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Interface roughness: a reason of inaccessibility of the negative resistance region in resonant-tunneling devices
Superlattices and Microstructures, Vol. 24, No. 1, 1998 Article No. sm960362
Interface roughness: a reason of inaccessibility of the negative resistance region in resonant-tunneling devices ´ski, A. Ma T. Figielski, T. Wosin ¸ kosa Institute of Physics, Polish Academy of Sciences, Al. Lotnik´ow 32, 02-668 Warszawa, Poland
(Received 15 July 1996)
Interface roughness in double-barrier resonant-tunneling devices affects the lateral electron motion in the quantum well and can give rise to subsidiary subbands or quasibound states in the well. We demonstrate that a shoulder frequently appearing beyond the principal resonance peak in the current–voltage characteristic can result from the resonant tunneling via those states. c 1998 Academic Press
Key words: semiconductor heterostructures GaAs/Alx Ga1−x As, quantum wells, resonant tunneling.
1. Introduction Double-barrier resonant-tunneling devices (RTDs) can exhibit negative differential resistance (NDR) in the current–voltage, I (V ), characteristics. They are promising devices which could operate as oscillators up to the Terahertz range. However, only RTDs with very low resonant–current densities display the smooth NDR regions required for the controllable performance of such devices. Usually, the resonant maximum of the tunnel current versus bias voltage of RTDs ends in either single-step or double-step bistability. In the latter case a shoulder appears between the two steps. The view has been generally taken that the shoulder represents no physical effect but is due to self-oscillations arising in the NDR region of the device [1, 2]. This explanation seemed to be only natural as similar shoulders could also appear in p–n tunnel (Esaki) diode characteristics. This view has been called into question by Berolo et al. [3] and by Figielski et al. [4] who have demonstrated that the shoulder frequently exhibits a regular structure independent of the oscillating conditions of the device. Such a structure is distinctly seen in the derivative of the current with respect to the bias voltage (Fig. 1) or, equivalently, in the ac photocurrent induced by a chopped illumination generating electron–hole pairs in the device [5, 6]. However, the reasons of the observed structure proposed by these authors, i.e. different resonance conditions corresponding to inhomogeneity of the well width [3] or the tunneling via coupled electron–LO phonon states in the well [4], can hardly be accepted. In this paper we propose that electrons moving in a quantum well can undergo diffraction from well-boundary roughness regularities that corresponds to their scattering into subsidiary subbands shifted up in energy with respect to the principal subband in the well. Tunneling through these subsidiary subbands contributes to the current beyond the principal resonance peak. 0749–6036/98/070069 + 06 $30.00/0
c 1998 Academic Press
70
Superlattices and Microstructures, Vol. 24, No. 1, 1998 6
I (mA)
4 2 A 0 40
0
dI /dV (mS)
20
B
–20 0.0
0.2
0.4
0.6
V (V) Fig. 1. Current A and differential conductance B versus bias voltage of the RTD described in [4], measured at a temperature of 80 K in a circuit with a 5 load resistor. The shoulder beyond the resonance peak exhibits periodic fine structure with the period 36 mV. The dashed line segments link the measuring points at the edges of the bistability region.
2. Experimental results First, we present some arguments in favour of the physical reality of the shoulder. It should be stated at the beginning that all the results presented here were obtained while a suitable resistance (shunt) was included in parallel with the RTD in order to suppress high-frequency oscillations in the current, or lowfrequency relaxation-type instabilities, otherwise observed with an oscilloscope connected across the device. The intrinsic I (V ) curve of the RTD was then calculated by subtracting, from the total current, the current flowing through the parallel resistance. We recall that if the parallel resistance is much smaller than the total resistance in series with the RTD in the measuring circuit, it represents the load resistance in the series equivalent of the circuit (Th´evenin’s theorem [7]). A capacitance (up to ∼ 1µF) connected across the devices had only a minor effect on the measured dc I (V ) curves. The I (V ) characteristic of one of the investigated RTDs (fabricated in Manchester [4]), shown in Fig. 1, exhibits a very broad shoulder with a regular fine structure. This structure was sequentially smeared out with increasing temperature. We varied the load resistance in the measuring circuit by including different shunt resistors in parallel with the device. In each case, the apparent I (V ) line in the region of bistability coincided with the load line. This fundamental observation, which has never been reported before in the literature, strongly points out the physical reality of the shoulder. Figure 1 refers to the lowest shunt resistance included, equal to 5. It is evident from this figure that the intrinsic negative differential resistance on the right-hand side of the principal resonance peak has to be lower in absolute value than 5. Instead, the maximal positive differential resistance on the left-hand side of this peak, read directly from the d I /d V curve (Fig. 1B), is ∼20. In conclusion, the principal peak has to be asymmetric, that is tipped over to the right, which is a common effect due to charge build-up in the quantum well under resonance conditions [8, 9]. Hence, we can also estimate the intrinsic half-width of the principal resonance peak to be of the order of a few tens of millivolts that is many times smaller than the extension of the broad shoulder. More information regarding this RTD can be found in [4]. Another example of the I (V ) curve of the investigated RTD (fabricated in Warsaw) is shown in Fig. 2.
Superlattices and Microstructures, Vol. 24, No. 1, 1998
71
5 B=0T
I (mA)
4 3 2 1
A
0
80 40
–40 0.5
0.6
0.7 V (V)
0.8
dI /dV (mS)
0 B
–80 0.9
Fig. 2. Current A and differential conductance B versus bias voltage of the RTD whose active part consists of 5.6 nm quantum well sandwiched between 3.7 nm-AlAs (emitter) and 5.6 nm-Al0,4 Ga0.6 As (collector) barriers, measured at 1.8 K with a 20 chip resistor connected directly across the device.
Here, a chip resistor of 20 was mounted directly on the sample holder which suppressed the instabilities but did not change the shoulder extension (similar results were obtained with a 5 resistor). In this case, we applied a magnetic field parallel to the tunnel current. We have expected that the magnetic field, which drastically modifies the lateral electron motion in the well, will also change the interaction of an electron with interfacial roughness. The observed effect is striking: an increasing magnetic field makes the shoulder become broader and exhibit a distinct structure; see Fig. 3. For comparison, we performed a similar experiment with a GaAs p–n tunnel diode. In the latter case the structureless shoulder remained unchanged up to the highest field applied; see Fig. 4.
3. Effect of interfacial roughness In the following, we demonstrate that interfacial roughness can give rise to subsidiary subbands (or localized states) in the quantum well and, consequently, to a characteristic shoulder in the I (V ) resonance curve. The GaAs/AlGaAs interface may be thought to consist of a random distribution of different islands of ± onemonolayer height, having sizes in the submicron range, adjacent to the otherwise smooth boundary [10]. Presumably, the interfacial roughness has a few spatial frequencies which can be separated by the Fourier transformation of the roughness profile. For the sake of simplicity we consider two orthogonal sets of spatial frequencies that correspond to 1/L i and 1/L j along two orthogonal axes lying in the plane of the quantum well, where the indexes i, j number particular frequencies in each set. Here, we do not differentiate between the two interfaces bordering the quantum well. Electrons entering the quantum well through the emitter barrier have all in-plane components of the wavevector, kEk , available in the Fermi sea of the emitter, and the in-plane component is conserved in tunneling. If the well boundaries were ideally smooth, an electron under resonance conditions would bounce back and forth between the two barriers while continuing its lateral motion. In reality, electrons which are elastically scattered at the boundary roughness undergo some diffusive reflection and, in particular, diffraction from the roughness regularities; see Fig. 5.
72
Superlattices and Microstructures, Vol. 24, No. 1, 1998 4 B = 13 T I (mA)
3 2 1 A 60
0
30
–30 0.5
0.6
0.7 V (V)
0.8
dI /dV (mS)
0 B
–60 0.9
Fig. 3. Same as in Fig. 2 after applying a magnetic field of 13 T parallel to the tunnel current.
I (µA)
20
A
B=0T
C
B = 13 T
15 10 5 100
0 D
0 –50
0.0
0.4
0.0 V (V)
0.4
dI/ dV (µS)
50 B
0.8
Fig. 4. Current A and differential conductance B versus bias voltage of GaAs p–n tunnel diode at 1.8 K C and D are as A and B but after applying a magnetic field of 13 T parallel to the tunnel current.
In this case, the Laue condition of diffraction can be written as: 1kEk = gEi j , where 1kEk denotes the change in the wavevector caused by the elastic scattering, and gEi j = [2πn/L i , 2π m/L j ], where n, and m are integers. An electron whose initial kinetic energy is sufficiently large to fulfil this condition can be scattered into one of the subsidiary subbands created by the well-boundary roughness regularity. These subbands are built of the states split off from the principal subband of the quantum well (the density of states will depend on the intensity of the diffracted wave) and are shifted up in energy with respect to the latter. So, in this case, even
Superlattices and Microstructures, Vol. 24, No. 1, 1998
73
Quantum well
A
C
B
Fig. 5. Two-dimensional outline of electron waves travelling in a quantum well: (A) primary wave, (B) specularly reflected wave, (C) diffracted wave.
when the RTD is biased beyond the principal resonance peak, electrons can still tunnel from the emitter into the subsidiary subbands without violation of the in-plane-momentum conservation rule. Actually, the two-dimensional array of equivalent scattering centres on the well boundaries may be quite irregular. Then, the subsidiary subbands would be badly defined and rather represent assemblages of broad, highly overlapping levels for which the k-vector is no more a good quantum number. Alternatively, such states might be regarded as resulting from interference between nearly free electron states with wavevector kEk and those with wavevector −kEk , that results in lateral localization of the electron wavefunction [11]. The intensity of the diffracted wave depends on the distance over which the wave interacts with the boundary, and thus, on the electron phase-coherence length. The latter is larger at low temperature when the electron– phonon scattering is substantially reduced. The distance of interaction depends also on the lifetime (dwell time) of an electron in the quantum well. For instance, the lifetime under resonance condition for the RTD shown in Fig. 1 has been calculated to be ∼3 ps. Then, the lateral distance in the quantum well covered by an electron which tunneled there from the Fermi energy in the emitter region is of the order of 1 µm. In that case, the electron ‘feels’ the well-boundary roughness. But the collector barrier may be designed to be much more transparent (which, however, lowers the resonant current density) so as to reach a regime in which the lateral distance is shorter than the island size. Then, the well-boundary roughness would be insignificant for device operation and no shoulder should appear beyond the principal resonance peak. There are numerous reports of asymmetric RTDs in the literature that are consistent with the above conclusion [9, 12]. The intensity of the diffracted wave could be enhanced by a spatial confinement of the electron wave propagating in the well. Such a confinement would eliminate a destructive interference of partial waves diffracted from phase-incoherent spatial frequencies of the roughness profile. The latter effect can probably be achieved in a quantizing magnetic field. Take into account the wavefunction 9 being the solution of the Schr¨odinger equation for two-dimensional electron motion in a perpendicular magnetic field, in the Landau gauge. Then, 9 is a product of two functions of which one describes a wave running in a certain direction (which is responsible for the diffraction) and the other describes a harmonic oscillator vibrating in the perpendicular direction (which leads to the cyclotron orbits and Landau levels). Thus the states 9 are extended in one lateral direction but are confined in another, orthogonal, direction [13]. This confinement could just be the reason for the observed effect of the magnetic field on the shoulder shape. In conclusion, we believe that the shoulder, commonly observed in RTDs beyond the principal resonance peak, cannot be solely explained as an artifact caused by self-oscillations in the NDR region of the device but is likely associated with quantum-well boundary roughness. Although this statement is more speculative than proved at the moment, it is worth further examining as it at least could offer a tool for controlling the interface roughness which would be complementary to optical methods. Acknowledgements—We are greatly indebted to Dr M. Kaniewska and Dr K. Regi´nski (Institute of Electron Technology, Warsaw) for fabrication of RTDs and to Professor A. E. Belyaev and Dr S. A. Vitusevich (Institute
74
Superlattices and Microstructures, Vol. 24, No. 1, 1998
of Semiconductor Physics, Kiev) for rendering the p–n tunnel diode available. This work was partly supported by the Committee for Scientific Research of Poland under Grant 2 P03B 035 08.
References [1] }T. C. L. G. Sollner, Phys. Rev. Lett. 59, 1622 (1987). [2] }T. J. Foster, M. L. Leadbeater, L. Eaves, M. Henini, O. H. Hughes, C. A. Payling, F. W. Sheard, P. E. Simmonds, and M. A. Pate, Phys. Rev. B39, 6205 (1989). [3] }O. Berolo, E. Fortin, and L. Allard, Superlatt. Microstr. 17, 163 (1995). [4] }T. Figielski, A. M¸akosa, T. Wosi´nski, P. C. Harness, and K. E. Singer, Solid State Commun. 91, 913 (1994). [5] }P. G. Richard, E. Fortin, and O. Berolo, Solid-State Electron. 33, 93 (1990). [6] }T. Figielski, A. M¸akosa, T. Wosi´nski, P. C. Harness, and K. E. Singer, Acta Phys. Polon. A84, 817 (1993). [7] }J. J. Brophy, Basic Electronics for Scientists (McGraw-Hill Co., New York, 1966). [8] }F. W. Sheard and G. A. Tombs, Appl. Phys. Lett. 52, 1228 (1988). [9] }C. Zhang, M. L. F. Lerch, A. D. Martin, P. E. Simmonds, and L. Eaves, Phys. Rev. Lett. 72, 3397 (1994). [10] }A. Ourmazd, in Semiconductor Interfaces at the Sub-Nanometer Scale, edited by H. W. M. Salemink and M. D. Pashley (Kluwer Academic Publishers, Dordrecht, 1993) p. 139. [11] }D. Z.-Y. Ting, S. K. Kirby, and T. C. McGill, Appl. Phys. Lett. 64, 2004 (1994). [12] }A. Zaslavsky, V. J. Goldman, and D. C. Tsui, Appl. Phys. Lett. 53, 1408 (1988). [13] }R. E. Prange, in The Quantum Hall Effect, edited by R. E. Prange and S. M. Girvin, Chapter 1 (SpringerVerlag, New York, 1987).
Interface roughness: a reason of inaccessibility of the negative resistance region in resonant-tunneling devices ´ski, A. Ma T. Figielski, T. Wosin ¸ kosa Institute of Physics, Polish Academy of Sciences, Al. Lotnik´ow 32, 02-668 Warszawa, Poland
(Received 15 July 1996)
Interface roughness in double-barrier resonant-tunneling devices affects the lateral electron motion in the quantum well and can give rise to subsidiary subbands or quasibound states in the well. We demonstrate that a shoulder frequently appearing beyond the principal resonance peak in the current–voltage characteristic can result from the resonant tunneling via those states. c 1998 Academic Press
Key words: semiconductor heterostructures GaAs/Alx Ga1−x As, quantum wells, resonant tunneling.
1. Introduction Double-barrier resonant-tunneling devices (RTDs) can exhibit negative differential resistance (NDR) in the current–voltage, I (V ), characteristics. They are promising devices which could operate as oscillators up to the Terahertz range. However, only RTDs with very low resonant–current densities display the smooth NDR regions required for the controllable performance of such devices. Usually, the resonant maximum of the tunnel current versus bias voltage of RTDs ends in either single-step or double-step bistability. In the latter case a shoulder appears between the two steps. The view has been generally taken that the shoulder represents no physical effect but is due to self-oscillations arising in the NDR region of the device [1, 2]. This explanation seemed to be only natural as similar shoulders could also appear in p–n tunnel (Esaki) diode characteristics. This view has been called into question by Berolo et al. [3] and by Figielski et al. [4] who have demonstrated that the shoulder frequently exhibits a regular structure independent of the oscillating conditions of the device. Such a structure is distinctly seen in the derivative of the current with respect to the bias voltage (Fig. 1) or, equivalently, in the ac photocurrent induced by a chopped illumination generating electron–hole pairs in the device [5, 6]. However, the reasons of the observed structure proposed by these authors, i.e. different resonance conditions corresponding to inhomogeneity of the well width [3] or the tunneling via coupled electron–LO phonon states in the well [4], can hardly be accepted. In this paper we propose that electrons moving in a quantum well can undergo diffraction from well-boundary roughness regularities that corresponds to their scattering into subsidiary subbands shifted up in energy with respect to the principal subband in the well. Tunneling through these subsidiary subbands contributes to the current beyond the principal resonance peak. 0749–6036/98/070069 + 06 $30.00/0
c 1998 Academic Press
70
Superlattices and Microstructures, Vol. 24, No. 1, 1998 6
I (mA)
4 2 A 0 40
0
dI /dV (mS)
20
B
–20 0.0
0.2
0.4
0.6
V (V) Fig. 1. Current A and differential conductance B versus bias voltage of the RTD described in [4], measured at a temperature of 80 K in a circuit with a 5 load resistor. The shoulder beyond the resonance peak exhibits periodic fine structure with the period 36 mV. The dashed line segments link the measuring points at the edges of the bistability region.
2. Experimental results First, we present some arguments in favour of the physical reality of the shoulder. It should be stated at the beginning that all the results presented here were obtained while a suitable resistance (shunt) was included in parallel with the RTD in order to suppress high-frequency oscillations in the current, or lowfrequency relaxation-type instabilities, otherwise observed with an oscilloscope connected across the device. The intrinsic I (V ) curve of the RTD was then calculated by subtracting, from the total current, the current flowing through the parallel resistance. We recall that if the parallel resistance is much smaller than the total resistance in series with the RTD in the measuring circuit, it represents the load resistance in the series equivalent of the circuit (Th´evenin’s theorem [7]). A capacitance (up to ∼ 1µF) connected across the devices had only a minor effect on the measured dc I (V ) curves. The I (V ) characteristic of one of the investigated RTDs (fabricated in Manchester [4]), shown in Fig. 1, exhibits a very broad shoulder with a regular fine structure. This structure was sequentially smeared out with increasing temperature. We varied the load resistance in the measuring circuit by including different shunt resistors in parallel with the device. In each case, the apparent I (V ) line in the region of bistability coincided with the load line. This fundamental observation, which has never been reported before in the literature, strongly points out the physical reality of the shoulder. Figure 1 refers to the lowest shunt resistance included, equal to 5. It is evident from this figure that the intrinsic negative differential resistance on the right-hand side of the principal resonance peak has to be lower in absolute value than 5. Instead, the maximal positive differential resistance on the left-hand side of this peak, read directly from the d I /d V curve (Fig. 1B), is ∼20. In conclusion, the principal peak has to be asymmetric, that is tipped over to the right, which is a common effect due to charge build-up in the quantum well under resonance conditions [8, 9]. Hence, we can also estimate the intrinsic half-width of the principal resonance peak to be of the order of a few tens of millivolts that is many times smaller than the extension of the broad shoulder. More information regarding this RTD can be found in [4]. Another example of the I (V ) curve of the investigated RTD (fabricated in Warsaw) is shown in Fig. 2.
Superlattices and Microstructures, Vol. 24, No. 1, 1998
71
5 B=0T
I (mA)
4 3 2 1
A
0
80 40
–40 0.5
0.6
0.7 V (V)
0.8
dI /dV (mS)
0 B
–80 0.9
Fig. 2. Current A and differential conductance B versus bias voltage of the RTD whose active part consists of 5.6 nm quantum well sandwiched between 3.7 nm-AlAs (emitter) and 5.6 nm-Al0,4 Ga0.6 As (collector) barriers, measured at 1.8 K with a 20 chip resistor connected directly across the device.
Here, a chip resistor of 20 was mounted directly on the sample holder which suppressed the instabilities but did not change the shoulder extension (similar results were obtained with a 5 resistor). In this case, we applied a magnetic field parallel to the tunnel current. We have expected that the magnetic field, which drastically modifies the lateral electron motion in the well, will also change the interaction of an electron with interfacial roughness. The observed effect is striking: an increasing magnetic field makes the shoulder become broader and exhibit a distinct structure; see Fig. 3. For comparison, we performed a similar experiment with a GaAs p–n tunnel diode. In the latter case the structureless shoulder remained unchanged up to the highest field applied; see Fig. 4.
3. Effect of interfacial roughness In the following, we demonstrate that interfacial roughness can give rise to subsidiary subbands (or localized states) in the quantum well and, consequently, to a characteristic shoulder in the I (V ) resonance curve. The GaAs/AlGaAs interface may be thought to consist of a random distribution of different islands of ± onemonolayer height, having sizes in the submicron range, adjacent to the otherwise smooth boundary [10]. Presumably, the interfacial roughness has a few spatial frequencies which can be separated by the Fourier transformation of the roughness profile. For the sake of simplicity we consider two orthogonal sets of spatial frequencies that correspond to 1/L i and 1/L j along two orthogonal axes lying in the plane of the quantum well, where the indexes i, j number particular frequencies in each set. Here, we do not differentiate between the two interfaces bordering the quantum well. Electrons entering the quantum well through the emitter barrier have all in-plane components of the wavevector, kEk , available in the Fermi sea of the emitter, and the in-plane component is conserved in tunneling. If the well boundaries were ideally smooth, an electron under resonance conditions would bounce back and forth between the two barriers while continuing its lateral motion. In reality, electrons which are elastically scattered at the boundary roughness undergo some diffusive reflection and, in particular, diffraction from the roughness regularities; see Fig. 5.
72
Superlattices and Microstructures, Vol. 24, No. 1, 1998 4 B = 13 T I (mA)
3 2 1 A 60
0
30
–30 0.5
0.6
0.7 V (V)
0.8
dI /dV (mS)
0 B
–60 0.9
Fig. 3. Same as in Fig. 2 after applying a magnetic field of 13 T parallel to the tunnel current.
I (µA)
20
A
B=0T
C
B = 13 T
15 10 5 100
0 D
0 –50
0.0
0.4
0.0 V (V)
0.4
dI/ dV (µS)
50 B
0.8
Fig. 4. Current A and differential conductance B versus bias voltage of GaAs p–n tunnel diode at 1.8 K C and D are as A and B but after applying a magnetic field of 13 T parallel to the tunnel current.
In this case, the Laue condition of diffraction can be written as: 1kEk = gEi j , where 1kEk denotes the change in the wavevector caused by the elastic scattering, and gEi j = [2πn/L i , 2π m/L j ], where n, and m are integers. An electron whose initial kinetic energy is sufficiently large to fulfil this condition can be scattered into one of the subsidiary subbands created by the well-boundary roughness regularity. These subbands are built of the states split off from the principal subband of the quantum well (the density of states will depend on the intensity of the diffracted wave) and are shifted up in energy with respect to the latter. So, in this case, even
Superlattices and Microstructures, Vol. 24, No. 1, 1998
73
Quantum well
A
C
B
Fig. 5. Two-dimensional outline of electron waves travelling in a quantum well: (A) primary wave, (B) specularly reflected wave, (C) diffracted wave.
when the RTD is biased beyond the principal resonance peak, electrons can still tunnel from the emitter into the subsidiary subbands without violation of the in-plane-momentum conservation rule. Actually, the two-dimensional array of equivalent scattering centres on the well boundaries may be quite irregular. Then, the subsidiary subbands would be badly defined and rather represent assemblages of broad, highly overlapping levels for which the k-vector is no more a good quantum number. Alternatively, such states might be regarded as resulting from interference between nearly free electron states with wavevector kEk and those with wavevector −kEk , that results in lateral localization of the electron wavefunction [11]. The intensity of the diffracted wave depends on the distance over which the wave interacts with the boundary, and thus, on the electron phase-coherence length. The latter is larger at low temperature when the electron– phonon scattering is substantially reduced. The distance of interaction depends also on the lifetime (dwell time) of an electron in the quantum well. For instance, the lifetime under resonance condition for the RTD shown in Fig. 1 has been calculated to be ∼3 ps. Then, the lateral distance in the quantum well covered by an electron which tunneled there from the Fermi energy in the emitter region is of the order of 1 µm. In that case, the electron ‘feels’ the well-boundary roughness. But the collector barrier may be designed to be much more transparent (which, however, lowers the resonant current density) so as to reach a regime in which the lateral distance is shorter than the island size. Then, the well-boundary roughness would be insignificant for device operation and no shoulder should appear beyond the principal resonance peak. There are numerous reports of asymmetric RTDs in the literature that are consistent with the above conclusion [9, 12]. The intensity of the diffracted wave could be enhanced by a spatial confinement of the electron wave propagating in the well. Such a confinement would eliminate a destructive interference of partial waves diffracted from phase-incoherent spatial frequencies of the roughness profile. The latter effect can probably be achieved in a quantizing magnetic field. Take into account the wavefunction 9 being the solution of the Schr¨odinger equation for two-dimensional electron motion in a perpendicular magnetic field, in the Landau gauge. Then, 9 is a product of two functions of which one describes a wave running in a certain direction (which is responsible for the diffraction) and the other describes a harmonic oscillator vibrating in the perpendicular direction (which leads to the cyclotron orbits and Landau levels). Thus the states 9 are extended in one lateral direction but are confined in another, orthogonal, direction [13]. This confinement could just be the reason for the observed effect of the magnetic field on the shoulder shape. In conclusion, we believe that the shoulder, commonly observed in RTDs beyond the principal resonance peak, cannot be solely explained as an artifact caused by self-oscillations in the NDR region of the device but is likely associated with quantum-well boundary roughness. Although this statement is more speculative than proved at the moment, it is worth further examining as it at least could offer a tool for controlling the interface roughness which would be complementary to optical methods. Acknowledgements—We are greatly indebted to Dr M. Kaniewska and Dr K. Regi´nski (Institute of Electron Technology, Warsaw) for fabrication of RTDs and to Professor A. E. Belyaev and Dr S. A. Vitusevich (Institute
74
Superlattices and Microstructures, Vol. 24, No. 1, 1998
of Semiconductor Physics, Kiev) for rendering the p–n tunnel diode available. This work was partly supported by the Committee for Scientific Research of Poland under Grant 2 P03B 035 08.
References [1] }T. C. L. G. Sollner, Phys. Rev. Lett. 59, 1622 (1987). [2] }T. J. Foster, M. L. Leadbeater, L. Eaves, M. Henini, O. H. Hughes, C. A. Payling, F. W. Sheard, P. E. Simmonds, and M. A. Pate, Phys. Rev. B39, 6205 (1989). [3] }O. Berolo, E. Fortin, and L. Allard, Superlatt. Microstr. 17, 163 (1995). [4] }T. Figielski, A. M¸akosa, T. Wosi´nski, P. C. Harness, and K. E. Singer, Solid State Commun. 91, 913 (1994). [5] }P. G. Richard, E. Fortin, and O. Berolo, Solid-State Electron. 33, 93 (1990). [6] }T. Figielski, A. M¸akosa, T. Wosi´nski, P. C. Harness, and K. E. Singer, Acta Phys. Polon. A84, 817 (1993). [7] }J. J. Brophy, Basic Electronics for Scientists (McGraw-Hill Co., New York, 1966). [8] }F. W. Sheard and G. A. Tombs, Appl. Phys. Lett. 52, 1228 (1988). [9] }C. Zhang, M. L. F. Lerch, A. D. Martin, P. E. Simmonds, and L. Eaves, Phys. Rev. Lett. 72, 3397 (1994). [10] }A. Ourmazd, in Semiconductor Interfaces at the Sub-Nanometer Scale, edited by H. W. M. Salemink and M. D. Pashley (Kluwer Academic Publishers, Dordrecht, 1993) p. 139. [11] }D. Z.-Y. Ting, S. K. Kirby, and T. C. McGill, Appl. Phys. Lett. 64, 2004 (1994). [12] }A. Zaslavsky, V. J. Goldman, and D. C. Tsui, Appl. Phys. Lett. 53, 1408 (1988). [13] }R. E. Prange, in The Quantum Hall Effect, edited by R. E. Prange and S. M. Girvin, Chapter 1 (SpringerVerlag, New York, 1987).