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Ballistic electron transport beyond 100 microm IN 2D electron systems
Surface Science 228 (1990) 283-285 North-Holland
BALLISTIC
ELECTRON
J. SPECTOR,
Received
TRANSPORT
H.L. STORMER,
AT& T Bell Laboratories,
283
BEYOND
K.W. BALDWIN,
100 pm IN 2D ELECTRON
L.N. PFEIFFER
SYSTEMS
and K.W. WEST
Murray Hill, NJ 07974, USA
2 June 1989: accepted
for publications
28 September
1989
We have observed ballistic electron transport beyond 100 pm in the 2D electron system of an ultra-high mobility GaA-(AlGa)As heterostructure employing a magnetic electron focusing technique through wide point contacts. The amplitude of the characteristic magneto-oscillations is found to depend exponentially on electron propagation distance with a decay length h, = 15 pm. X, differs by about a factor of 2 from the mobility mean free path of 28 pm. This may indicate the determination of a different angular average of scattering events in focusing experiments as compared to standard mobility measurements.
In recent experiments, van Houten et al. [ll] have demonstrated coherent electron focusing in a two-dimensional electron system. The electrons were injected by a point contact (emitter and refocused by a perpendicular magnetic field at a second point contact (collector) separated 3 pm from the emitter. They have shown that the injected electrons are specularly reflected from the boundary between the emitter and collector. As discussed by Beenakker et al. [2], the magnetic field B acts to focus an angular distribution of injected electrons along caustics which have points of intersection with the boundary joining the emitter and collector separated by integral multiples of the cyclotron diameter dcyc = 2Ak,/eB (kF is the Fermi wave vector). This results in the collector voltage having a period dependence in B with period 2Ak,/ed, where d is the distance between the emitter and collector. These studies have investigated the properties of coherent electron focusing with point contacts narrow enough to be in the quantum regime and emitter-collector distances of 3 pm or less. We have employed this method to study ballistic electron propagation over macroscopic emitter-collector distances using rather wide point contacts that were not in the quantum regime. Our recent advances in molecular beam epitaxial growth techniques have resulted in 2D electron systems with 0039-6028/90/$03.50 (North-Holland)
0 Elsevier Science Publishers
B.V.
mobilities of a few times lo6 cm2/V . s. These 2D electron systems have allowed us to study ballistic electron transport at distances up to 100 pm. Using standard photolithographic techniques, Ti-Au gates were patterned into focusing arrays with emitter-collector distances from 1 to 64 pm, and geometrical point contact openings of 1 pm (fig. 1). This pattern was deposited onto a single interface Al,,,Ga,.,, As/GaAs heterostructure
Fig. 1. Gate and contact pattern for electron focusing experiments. I, represents the emitter current source and V, represents the collector voltage. The specific configuration indicated was used for the 32 pm emitter-collector measurements.
which was &doped with Si in the AlGaAs region a distance of 700 A from the interface. The overall distance between the 2D electron system and the surface was 5000 A. A mobility of 5.5 X 10’ cm’/V s and an electron density n of 1.1 X 10” cm ’ were determined in a separate specimen from the same wafer under similar conditions. For the focusing experiments a negative bias voltage was applied to the gates so as to just deplete electrons below the gates. while leaving the point contact openings close to 1 pm. With a Fermi wavelength X,. = 2r/k,. = 0.08 pm, the point contact width exceeded h, by about a factor of 10. Coherent electron focusing spectra were measured at a temperature of 0.3 K in a four point Iongitudinal resistance configuration. Fig. 1 shows specifically the configuration for an emittercollector distance of 32 pm. The arrows indicate the positions of the point contacts. Standard four point AC lock-in techniques were employed. The results coincided with data from DC measurements taken in the same configuration. An emitter current (I,) of 100 nA was drawn between contacts 1 and 4 while the collector voltage (V,) was measured between contacts 2 and 3. At such current leveis the transport remained well within the linear regime. The point contact resistance were approximately 500 Q implying that the electrons were injected at an energy 50 PeV above the Fermi energy. The magnetic field B was applied perpetidicular to the 2D electron system in a direction so as to bend the trajectories of the in jcctod clcctrons toward the collector. Fig. 2 shows a composite of the measured focusing voltage as a function of B with emitter collector distances d from 4 to 64 pm. The vertical scale of each curve has been adjusted so as to enable comparison of the periods. The spectra for (1 less than 4 pm was not well resolved because the point contact openings were comparable to the emitter-collector distance. Note that for a given d. at peak number p, the average path length an electron propagates before hitting the collector is equal to the number of semicircular orbits times the arc length of each orbit = p( m/2)(d/p) = (IT/~. which is independent of p. Hence, the fact that the oscillations can be seen with an emitter collector distance of 64 pm implies that ballistic
electrons can be detected in our 2D system after propagating 57/2 x 64 pm = 100 pm. The period A,% of the oscillations in B is related to d, via the relation AH = hk,-/ed. The oscillations in nearly all of the curves die out beyond about 700 gauss. This is roughly equal to the field at which the cyclotron diameter is equal to the width of the contacts. at which point all electrons strike the collector and the oscillations are expected to vanish. It is not clear why the 64 pm data die out at considerably lower magnetic field. It may be that additional oscillations are buried in the noise or that there is a remnant diffuse component to the reflections causing the focusing peaks to become washed out after a sufficiently large number of reflections. As has previously been observed [3]. the focusing voltage assumes negative values over some intervals of B, probably caused by an inversion ot carrier collection efficiency between the point contact collector and the large ohmic contact #3 in fig. 1. Apart from the variation in periodicity for different d. a precipitous drop in the amplitude of the oscillations with increasing d is the most striking aspect of the data of fig. 2. Although we have not developed a detailed understanding of the shape of the oscillations and their envelopes we are able to adopt a simple measure for the strength of the oscillations by selecting their peak-to-peak amplitude. In fig. 3 we show the strength of the oscillations versus d on a semi-logarithmic plot. The line is a least square fit to an exponential of
J. Spector et al. / Electron transport in 20 electron systems
1 /e
LENGTH = 9 i-pm
m-40~
70 d(pm)
Fig. 3. Maximum peak amplitude of the magneto-oscillations, normalized to the emitter voltage, as a function of emitter-collector distance d.
the form ee““, with A, = 10 pm. The data fit extremely well to such an exponential dependence over almost three orders of magnitude. The uncertainties in the determination of the amplitudes of the oscillations are sufficiently small as to not significantly affect the fit. A, = 10 pm implies an arc decay length A, = (7r/2)A, = 15 pm. The mean free path as deduced from mobility measurements the Fermi velocity A, = rvr where r = m*p/e, u r= Ak,/m” and m* = the effective mass of the electrons. For our sample this yields A, = 28 pm, a factor of 2 greater than A,. It is not unreasonable that A, is less than A,, because A,, is the path length after which an electron has lost all its momentum in the direction of its original trajectory. This may require as few as one large angle scattering event or many small angle scattering events [4,5]. In the electron focusing experiment
285
on the other hand all scattering processes may be equally effective in diverting a carrier from its trajectory towards the distant collector. In this way, A, represents a different type of angular averaging than does A, from traditional mobility measurements. In conclusion, we have shown that in sufficiently clean 2D electron systems ballistic electrons can be propagated over distances greater than 100 pm. The decay of the amplitude of the oscillations in the electron focusing spectra follows a strict exponential dependence with propagation distance which is a factor of two less than the mobility mean free path. We would like to acknowledge helpful discussions with J.P. Eisenstein, P.B. Littlewood and H.U. Baranger.
References 111H. van
Houten, B.J. van Wees, J.E. Mooij, C.W.J. Beenakker, J.G. Williamson and C.T. Foxon, Europhys. Lett. 5 (1988) 721. 121C.W.J. Beenakker. H. van Houten and B.J. van Wees, Festkorperprobleme, Vol. 29 of Advances in Solid State Physics, Ed. U. Rossler (Pergamon/Vieweg, Braunschweig, 1989). J.G. Williams, M.E.I. [31 H. van Houten, C.W.J. Beenakker, Broekaart, P.H.M. van Loosdrecht, B.J. van Wees, J.E. Mooij, C.T. Foxon and J.J. Harris, Phys. Rev. B 39 (1989) 8556. [41 S. Das Sarma and Frank Stern. Phys. Rev. B 32 (1985) 8442. R.J. Higgins, R.K. Goodall, P.R. Jay, M. [51 J.P. Harrang, Laviron and P. Delescluse, Phys. Rev. B 32 (1985) 8126.
BALLISTIC
ELECTRON
J. SPECTOR,
Received
TRANSPORT
H.L. STORMER,
AT& T Bell Laboratories,
283
BEYOND
K.W. BALDWIN,
100 pm IN 2D ELECTRON
L.N. PFEIFFER
SYSTEMS
and K.W. WEST
Murray Hill, NJ 07974, USA
2 June 1989: accepted
for publications
28 September
1989
We have observed ballistic electron transport beyond 100 pm in the 2D electron system of an ultra-high mobility GaA-(AlGa)As heterostructure employing a magnetic electron focusing technique through wide point contacts. The amplitude of the characteristic magneto-oscillations is found to depend exponentially on electron propagation distance with a decay length h, = 15 pm. X, differs by about a factor of 2 from the mobility mean free path of 28 pm. This may indicate the determination of a different angular average of scattering events in focusing experiments as compared to standard mobility measurements.
In recent experiments, van Houten et al. [ll] have demonstrated coherent electron focusing in a two-dimensional electron system. The electrons were injected by a point contact (emitter and refocused by a perpendicular magnetic field at a second point contact (collector) separated 3 pm from the emitter. They have shown that the injected electrons are specularly reflected from the boundary between the emitter and collector. As discussed by Beenakker et al. [2], the magnetic field B acts to focus an angular distribution of injected electrons along caustics which have points of intersection with the boundary joining the emitter and collector separated by integral multiples of the cyclotron diameter dcyc = 2Ak,/eB (kF is the Fermi wave vector). This results in the collector voltage having a period dependence in B with period 2Ak,/ed, where d is the distance between the emitter and collector. These studies have investigated the properties of coherent electron focusing with point contacts narrow enough to be in the quantum regime and emitter-collector distances of 3 pm or less. We have employed this method to study ballistic electron propagation over macroscopic emitter-collector distances using rather wide point contacts that were not in the quantum regime. Our recent advances in molecular beam epitaxial growth techniques have resulted in 2D electron systems with 0039-6028/90/$03.50 (North-Holland)
0 Elsevier Science Publishers
B.V.
mobilities of a few times lo6 cm2/V . s. These 2D electron systems have allowed us to study ballistic electron transport at distances up to 100 pm. Using standard photolithographic techniques, Ti-Au gates were patterned into focusing arrays with emitter-collector distances from 1 to 64 pm, and geometrical point contact openings of 1 pm (fig. 1). This pattern was deposited onto a single interface Al,,,Ga,.,, As/GaAs heterostructure
Fig. 1. Gate and contact pattern for electron focusing experiments. I, represents the emitter current source and V, represents the collector voltage. The specific configuration indicated was used for the 32 pm emitter-collector measurements.
which was &doped with Si in the AlGaAs region a distance of 700 A from the interface. The overall distance between the 2D electron system and the surface was 5000 A. A mobility of 5.5 X 10’ cm’/V s and an electron density n of 1.1 X 10” cm ’ were determined in a separate specimen from the same wafer under similar conditions. For the focusing experiments a negative bias voltage was applied to the gates so as to just deplete electrons below the gates. while leaving the point contact openings close to 1 pm. With a Fermi wavelength X,. = 2r/k,. = 0.08 pm, the point contact width exceeded h, by about a factor of 10. Coherent electron focusing spectra were measured at a temperature of 0.3 K in a four point Iongitudinal resistance configuration. Fig. 1 shows specifically the configuration for an emittercollector distance of 32 pm. The arrows indicate the positions of the point contacts. Standard four point AC lock-in techniques were employed. The results coincided with data from DC measurements taken in the same configuration. An emitter current (I,) of 100 nA was drawn between contacts 1 and 4 while the collector voltage (V,) was measured between contacts 2 and 3. At such current leveis the transport remained well within the linear regime. The point contact resistance were approximately 500 Q implying that the electrons were injected at an energy 50 PeV above the Fermi energy. The magnetic field B was applied perpetidicular to the 2D electron system in a direction so as to bend the trajectories of the in jcctod clcctrons toward the collector. Fig. 2 shows a composite of the measured focusing voltage as a function of B with emitter collector distances d from 4 to 64 pm. The vertical scale of each curve has been adjusted so as to enable comparison of the periods. The spectra for (1 less than 4 pm was not well resolved because the point contact openings were comparable to the emitter-collector distance. Note that for a given d. at peak number p, the average path length an electron propagates before hitting the collector is equal to the number of semicircular orbits times the arc length of each orbit = p( m/2)(d/p) = (IT/~. which is independent of p. Hence, the fact that the oscillations can be seen with an emitter collector distance of 64 pm implies that ballistic
electrons can be detected in our 2D system after propagating 57/2 x 64 pm = 100 pm. The period A,% of the oscillations in B is related to d, via the relation AH = hk,-/ed. The oscillations in nearly all of the curves die out beyond about 700 gauss. This is roughly equal to the field at which the cyclotron diameter is equal to the width of the contacts. at which point all electrons strike the collector and the oscillations are expected to vanish. It is not clear why the 64 pm data die out at considerably lower magnetic field. It may be that additional oscillations are buried in the noise or that there is a remnant diffuse component to the reflections causing the focusing peaks to become washed out after a sufficiently large number of reflections. As has previously been observed [3]. the focusing voltage assumes negative values over some intervals of B, probably caused by an inversion ot carrier collection efficiency between the point contact collector and the large ohmic contact #3 in fig. 1. Apart from the variation in periodicity for different d. a precipitous drop in the amplitude of the oscillations with increasing d is the most striking aspect of the data of fig. 2. Although we have not developed a detailed understanding of the shape of the oscillations and their envelopes we are able to adopt a simple measure for the strength of the oscillations by selecting their peak-to-peak amplitude. In fig. 3 we show the strength of the oscillations versus d on a semi-logarithmic plot. The line is a least square fit to an exponential of
J. Spector et al. / Electron transport in 20 electron systems
1 /e
LENGTH = 9 i-pm
m-40~
70 d(pm)
Fig. 3. Maximum peak amplitude of the magneto-oscillations, normalized to the emitter voltage, as a function of emitter-collector distance d.
the form ee““, with A, = 10 pm. The data fit extremely well to such an exponential dependence over almost three orders of magnitude. The uncertainties in the determination of the amplitudes of the oscillations are sufficiently small as to not significantly affect the fit. A, = 10 pm implies an arc decay length A, = (7r/2)A, = 15 pm. The mean free path as deduced from mobility measurements the Fermi velocity A, = rvr where r = m*p/e, u r= Ak,/m” and m* = the effective mass of the electrons. For our sample this yields A, = 28 pm, a factor of 2 greater than A,. It is not unreasonable that A, is less than A,, because A,, is the path length after which an electron has lost all its momentum in the direction of its original trajectory. This may require as few as one large angle scattering event or many small angle scattering events [4,5]. In the electron focusing experiment
285
on the other hand all scattering processes may be equally effective in diverting a carrier from its trajectory towards the distant collector. In this way, A, represents a different type of angular averaging than does A, from traditional mobility measurements. In conclusion, we have shown that in sufficiently clean 2D electron systems ballistic electrons can be propagated over distances greater than 100 pm. The decay of the amplitude of the oscillations in the electron focusing spectra follows a strict exponential dependence with propagation distance which is a factor of two less than the mobility mean free path. We would like to acknowledge helpful discussions with J.P. Eisenstein, P.B. Littlewood and H.U. Baranger.
References 111H. van
Houten, B.J. van Wees, J.E. Mooij, C.W.J. Beenakker, J.G. Williamson and C.T. Foxon, Europhys. Lett. 5 (1988) 721. 121C.W.J. Beenakker. H. van Houten and B.J. van Wees, Festkorperprobleme, Vol. 29 of Advances in Solid State Physics, Ed. U. Rossler (Pergamon/Vieweg, Braunschweig, 1989). J.G. Williams, M.E.I. [31 H. van Houten, C.W.J. Beenakker, Broekaart, P.H.M. van Loosdrecht, B.J. van Wees, J.E. Mooij, C.T. Foxon and J.J. Harris, Phys. Rev. B 39 (1989) 8556. [41 S. Das Sarma and Frank Stern. Phys. Rev. B 32 (1985) 8442. R.J. Higgins, R.K. Goodall, P.R. Jay, M. [51 J.P. Harrang, Laviron and P. Delescluse, Phys. Rev. B 32 (1985) 8126.