- Email: [email protected]
Magnetic field quenching of miniband conduction in quasi-one-dimensional superlattices
Physica B 272 (1999) 190}193
Magnetic "eld quenching of miniband conduction in quasi-one-dimensional superlattices L. Eaves!,*, H.M. Murphy!, A. Nogaret!,1, S.T. Stoddart!, P.C. Main!, M. Henini!, N. Mori", C. Hamaguchi", D.K. Maude#, J.-C. Portal# !School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, UK "Department of Electronic Engineering, Osaka University, Suita City, Osaka 565, Japan #LCMI-CNRS, 38042 Grenoble and INSA-CNRS, 31077 Toulouse, France
Abstract We investigate miniband conduction in GaAs/Al Ga As superlattices at low temperatures and at high magnetic x 1~x "elds up to 23 T. The "elds are applied perpendicular to the tunnel barriers. The current #owing through the sample is strongly suppressed by the magnetic "eld over the voltage range corresponding to the miniband conduction peak. This e!ect is related to the formation of quasi-one-dimensional Landau level minibands when the cyclotron energy exceeds the miniband width. ( 1999 Elsevier Science B.V. All rights reserved. Keywords: Superlattice; Miniband; Magneto-transport
It is now almost 30 years since the pioneering work of Esaki and Tsu [1] on electron transport through semiconductor superlattices. During the last decade, there have been a large number of new measurements which have investigated the electrical and photonic properties of this fundamental solid state structure. These include electrical transport experiments [2}5], optical four-wave mixing [6] and THz [7,8] measurements. DC miniband conduction in a superlattice is characterised by the presence of a peak and hence negative di!erential conductance (NDC) in the current}voltage characteristics, I(<). The form and
* Corresponding author. Fax: #44-115-951-5180. E-mail address: [email protected] (L. Eaves) 1 Present address: Department of Physics, University of Bath, Bath BA2 7AY, UK.
voltage position of the peak can be accurately described by a semi-classical Drude model [1]. The peak occurs when u q&1, where u "eFd/+, q is B B the scattering time, F is the electric "eld applied along the growth direction and d is the superlattice period. In this paper we investigate the e!ect of high magnetic "elds (up to 23 T) on DC miniband conduction in undoped superlattices at low temperatures. The "eld, B, is applied parallel to the superlattice growth direction and has the e!ect of suppressing the peak in I(<) due to the miniband. Landau quantisation of the electron motion in the plane of the wells acts to suppress the miniband conduction when the cyclotron energy, +u , exceeds the miniband width D. # Under these conditions, the localisation of the electrons in the plane e!ectively creates an array of 1D superlattices.
0921-4526/99/$ - see front matter ( 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 0 2 6 8 - 9
L. Eaves et al. / Physica B 272 (1999) 190}193
191
Fig. 1. Conduction band diagram for an undoped GaAs/ Al Ga As superlattice under zero applied external voltage. 0.4 0.6 The shaded regions of the diagram represent the minibands of the superlattice.
The superlattices were grown by molecular beam epitaxy on (1 0 0) n`-GaAs substrates and then fabricated into circular pillars of varying diameters (typically 5}100 lm). Each sample contained an N-period superlattice, separated from two heavily Si-doped GaAs contact electrodes by 10.2 nm wide undoped GaAs spacer layers. Each period consisted of an Al Ga As barrier of width b and a x 1~x GaAs well of width w, giving a period d"b#w. A series of di!erent structures was investigated. Here we focus on a device with the following parameters: N"19, b"2.08 nm, w"9.72 nm, d" 11.8 nm having a barrier composition corresponding to x"0.4. The period was con"rmed by X-ray di!raction. An e!ective mass calculation for this structure gives the energy width of the "rst and second minibands as D "12.1 and D "50.3 meV, 1 2 respectively, separated by a minigap E "98 meV. ' Fig. 1 shows a schematic band diagram for the device under zero applied voltage. The I(<) characteristics at ¹"4.2 K for "elds from B"0 to 23 T are shown in Fig. 2. At zero "eld a peak is observed at 95 mV, followed by a region of NDC. This is characteristic of miniband conduction. The small shoulder, observed at a slightly higher voltage than the main peak, is due to current oscillations associated with the NDC. They can be
Fig. 2. I(<) characteristics at ¹"4.2 K in various B up to 23 T for a 19-period undoped GaAs/(AlGa)As superlattice (for details, see text). The magnetic "eld is applied perpendicular to the quantum wells of the superlattice (B parallel to the current density, J). Also shown as a dashed line is an I(<) at ¹"200 K and B"23 T.
reduced by the addition of load resistors to the measurement circuit but are di$cult to eliminate. At the current peak, the potential drop across each period of the superlattice is 4.5 mV (allowing for some voltage being dropped across the spacer layers) which is much less than D . Hence, the 1 degree of Wannier}Stark localisation is small and a classical Drude model should be valid in this regime. At higher <, in line with earlier work [2,3] the current increases due to non-resonant processes. Increasing B at low ¹ strongly suppresses the miniband conduction peak; at 23 T, the peak value of current is 18 times smaller than at zero "eld. At the highest "eld of 23 T, the e!ect of increasing the sample temperature to ¹"200 K is to restore partially the miniband conduction and the amplitude of the current peak. The same qualitative behaviour has been seen in several devices of di!erent d, b, w and x. The I(<) characteristics of the superlattice in high magnetic "eld are quite di!erent from those of
192
L. Eaves et al. / Physica B 272 (1999) 190}193
Fig. 3. I(<) characteristics at ¹"4.2 K in various BEJ for a double-barrier structure. Al Ga As barriers of thickness 0.4 0.6 8.5 nm enclosed a 7.6 nm GaAs quantum well. The main electron tunnelling peak is labelled E1, and LO and LO are optic 0 1 phonon-assisted tunnelling satellites as discussed in the text.
double barrier resonant tunnelling devices. Fig. 3 shows a series of I(<) curves taken at a temperature of 4 K of an n}i}n double barrier diode, based on GaAs/(AlGa)As. The two (AlGa)As barriers, [Al]"40%, of thickness 8.5 nm, enclose a GaAs quantum well of thickness 7.6 nm. These layers are incorporated in an undoped (intrinsic"i) region between two n-doped GaAs layers with graded doping. The magnetic "eld was applied perpendicular to the plane of the barriers. It can be seen that the magnetic "eld has only a very weak e!ect on the principal electron resonant tunnelling peak, E1 } the amplitude of the peak in the current and its voltage position are only very weakly in#uenced by magnetic "eld. The "eld has a stronger e!ect on the optic phonon-assisted tunnelling satellite transition, LO , and associated lines such as LO , 0 1 which correspond to a change of Landau level index *n"1 as the electron tunnels from the emitter accumulation layer into the quantum well. Similar experiments on triple barrier RTDs again show
that the e!ect on the tunnel current of a magnetic "eld in this orientation is rather weak. We can therefore conclude that the observed strong suppression of the miniband conduction peak at high magnetic "elds and low temperatures is a speci"c property of the superlattice and of the miniband conduction process. To explain the suppression of the miniband conduction by the "eld, we need to consider the various scattering mechanisms in the superlattice. The simple Esaki}Tsu model [1] implicitly assumes the presence of inelastic scattering, speci"ed by a single collision time q. More complex models, which use Monte-Carlo techniques to solve the full threedimensional Boltzmann equation, take into account all of the main scattering mechanisms [9]. The e!ect of the magnetic "eld is to quantise the in-plane motion into Landau levels. This leads to a series of quasi-one-dimensional minibands, separated from each other by +u , as shown in Fig. 4. # We can neglect the e!ect of the upper miniband, since the minigap E is much greater than both D ' 1 and +u "40 meV at the top "eld of 23 T. As can # be seen from Fig. 4 there are two distinct regimes of magnetic "eld: when +u (D the quasi 1D # 1 minibands of di!erent n are nested and elastic scattering processes involving a change of n can occur, for example as shown by the line A to B in Fig. 4(a); for +u 'D the minibands become energetically # 1 decoupled and elastic transitions between the minibands are forbidden (Fig. 4(b)). Elastic transitions can arise from the presence of residual ionised impurities in the superlattice or from monolayer #uctuations in the quantum wells. Acoustic phonon scattering processes are strictly inelastic but since the width of the minizone is 2p/d, the energy involved in such processes is typically less than 2p+v/d&1 meV where v is the sound velocity of longitudinal acoustic phonons at the zone centre. This energy is small compared to both D and +u and such processes may be considered 1 # to be quasi-elastic. Hence, the dominant inelastic process is longitudinal optic (LO) phonon scattering, which consists only of emission processes at low temperatures. We can now explain the observed results within a semi-classical picture. The centre of the distribution of electrons, f (k ), within a miniband in k Z
L. Eaves et al. / Physica B 272 (1999) 190}193
Fig. 4. Schematic energy dispersion curves showing possible low-temperature electron transport mechanisms for a superlattice miniband in a magnetic "eld applied parallel to z: (a) for +u (D: arrows from A to B, B to C and C to D represent # impurity scattering, free #ight of the electron, and LO phonon emission, respectively (b) for +u 'D where the electron is # con"ned to the n"0 Landau level.
space will give the average displacement of the electrons at time t. At low ¹ and B when +u ( # D , k can be randomized as shown in Fig. 4(a). 1 Z An electron at k "0 accelerated by applied elecZ tric "eld can be elastically scattered into higher Landau levels. This shifts the centre of the f (k ) Z distribution to eFq/+, where q is an average scattering time, resulting in a non-zero drift velocity and hence conduction. The electron can subsequently relax via LO phonon emission (process C}D in Fig. 4(a)). At su$ciently high B, when +u 'D (see Fig. # 1 4(b)), k can no longer be randomized. The electron Z
193
undergoes elastic and quasi-elastic scattering processes within the n"0 miniband whose width, D , 1 is much less then the LO phonon energy, +u . LO Thus, at su$ciently high magnetic "elds and low temperatures, the electron is constrained to move and scatter between states of equal and opposite k within the n"0 miniband. The centre of the f (k ) Z distribution therefore remains at k "0 for any Z strength of applied electric "eld, leading to the strong suppression of miniband conduction under these conditions. At high temperature the miniband conduction peak is partially restored, we believe due to population of higher Landau level minibands and to the increase in e$ciency of acoustic phonon scattering as an inelastic process when k ¹&D . Electrons B 1 can consequently be inelastically scattered within a miniband. This once more allows the randomization of k shifting the centre of the f (k ) distribuZ Z tion to a non-zero value, resulting in conduction. This qualitative model is supported by a detailed Monte Carlo simulation which will be published elsewhere.
Acknowledgements This work is supported by the United Kingdom Engineering and Physical Sciences Research Council (EPSRC) and the European Union. L.E. is grateful for an advanced fellowship from the EPSRC.
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
L. Esaki, R. Tsu, IBM Res. Dev. 14 (1970) 61. A. Sibille et al., Phys. Rev. Lett. 64 (1990) 52. L. Canali et al., Phys. Rev. Lett. 76 (1996) 3618. H.J. Hutchinson et al., J. Appl. Phys. 75 (1994) 320. A. Sibille et al., Europhys. Lett. 13 (1990) 279. J. Feldmann et al., Phys. Rev. B 46 (1992) 7252. K. Unterrainer et al., Phys. Rev. Lett. 76 (1996) 2973. C. Waschke et al., Phys. Rev. Lett. 70 (1993) 3319. M. Artaki, K. Hess, Superlattices Microstruct. 1 (1985) 489.
Magnetic "eld quenching of miniband conduction in quasi-one-dimensional superlattices L. Eaves!,*, H.M. Murphy!, A. Nogaret!,1, S.T. Stoddart!, P.C. Main!, M. Henini!, N. Mori", C. Hamaguchi", D.K. Maude#, J.-C. Portal# !School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, UK "Department of Electronic Engineering, Osaka University, Suita City, Osaka 565, Japan #LCMI-CNRS, 38042 Grenoble and INSA-CNRS, 31077 Toulouse, France
Abstract We investigate miniband conduction in GaAs/Al Ga As superlattices at low temperatures and at high magnetic x 1~x "elds up to 23 T. The "elds are applied perpendicular to the tunnel barriers. The current #owing through the sample is strongly suppressed by the magnetic "eld over the voltage range corresponding to the miniband conduction peak. This e!ect is related to the formation of quasi-one-dimensional Landau level minibands when the cyclotron energy exceeds the miniband width. ( 1999 Elsevier Science B.V. All rights reserved. Keywords: Superlattice; Miniband; Magneto-transport
It is now almost 30 years since the pioneering work of Esaki and Tsu [1] on electron transport through semiconductor superlattices. During the last decade, there have been a large number of new measurements which have investigated the electrical and photonic properties of this fundamental solid state structure. These include electrical transport experiments [2}5], optical four-wave mixing [6] and THz [7,8] measurements. DC miniband conduction in a superlattice is characterised by the presence of a peak and hence negative di!erential conductance (NDC) in the current}voltage characteristics, I(<). The form and
* Corresponding author. Fax: #44-115-951-5180. E-mail address: [email protected] (L. Eaves) 1 Present address: Department of Physics, University of Bath, Bath BA2 7AY, UK.
voltage position of the peak can be accurately described by a semi-classical Drude model [1]. The peak occurs when u q&1, where u "eFd/+, q is B B the scattering time, F is the electric "eld applied along the growth direction and d is the superlattice period. In this paper we investigate the e!ect of high magnetic "elds (up to 23 T) on DC miniband conduction in undoped superlattices at low temperatures. The "eld, B, is applied parallel to the superlattice growth direction and has the e!ect of suppressing the peak in I(<) due to the miniband. Landau quantisation of the electron motion in the plane of the wells acts to suppress the miniband conduction when the cyclotron energy, +u , exceeds the miniband width D. # Under these conditions, the localisation of the electrons in the plane e!ectively creates an array of 1D superlattices.
0921-4526/99/$ - see front matter ( 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 0 2 6 8 - 9
L. Eaves et al. / Physica B 272 (1999) 190}193
191
Fig. 1. Conduction band diagram for an undoped GaAs/ Al Ga As superlattice under zero applied external voltage. 0.4 0.6 The shaded regions of the diagram represent the minibands of the superlattice.
The superlattices were grown by molecular beam epitaxy on (1 0 0) n`-GaAs substrates and then fabricated into circular pillars of varying diameters (typically 5}100 lm). Each sample contained an N-period superlattice, separated from two heavily Si-doped GaAs contact electrodes by 10.2 nm wide undoped GaAs spacer layers. Each period consisted of an Al Ga As barrier of width b and a x 1~x GaAs well of width w, giving a period d"b#w. A series of di!erent structures was investigated. Here we focus on a device with the following parameters: N"19, b"2.08 nm, w"9.72 nm, d" 11.8 nm having a barrier composition corresponding to x"0.4. The period was con"rmed by X-ray di!raction. An e!ective mass calculation for this structure gives the energy width of the "rst and second minibands as D "12.1 and D "50.3 meV, 1 2 respectively, separated by a minigap E "98 meV. ' Fig. 1 shows a schematic band diagram for the device under zero applied voltage. The I(<) characteristics at ¹"4.2 K for "elds from B"0 to 23 T are shown in Fig. 2. At zero "eld a peak is observed at 95 mV, followed by a region of NDC. This is characteristic of miniband conduction. The small shoulder, observed at a slightly higher voltage than the main peak, is due to current oscillations associated with the NDC. They can be
Fig. 2. I(<) characteristics at ¹"4.2 K in various B up to 23 T for a 19-period undoped GaAs/(AlGa)As superlattice (for details, see text). The magnetic "eld is applied perpendicular to the quantum wells of the superlattice (B parallel to the current density, J). Also shown as a dashed line is an I(<) at ¹"200 K and B"23 T.
reduced by the addition of load resistors to the measurement circuit but are di$cult to eliminate. At the current peak, the potential drop across each period of the superlattice is 4.5 mV (allowing for some voltage being dropped across the spacer layers) which is much less than D . Hence, the 1 degree of Wannier}Stark localisation is small and a classical Drude model should be valid in this regime. At higher <, in line with earlier work [2,3] the current increases due to non-resonant processes. Increasing B at low ¹ strongly suppresses the miniband conduction peak; at 23 T, the peak value of current is 18 times smaller than at zero "eld. At the highest "eld of 23 T, the e!ect of increasing the sample temperature to ¹"200 K is to restore partially the miniband conduction and the amplitude of the current peak. The same qualitative behaviour has been seen in several devices of di!erent d, b, w and x. The I(<) characteristics of the superlattice in high magnetic "eld are quite di!erent from those of
192
L. Eaves et al. / Physica B 272 (1999) 190}193
Fig. 3. I(<) characteristics at ¹"4.2 K in various BEJ for a double-barrier structure. Al Ga As barriers of thickness 0.4 0.6 8.5 nm enclosed a 7.6 nm GaAs quantum well. The main electron tunnelling peak is labelled E1, and LO and LO are optic 0 1 phonon-assisted tunnelling satellites as discussed in the text.
double barrier resonant tunnelling devices. Fig. 3 shows a series of I(<) curves taken at a temperature of 4 K of an n}i}n double barrier diode, based on GaAs/(AlGa)As. The two (AlGa)As barriers, [Al]"40%, of thickness 8.5 nm, enclose a GaAs quantum well of thickness 7.6 nm. These layers are incorporated in an undoped (intrinsic"i) region between two n-doped GaAs layers with graded doping. The magnetic "eld was applied perpendicular to the plane of the barriers. It can be seen that the magnetic "eld has only a very weak e!ect on the principal electron resonant tunnelling peak, E1 } the amplitude of the peak in the current and its voltage position are only very weakly in#uenced by magnetic "eld. The "eld has a stronger e!ect on the optic phonon-assisted tunnelling satellite transition, LO , and associated lines such as LO , 0 1 which correspond to a change of Landau level index *n"1 as the electron tunnels from the emitter accumulation layer into the quantum well. Similar experiments on triple barrier RTDs again show
that the e!ect on the tunnel current of a magnetic "eld in this orientation is rather weak. We can therefore conclude that the observed strong suppression of the miniband conduction peak at high magnetic "elds and low temperatures is a speci"c property of the superlattice and of the miniband conduction process. To explain the suppression of the miniband conduction by the "eld, we need to consider the various scattering mechanisms in the superlattice. The simple Esaki}Tsu model [1] implicitly assumes the presence of inelastic scattering, speci"ed by a single collision time q. More complex models, which use Monte-Carlo techniques to solve the full threedimensional Boltzmann equation, take into account all of the main scattering mechanisms [9]. The e!ect of the magnetic "eld is to quantise the in-plane motion into Landau levels. This leads to a series of quasi-one-dimensional minibands, separated from each other by +u , as shown in Fig. 4. # We can neglect the e!ect of the upper miniband, since the minigap E is much greater than both D ' 1 and +u "40 meV at the top "eld of 23 T. As can # be seen from Fig. 4 there are two distinct regimes of magnetic "eld: when +u (D the quasi 1D # 1 minibands of di!erent n are nested and elastic scattering processes involving a change of n can occur, for example as shown by the line A to B in Fig. 4(a); for +u 'D the minibands become energetically # 1 decoupled and elastic transitions between the minibands are forbidden (Fig. 4(b)). Elastic transitions can arise from the presence of residual ionised impurities in the superlattice or from monolayer #uctuations in the quantum wells. Acoustic phonon scattering processes are strictly inelastic but since the width of the minizone is 2p/d, the energy involved in such processes is typically less than 2p+v/d&1 meV where v is the sound velocity of longitudinal acoustic phonons at the zone centre. This energy is small compared to both D and +u and such processes may be considered 1 # to be quasi-elastic. Hence, the dominant inelastic process is longitudinal optic (LO) phonon scattering, which consists only of emission processes at low temperatures. We can now explain the observed results within a semi-classical picture. The centre of the distribution of electrons, f (k ), within a miniband in k Z
L. Eaves et al. / Physica B 272 (1999) 190}193
Fig. 4. Schematic energy dispersion curves showing possible low-temperature electron transport mechanisms for a superlattice miniband in a magnetic "eld applied parallel to z: (a) for +u (D: arrows from A to B, B to C and C to D represent # impurity scattering, free #ight of the electron, and LO phonon emission, respectively (b) for +u 'D where the electron is # con"ned to the n"0 Landau level.
space will give the average displacement of the electrons at time t. At low ¹ and B when +u ( # D , k can be randomized as shown in Fig. 4(a). 1 Z An electron at k "0 accelerated by applied elecZ tric "eld can be elastically scattered into higher Landau levels. This shifts the centre of the f (k ) Z distribution to eFq/+, where q is an average scattering time, resulting in a non-zero drift velocity and hence conduction. The electron can subsequently relax via LO phonon emission (process C}D in Fig. 4(a)). At su$ciently high B, when +u 'D (see Fig. # 1 4(b)), k can no longer be randomized. The electron Z
193
undergoes elastic and quasi-elastic scattering processes within the n"0 miniband whose width, D , 1 is much less then the LO phonon energy, +u . LO Thus, at su$ciently high magnetic "elds and low temperatures, the electron is constrained to move and scatter between states of equal and opposite k within the n"0 miniband. The centre of the f (k ) Z distribution therefore remains at k "0 for any Z strength of applied electric "eld, leading to the strong suppression of miniband conduction under these conditions. At high temperature the miniband conduction peak is partially restored, we believe due to population of higher Landau level minibands and to the increase in e$ciency of acoustic phonon scattering as an inelastic process when k ¹&D . Electrons B 1 can consequently be inelastically scattered within a miniband. This once more allows the randomization of k shifting the centre of the f (k ) distribuZ Z tion to a non-zero value, resulting in conduction. This qualitative model is supported by a detailed Monte Carlo simulation which will be published elsewhere.
Acknowledgements This work is supported by the United Kingdom Engineering and Physical Sciences Research Council (EPSRC) and the European Union. L.E. is grateful for an advanced fellowship from the EPSRC.
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
L. Esaki, R. Tsu, IBM Res. Dev. 14 (1970) 61. A. Sibille et al., Phys. Rev. Lett. 64 (1990) 52. L. Canali et al., Phys. Rev. Lett. 76 (1996) 3618. H.J. Hutchinson et al., J. Appl. Phys. 75 (1994) 320. A. Sibille et al., Europhys. Lett. 13 (1990) 279. J. Feldmann et al., Phys. Rev. B 46 (1992) 7252. K. Unterrainer et al., Phys. Rev. Lett. 76 (1996) 2973. C. Waschke et al., Phys. Rev. Lett. 70 (1993) 3319. M. Artaki, K. Hess, Superlattices Microstruct. 1 (1985) 489.