelength lF | 40 nm. An electrostatic depletion of 50–100 nm around each gate edge is calculated (based on Aharonov–Bohm oscillations observed at high magnetic fields [12]) leading to billiard width w d , area A d and energy level spacing DE as shown in Table 1. Experiments were performed with samples thermally linked to the mixing chamber of a dilution refrigerator allowing electron temperatures ranging from 4.2 K down to | 30 mK. Electrical measurements were performed using low frequency AC lock-in techniques in a four-terminal configuration to eliminate lead resistance from measurements. All measurements were made with low sample current (Iex , 10 nA) to avoid electron heating. The temperature dependence studies were performed on a single thermal cycle. The fractal dimension of the MCF data is analysed using a box counting technique [3,9]. We begin by superimposing a mesh of squares of width DB 3 DB over the data. For a particular DB we then count the minimum number of boxes N(DB) required to completely cover the trace [3,9]. This count is
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Fig. 1. (a) Magneto-conductance fluctuations (MCF) obtained from a 0.6-mm billiard with temperatures (from top): 2.1 K, 1.3 K, 900 mK, 220 mK and (b) (from top) 4.2 K, 1.4 K, 480 mK, 80 mK for the 1-mm billiard. Inset is a scanning electron micrograph of the square billiard geometry used throughout this investigation.
repeated as the square width DB is gradually reduced from its largest size, the width of the data trace, down to its minimum size, the spacing between consecutive data points DBR . The fractal dimension is then given by [3], 2 log NsDBd DF 5 lim ]]]]. log DB DB →DB R
(1)
In practice, DF is obtained by a linear fit to a plot of 2log N(DB) vs. log DB. A modified box counting technique known as the variation method [13] has been used in this work and is discussed in detail elsewhere [14]. Firstly we will deal with the effect of temperature on the fractal MCF. At zero temperature we have infinite phase coherence and, induced by the soft-wall potential profile, a power-law areal distribution which generates the fractal behaviour [7]. Raising the temperature reduces the electron phase coherence length lf and restricts the trajectories which can participate in quantum interference Table 1 Values for lithographic billiard width w, depleted billiard width w d and area A d , and energy level spacing DE based on the assumption of a 50–100 nm electrostatic depletion edge around the gates Lithographic width w
Depleted width w d
Billiard area A d
Energy level spacing DE
mm mm mm mm
0.3 mm 0.5 mm 0.85 mm 1.8 mm
9310 214 m 2 2.5310 213 m 2 7.2310 213 m 2 3.2310 212 m 2
80 meV 29 meV 10 meV 2.3 meV
0.4 0.6 1.0 2.0
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A.P. Micolich et al. / Microelectronic Engineering 51 – 52 (2000) 241 – 247
processes to those with length l , lf . In particular, contributions to the MCF from the long trajectory, large area loops are reduced relative to the short trajectory, small area loops. Therefore, considering the Aharonov–Bohm relation DB 5 h /eA [15], the smaller period (larger area) MCF components will be damped relative to large period (smaller area) MCF components when the phase coherence length is reduced. This is clearly observed in the MCF data of Fig. 1a and b as the temperature is increased. There are two possibilities for how the temperature-induced redistribution of trajectory area contributions affects the fractal behaviour. If the redistribution of contributing loops causes deviations from a power-law, fractal behaviour will no longer be observed. This may occur over the full range of the fractal behaviour, or only in the vicinity of the upper and lower limits which reduces the range over which fractal behaviour is observed. However, if the redistribution results in a new power law (i.e. with a different exponent) then the fractal nature of the MCF is retained and a different DF value will be apparent. Results of our analysis, shown in Fig. 2b, demonstrate that the fractal nature of the MCF is preserved and that the fractal limits do not shift. However, DF changes from 1.47 to 1.35 as T is raised from 0.22 K to 2.1 K. To examine this behaviour further we look at how electron phase coherence time tf varies with
Fig. 2. (a) Phase coherence time tf versus temperature for the 0.6- and 1.0-mm billiards. tf increases with decreasing temperature and saturates once the discrete energy level structure of the billiard becomes resolved. (b) DF versus temperature for the same data sets displays a remarkably similar trend, with DF also saturating at T Q .
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temperature. Fig. 2a presents the temperature dependence of tf extracted from the MCF using an autocorrelation field analysis [16]. As the temperature is increased, tf decreases and by extrapolation of this trend we find that at temperatures |10 K the phase coherence length will become significantly smaller than the billiard width. Hence, trajectories traversing the billiard will no longer remain phase coherent and a transition to classical behaviour will occur. This is evident in Fig. 2b where we observe a steady decrease in DF as quantum interference is removed. Following this decrease we may expect a fractal dimension of one and non-fractal MCF in the high temperature limit. Billiard experiments at temperatures between 1 and |10 K are planned to investigate this further. As T is decreased, tf increases and saturates at 420 mK and 125 mK for the 0.6 and 1 mm billiards, respectively. This saturation occurs once the thermal smearing k B T is sufficiently reduced that the discrete energy level structure of the billiard, arising from quantum confinement of the electrons, is resolved (ie. k B T ,DE, where DE is the energy level spacing of the billiard) [17,18]. The temperature T Q where this occurs is given by [17,18]: DE 1 2p " 2 ] ] ]]. TQ 5 5 kB k B m*A d
(2)
The saturation temperatures we observe are very close to those calculated using Eq. (2) for the billiards investigated. From the saturation of tf we can determine the temperature at which the crossover from semiclassical to quantum behaviour occurs (i.e. T Q ). This is accompanied by a similar saturation in DF , with the saturated DF value depending on billiard size as shown in Fig. 2b. For temperatures in the saturation regime, reducing the size of the billiard leads to an increase in the energy level spacing DE. This further enhances the discrete nature of the billiard energy levels causing a transition to a more periodic character in the MCF, as shown in Fig. 3. In this case, as with increasing temperature, there are two possible effects that might be expected due to size-induced
Fig. 3. MCF for 0.4, 0.6, 1.0 and 2.0 mm billiards, each with the geometry shown in Fig. 1 (inset). All sets are at base temperature (|30 mK). The MCF takes on a more periodic appearance as billiard width w d is reduced and the spacing between discrete energy levels DE in the billiard is increased.
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Fig. 4. Fractal dimension measured for the sets displayed in Fig. 3 as a function of temperature. DF clearly decreases with decreasing billiard width w d .
changes in the MCF — the suppression of fractal behaviour due to deviations from power-law behaviour or modifications to the power law distribution which induce a change in fractal dimension. In Fig. 4 we observe that the fractal behaviour is maintained but that the fractal dimension decreases as billiard size is reduced. Extrapolating this trend, a fractal dimension of one should occur at a billiard width of |0.2 mm. However, due to the spatial separation between the 2DEG and the surface gates in these billiards, fringing fields lead to depleted 2DEG regions which do not closely match the surface-gate pattern. This makes surface-gate billiards smaller than 0.3 mm difficult to accurately define. We are exploring other possible techniques such as in-plane gates [19,20] to define billiards in high electron mobility environments.
References [1] C.W.J. Beenakker, H. van Houten, Quantum transport in semiconductor nanostructures, Solid State Phys. 44 (1991) 1. [2] C.M. Marcus, A.J. Rimberg, R.M. Westervelt, P.F. Hopkins, A.C. Gossard, Conductance fluctuations and chaotic scattering in ballistic microstructures, Phys. Rev. Lett. 69 (1992) 506. [3] A.P. Micolich, R.P. Taylor, R. Newbury, J.P. Bird, R. Wirtz, C.P. Dettmann, Y. Aoyagi, T. Sugano, Geometry-induced fractal behaviour in a semiconductor billiard, J. Phys. Condens. Matter 10 (1998) 1339. [4] A.S. Sachrajda, R. Ketzmerick, C. Gould, Y. Feng, P.J. Kelly, A. Delage, Z. Wasilewski, Fractal conductance fluctuations in a soft-wall stadium and a Sinai billiard, Phys. Rev. Lett. 80 (1998) 1948. [5] R.P. Taylor, The role of surface-gate technology for AlGaAs / GaAs nanostructures, Nanotechnology 5 (1994) 183. [6] T.M. Fromhold, C.R. Tench, R.P. Taylor, A.P. Micolich, R. Newbury, Self-similar conductance fluctuations in a Sinai billiard with a mixed chaotic phase space, Physica B 249–251 (1998) 334. [7] R. Ketzmerick, Fractal conductance fluctuations in generic chaotic cavities, Phys. Rev. B 54 (1996) 10841. [8] R.P. Taylor, A.P. Micolich, R. Newbury, J.P. Bird, T.M. Fromhold, J. Cooper, Y. Aoyagi, T. Sugano, Exact and statistical self-similarity in magnetoconductance fluctuations: a unified picture, Phys. Rev. B 58 (1998) 11107. [9] B.B. Mandelbrot, The Fractal Geometry of Nature, Freeman, San Francisco, 1982.
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[10] D. Avnir, O. Biham, D. Lidar, O. Malcai, Is the geometry of nature fractal?, Science 279 (1998) 39. [11] H. Hegger, B. Huckestein, K. Hecker, M. Janssen, A. Freimuth, G. Reckziegel, R. Tuzinski, Fractal conductance fluctuations in gold nanowires, Phys. Rev. Lett. 77 (1996) 3885. ´ [12] J.P. Bird, M. Stopa, R.P. Taylor, R. Newbury, Y. Aoyagi, T. Sugano, Aharonov–Bohm oscillations in quantum dots: precise departures from h / e periodicity, Superlatt. Microstruct. 22 (1997) 57. [13] B. Dubuc, J.F. Quiniou, C. Roques-Carmes, C. Tricot, S.W. Zucker, Evaluating the fractal dimension of profiles, Phys. Rev. A 39 (1989) 1500. [14] A.P. Micolich, R.P. Taylor, R. Newbury, J.P. Bird, Y. Aoyagi, T. Sugano, Temperature dependent fractal dimension of magneto-conductance fluctuations in semiconductor billiards, Superlatt. Microstruct. 25 (1999) 157. [15] Y. Aharonov, D. Bohm, Significance of electromagnetic potentials in the quantum theory, Phys. Rev. 115 (1959) 485. [16] J.P. Bird, K. Ishibashi, D.K. Ferry, R. Newbury, D.M. Olatona, Y. Ochiai, Y. Aoyagi, T. Sugano, Phase breaking in ballistic quantum dots: a correlation field analysis, Surf. Sci. 361–362 (1996) 730. [17] J.P. Bird, H. Linke, J. Cooper, A.P. Micolich, D.K. Ferry, R. Akis, Y. Ochiai, R.P. Taylor, R. Newbury, P. Omling, Y. Aoyagi, T. Sugano, Phase breaking as a probe of the intrinsic level spectrum of open quantum dots, Phys. Stat. Sol. (b) 204 (1997) 314. [18] R.M. Clarke, I.H. Chan, C.M. Marcus, C.I. Duruoz, J.S. Harris Jr., K. Campman, A.C. Gossard, Temperature dependence of phase breaking in ballistic quantum dots, Phys. Rev. B 52 (1995) 2656. [19] K.M. Connolly, D.P. Pivin Jr., D.K. Ferry, H.H. Wieder, Conductance fluctuations in a ballistic in-plane gated InGaAs quantum dot, Superlatt. Microstruct. 20 (1996) 96. [20] H. Pothier, J. Weis, R.J. Haug, K. Von Klitzing, K. Ploog, Realization of an in-plane-gate single-electron transistor, Appl. Phys. Lett. 62 (1993) 3174.
Temperature and size dependence of fractal MCF in semiconductor billiards a, a,e b a c c A.P. Micolich *, R.P. Taylor , J.P. Bird , R. Newbury , T.M. Fromhold , C.R. Tench , a d d H. Linke , Y. Aoyagi , T. Sugano a
School of Physics, University of New South Wales, Sydney, NSW, 2052, Australia Center for Solid State Research, Arizona State University, Tempe, AZ 85287 -6206, USA c School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2 RD, UK d Nanoelectronics Materials Laboratory, RIKEN, 2 -1 Hirosawa, Wako-shi, Saitama 351 -01, Japan e Physics Department, University of Oregon, Eugene, OR 97403 -1274, USA b
Abstract Fractal behaviour in the magneto-conductance fluctuations of mesoscopic systems has been predicted using semiclassical theory and observed experimentally in semiconductor billiards. Surface-gate billiards provide an ideal environment for investigations of fractal behaviour due to the ease with which parameters in the experimental system can be tuned. In this study we vary temperature and billiard size to investigate the classical and quantum limits of the semiclassical theory for fractal magneto-conductance fluctuations. 2000 Elsevier Science B.V. All rights reserved. Keywords: Fractal conductance fluctuations; Mesoscopic semiconductor billiards
Modern semiconductor growth and fabrication technologies have allowed extensive investigations of low-dimensional electron systems in recent decades [1]. Lately, considerable interest has focused on semiconductor billiards, where material-induced length scales, such as the electron mean free path and the inelastic scattering length, exceed the electron confinement length and, as a consequence, phase-coherent ballistic transport effects are observed [1–4]. Sub-micron billiards can be defined in the two-dimensional electron gas (2DEG) of an AlGaAs / GaAs heterostructure using patterned metal gates deposited on the heterostructure surface. Application of a negative bias to these surface-gates leads to electrostatic depletion of the 2DEG in the regions directly below the gates and the formation of a billiard whose geometry closely resembles that of the patterned surface-gates [5]. At low temperatures, reproducible fluctuations in billiard conductance, due to electron quantum interference effects, are observed as a function of magnetic field [2–4]. These magneto-conductance fluctuations (MCF) provide an important probe of electron transport through the billiard. *Corresponding author. Fax: 161-2-9385-6060. E-mail address: [email protected] (A.P. Micolich) 0167-9317 / 00 / $ – see front matter PII: S0167-9317( 99 )00484-0
2000 Elsevier Science B.V. All rights reserved.
242
A.P. Micolich et al. / Microelectronic Engineering 51 – 52 (2000) 241 – 247
Whilst traditionally it has been assumed that the confining potential of semiconductor billiards may be adequately approximated by an infinite square-well potential profile, recent studies of more realistic ‘soft-wall’ billiard potentials have received significant attention. These ‘soft-wall’ potentials have been shown to generate a mixed phase-space, containing regions of both chaotic and stable (i.e. non-chaotic) trajectories [6]. It has been speculated that the boundary between chaotic and non-chaotic regions generates a power-law distribution of the area A enclosed by electron trajectories [7], in contrast to the exponential distribution expected for a purely chaotic phase-space. A semiclassical theory [7], which monitors the phase accumulated by electron waves travelling along the classical trajectories, predicts that this power-law area distribution generates fractal MCF in billiard electron transport experiments. One of the key properties of fractal MCF is self-similarity. In the MCF of square billiards statistical self-similarity is observed, where the statistical properties of the fluctuations are the same at different magnetic field scales [8]. The consequence of fluctuations at different scales is that the data trace occupies more space than a smooth one-dimensional line but less space than a two-dimensional plane. The fractal dimension DF quantifies this space-filling property [9]. Hence fractal MCF satisfy 1 , DF , 2 [3]. Another important factor is the highest and lowest magnifications over which fractal behaviour can be observed. Whilst mathematically-generated fractals (e.g. the Mandelbrot Set) occur over an infinite range of magnifications, fractals in physical systems have an upper and lower limit beyond which fractal behaviour is no longer observed (e.g. a coastline where the upper limit is the length of the coastline and the lower limit the scale of the grains of sand comprising the coastline) [3,9,10]. The presence of fractal behaviour in the conductance fluctuations of mesoscopic systems is now well-established following studies of quasi-ballistic gold nanowires [11] and ballistic semiconductor billiards [3,4]. Unlike many natural systems, the advantage of studying fractals in semiconductor billiards is the ease with which we can tune various parameters in the experimental system and examine how this effects the fractal behaviour. In this study we focus on temperature and billiard size. These two parameters allow us to investigate the classical and quantum limits of the semiclassical theory for fractal conductance fluctuations [7]. Experiments were performed using billiards defined by the gate geometry shown in Fig. 1 (inset). Four different sized square billiards (w 3 w) were used in this study with lithographic widths w 5 0.4, 0.6, 1.0 and 2.0 mm. In all cases the billiards are significantly smaller than the electronic mean free path of the material ( . 5 mm), ensuring ballistic transport within each billiard. Typical carrier concentrations (hs ) and mobilities ( m ) for these billiards are 4–5 3 10 15 m 2 and 20–70 m 2 / V s respectively giving a Fermi wavelength lF | 40 nm. An electrostatic depletion of 50–100 nm around each gate edge is calculated (based on Aharonov–Bohm oscillations observed at high magnetic fields [12]) leading to billiard width w d , area A d and energy level spacing DE as shown in Table 1. Experiments were performed with samples thermally linked to the mixing chamber of a dilution refrigerator allowing electron temperatures ranging from 4.2 K down to | 30 mK. Electrical measurements were performed using low frequency AC lock-in techniques in a four-terminal configuration to eliminate lead resistance from measurements. All measurements were made with low sample current (Iex , 10 nA) to avoid electron heating. The temperature dependence studies were performed on a single thermal cycle. The fractal dimension of the MCF data is analysed using a box counting technique [3,9]. We begin by superimposing a mesh of squares of width DB 3 DB over the data. For a particular DB we then count the minimum number of boxes N(DB) required to completely cover the trace [3,9]. This count is
A.P. Micolich et al. / Microelectronic Engineering 51 – 52 (2000) 241 – 247
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Fig. 1. (a) Magneto-conductance fluctuations (MCF) obtained from a 0.6-mm billiard with temperatures (from top): 2.1 K, 1.3 K, 900 mK, 220 mK and (b) (from top) 4.2 K, 1.4 K, 480 mK, 80 mK for the 1-mm billiard. Inset is a scanning electron micrograph of the square billiard geometry used throughout this investigation.
repeated as the square width DB is gradually reduced from its largest size, the width of the data trace, down to its minimum size, the spacing between consecutive data points DBR . The fractal dimension is then given by [3], 2 log NsDBd DF 5 lim ]]]]. log DB DB →DB R
(1)
In practice, DF is obtained by a linear fit to a plot of 2log N(DB) vs. log DB. A modified box counting technique known as the variation method [13] has been used in this work and is discussed in detail elsewhere [14]. Firstly we will deal with the effect of temperature on the fractal MCF. At zero temperature we have infinite phase coherence and, induced by the soft-wall potential profile, a power-law areal distribution which generates the fractal behaviour [7]. Raising the temperature reduces the electron phase coherence length lf and restricts the trajectories which can participate in quantum interference Table 1 Values for lithographic billiard width w, depleted billiard width w d and area A d , and energy level spacing DE based on the assumption of a 50–100 nm electrostatic depletion edge around the gates Lithographic width w
Depleted width w d
Billiard area A d
Energy level spacing DE
mm mm mm mm
0.3 mm 0.5 mm 0.85 mm 1.8 mm
9310 214 m 2 2.5310 213 m 2 7.2310 213 m 2 3.2310 212 m 2
80 meV 29 meV 10 meV 2.3 meV
0.4 0.6 1.0 2.0
244
A.P. Micolich et al. / Microelectronic Engineering 51 – 52 (2000) 241 – 247
processes to those with length l , lf . In particular, contributions to the MCF from the long trajectory, large area loops are reduced relative to the short trajectory, small area loops. Therefore, considering the Aharonov–Bohm relation DB 5 h /eA [15], the smaller period (larger area) MCF components will be damped relative to large period (smaller area) MCF components when the phase coherence length is reduced. This is clearly observed in the MCF data of Fig. 1a and b as the temperature is increased. There are two possibilities for how the temperature-induced redistribution of trajectory area contributions affects the fractal behaviour. If the redistribution of contributing loops causes deviations from a power-law, fractal behaviour will no longer be observed. This may occur over the full range of the fractal behaviour, or only in the vicinity of the upper and lower limits which reduces the range over which fractal behaviour is observed. However, if the redistribution results in a new power law (i.e. with a different exponent) then the fractal nature of the MCF is retained and a different DF value will be apparent. Results of our analysis, shown in Fig. 2b, demonstrate that the fractal nature of the MCF is preserved and that the fractal limits do not shift. However, DF changes from 1.47 to 1.35 as T is raised from 0.22 K to 2.1 K. To examine this behaviour further we look at how electron phase coherence time tf varies with
Fig. 2. (a) Phase coherence time tf versus temperature for the 0.6- and 1.0-mm billiards. tf increases with decreasing temperature and saturates once the discrete energy level structure of the billiard becomes resolved. (b) DF versus temperature for the same data sets displays a remarkably similar trend, with DF also saturating at T Q .
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245
temperature. Fig. 2a presents the temperature dependence of tf extracted from the MCF using an autocorrelation field analysis [16]. As the temperature is increased, tf decreases and by extrapolation of this trend we find that at temperatures |10 K the phase coherence length will become significantly smaller than the billiard width. Hence, trajectories traversing the billiard will no longer remain phase coherent and a transition to classical behaviour will occur. This is evident in Fig. 2b where we observe a steady decrease in DF as quantum interference is removed. Following this decrease we may expect a fractal dimension of one and non-fractal MCF in the high temperature limit. Billiard experiments at temperatures between 1 and |10 K are planned to investigate this further. As T is decreased, tf increases and saturates at 420 mK and 125 mK for the 0.6 and 1 mm billiards, respectively. This saturation occurs once the thermal smearing k B T is sufficiently reduced that the discrete energy level structure of the billiard, arising from quantum confinement of the electrons, is resolved (ie. k B T ,DE, where DE is the energy level spacing of the billiard) [17,18]. The temperature T Q where this occurs is given by [17,18]: DE 1 2p " 2 ] ] ]]. TQ 5 5 kB k B m*A d
(2)
The saturation temperatures we observe are very close to those calculated using Eq. (2) for the billiards investigated. From the saturation of tf we can determine the temperature at which the crossover from semiclassical to quantum behaviour occurs (i.e. T Q ). This is accompanied by a similar saturation in DF , with the saturated DF value depending on billiard size as shown in Fig. 2b. For temperatures in the saturation regime, reducing the size of the billiard leads to an increase in the energy level spacing DE. This further enhances the discrete nature of the billiard energy levels causing a transition to a more periodic character in the MCF, as shown in Fig. 3. In this case, as with increasing temperature, there are two possible effects that might be expected due to size-induced
Fig. 3. MCF for 0.4, 0.6, 1.0 and 2.0 mm billiards, each with the geometry shown in Fig. 1 (inset). All sets are at base temperature (|30 mK). The MCF takes on a more periodic appearance as billiard width w d is reduced and the spacing between discrete energy levels DE in the billiard is increased.
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A.P. Micolich et al. / Microelectronic Engineering 51 – 52 (2000) 241 – 247
Fig. 4. Fractal dimension measured for the sets displayed in Fig. 3 as a function of temperature. DF clearly decreases with decreasing billiard width w d .
changes in the MCF — the suppression of fractal behaviour due to deviations from power-law behaviour or modifications to the power law distribution which induce a change in fractal dimension. In Fig. 4 we observe that the fractal behaviour is maintained but that the fractal dimension decreases as billiard size is reduced. Extrapolating this trend, a fractal dimension of one should occur at a billiard width of |0.2 mm. However, due to the spatial separation between the 2DEG and the surface gates in these billiards, fringing fields lead to depleted 2DEG regions which do not closely match the surface-gate pattern. This makes surface-gate billiards smaller than 0.3 mm difficult to accurately define. We are exploring other possible techniques such as in-plane gates [19,20] to define billiards in high electron mobility environments.
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