- Email: [email protected]
Atomically precise modulated two-dimensional electron gas exhibiting stable negative differential resistance
Available online at www.sciencedirect.com
Physica E 22 (2004) 733 – 736 www.elsevier.com/locate/physe
Atomically precise modulated two-dimensional electron gas exhibiting stable negative di%erential resistance T. Feila;∗ , B. Riederb , W. Wegscheidera , J. Kellerb , M. Bichlerc , D. Schuhc , G. Abstreiterc a Institut
fur Experimentalphysik, Universitat Regensburg, Regenburg 93040, Germany fur Theoretische Physik, Universitat Regensburg, Regensburg 93040, Germany c Walter Schottky Institut, TU M unchen, Am Coulombwall, Garching 85748, Germany
b Institut
Abstract The electron transport through a short period modulated two-dimensional electron system exhibits clear negative di%erential conductance (NDC). The modulation parameters place the NDC peak clearly into the validity range of miniband conduction and it is therefore attributed to Bloch oscillations of electrons in the lowest miniband. In contrast to conventional superlattices the NDC in our device is stable for low carrier concentrations due to the reduced dimensionality of our system. With increasing electron concentration instabilities occur in the I –V -traces at the onset of NDC. Numerical simulations con7rm this transition and a stability criterion depending on the system parameters is given. Further evidence for stable Bloch oscillations is given by coupling the device to an external high frequency 7eld. For 7xed frequency a clear suppression and shift of the peak current are observed with increasing intensity of the radiation. Both facts are predicted by the semiclassical theory when Bloch oscillations are frequency modulated by an external AC electric 7eld. ? 2003 Elsevier B.V. All rights reserved. PACS: 73.21.Cd; 73.63.Hs Keywords: Cleaved-edge-overgrowth; Superlattice; Bloch oscillations; Charge domain stability
1. Introduction The observability of Bloch oscillations in superlattices predicted by Esaki and Tsu [1] led to extensive studies of electron dynamics in superlattices over the last 30 years. Although the existence of Bloch oscillations was proven by optical excitation experiments [2,3] the formation of charge domains has inhibited the investigation of electrically-driven oscillators based ∗
Corresponding author. E-mail address: [email protected] (T. Feil).
on Bloch oscillations. The signature of such charge domains are instabilities in the I –V -characteristics of superlattices in the negative di%erential conductance (NDC) region. Our device in contrast exhibits a parameter space in which such instabilities are absent and we believe, therefore, that it allows us to study the e%ects of electromagnetic 7eld absorption and ampli7cation through Bloch oscillations. 2. System The device we are studying is depicted in Fig. 1. It is realized with the cleaved-edge-overgrowth method [4]. In a 7rst growth step a superlattice consisting of
1386-9477/$ - see front matter ? 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2003.12.111
T. Feil et al. / Physica E 22 (2004) 733 – 736
n++ contact layer GaAs/AlGaAs superlattice (100 periods) n++ contact layer
AlGaAs barrier n++ gate contact
734
description for our device holds up to voltages of about 200 mV and crosses then over into the regime of Wannier–Stark-hopping. But for lower voltages, which we will discuss here, we can apply the semiclassical theory. 3. Miniband transport
Substrate 2DEG Fig. 1. Through the combination of two perpendicular growth steps a gate controlled 2DES modulated by a short period superlattice is realized.
alternating layers of GaAs and Al0:32 Ga0:68 As with respective thicknesses of 12 and 3 nm is sandwiched between two highly doped GaAs contact layers. In a second growth step the sample is cleaved in situ along the [1 1 0] plane and then overgrown with an Al0:32 Ga0:68 As barrier followed by an additional gate contact. The result is a two-dimensional electron system (2DES) whose density can be varied by the applied gate voltage UG and which is modulated by the superlattice of the 7rst growth step. The measurements are made with the contacts outside the superlattice in two point geometry, where the contact resistance is avoided by putting two contacts on each of the two highly doped layers. The basic design was introduced in Ref. [5] and modi7ed by replacing the modulation doping with a gate structure in Ref. [6]. The resulting electronic structure consists of subbands produced by the gate con7nement potential which are split into minibands and minigaps due to the superlattice modulation. It is calculated with a two-dimensional SchrHodinger–Poisson-solver [7]. The two lowest cosine like minibands both have a width of 3:3 meV and are separated by a gap of about 7 meV. At an energetic position of 2 meV above the second miniband a continuous spectrum begins which consists of the overlapping miniband states of the higher subbands of the gate con7nement. For the following discussion we will focus only on the lowest miniband since the 7rst minigap is large compared to the miniband width and therefore neglect tunneling between minibands. A comparison of the device parameters with [8] shows that the semiclassical miniband conduction
The semiclassical transport description through a miniband of width when an electric 7eld F is applied along a superlattice of period d starts with the equations of motion given by dk eF 1 dE = ; v= ; (1) dt ˝ ˝ dk where the dispersion is given by the tight binding approximation E(k) = (1 − cos(kd)): (2) 2 Solving the Boltzmann equation combined with (1) in case of a constant electric 7eld leads to a drift velocity of the form [9] F1 (=2kB T ) Jd !B : (3) vD = F0 (=2kB T ) 2˝ 1 + !B2 2 The parameters F0 and F1 are, respectively, the zeroth and 7rst-order Fourier coeLcients of the appropriate distribution function of the electrons at temperature T; !B = eFd=˝ is the Bloch frequency and the scattering time. The result closely resembles the early prediction of Esaki and Tsu [1] and also includes the fact that a 7nite number of electrons is distributed in the system. In our device the distribution changes from the Boltzmann form to the Fermi–Dirac one for a theoretically calculated density of about 1010 cm−2 which roughly corresponds to a gate voltage of 0:3 V. The liquid helium temperature I –V -traces of our device are shown in Fig. 2. NDC is clearly observed for all gate voltages. While the I –V -traces exhibit an instability for large carrier densities the characteristics for small densities are smooth and stable. From the peaks of the low density traces a scattering time of about 2 ps and a corresponding peak Bloch frequency of about 100 GHz are deduced. An estimation of the transition density from stable to unstable calculated from the peak current Ip = wenvD |!B =1 , where w is the width of the sample, gives a value of about 2 × 1010 cm−2 .
T. Feil et al. / Physica E 22 (2004) 733 – 736
735
220 200
SL-axis
unstable
180 160
Current [µA]
140
carrier fluctuation
UG=650mV UG=600mV
120 100
UG=550mV
80
UG=500mV
sheet-like
60
UG=450mV
40
UG=400mV UG=350mV UG=300mV
20
wire-like
0 -20
-0.1
3D
stable
-40 0.0
0.1
0.2
0.3
0.4
0.5
Voltage [V]
Fig. 2. I –V -characteristics of the modulated 2DES. For low gate voltages and therefore, smaller densities the traces are stable. Above UGate = 450 mV an instability occurs at which the device jumps between two stable points.
Conventional 3D superlattices do not exhibit such stable I –V -characteristics. Once the voltage reaches the regime of NDC carrier Muctuations in the device no longer di%use but start to grow until stable charge domains are formed. The fact of stability in our device can be traced back to the reduced dimensionality of our 2D system. Both numerical simulations with the drift–di%usion-model (DDM) and the Boltzmann equation were performed. In both cases a Muctuation was introduced in the system. Then the new resulting electric 7eld distribution was calculated with the Poisson equation. This new potential in turn gives then a new density distribution and an iterative computation of these steps leads to the time evolution of the introduced density Muctuation. The di%erence between a 3D superlattice and our system is that in the 7rst case the electrons are free particles in both perpendicular directions to the superlattice axis, while there is only one such dimension in the 2D device. Owing to the fact that the electron mobility is much higher in the free directions of the systems a density Muctuation in a conventional superlattice will have sheet like geometry. Due to the gate con7nement there is only one free direction in our system, and therefore the Muctuation will have a wire-like geometry. But the 7eld of an electrically charged wire falls of much more rapidly than that of a charged plate. The two di%erent situations are shown schematically in Fig. 3. An approximate lower limit stability criterion can be derived in the following
E=const.
E~1/r
2D
Fig. 3. The drawing shows the di%erent geometries for Muctuations in a conventional superlattice compared to the lower dimensional device.
way. We assume Muctuations proportional to ei(kz−!t) for the 7eld, density and current density added to a stationary state with j0 = en0 vD (E0 ). The DDM di%erential equation and the continuity equation then correlate the frequency ! with the device parameters. The growth or suppression of the Muctuation then depends on the sign of the imaginary part of !. The resulting dispersion of the Muctuation is given by the equation
2
−bk
! = kvD (E0 ) − i Dk + (1 − e
(E0 ) ) 0
;
(4)
where b is the width of the 2DES which is typically of the order of 10 nm. In contrast to 3D superlattices an additional term −e−bk ((E0 )=0 ) appears in the dispersion. When we assume that the Muctuation wavelength is limited by the sample length L which is typically 1 m then the bk is about 10−2 . With 1−e−bk ≈ bk for small bk we then see that the growth term is reduced by a factor of 100 compared to the 3D system and a lower limit transition density of about 109 cm−2 results. This is an order of magnitude smaller than the experimental one. This is due to the fact that we assumed beforehand that the Muctuation already started out in the form of a charged wire. But this charged wire is only formed after a short-time period in which a local Muctuation has spread out along the free carrier direction. During this initial period the electric 7eld is even more reduced having an even smaller dimensionality and therefore we can expect a higher experimental transition density.
736
T. Feil et al. / Physica E 22 (2004) 733 – 736
4. Frequency modulation of Bloch oscillations
46
(6)
where F0 and F1 are again the Fourier coeLcients of the distribution and Jn is the nth order Bessel function. For ! ¡ !B , the case we are interested in here, the peak in the I –V -traces is reduced and shifted to higher 7elds due to the external high frequency 7eld. In our experimental setup the RF frequency is guided along a rectangular waveguide into the helium bath. The waveguide is adjustably shortened at the end so that the device on which a short antenna has been mounted is located =4 away from the shortened end. While the high frequency electric 7eld is coupled to the device, static I –V -characteristics are measured. In order to avoid strong geometry dependences I –V -traces for 7xed frequency but with increasing input power are compared. The experimental result for f = 30 GHz is shown in Fig. 4 together with theoretical results from Eq. (6). Both the suppression and the shift of the current peak can be clearly seen. In order to check that the trace altering is not due to electron heating or a complete heating of the sample we need to study the inMuence of both e%ects on the I –V -traces. From Eq. (6) we directly see that a higher electron temperature a%ects the I –V -traces only through the changes in the Fourier coeLcients of the distribution function. Therefore a rising electron temperature would in-
no RF Power 0dBm Power 2dBm Power 4dBm Power 6dBm Power 8dBm Power 10dBm
increasing RF Power
44
Current [µA]
An interesting experiment in order to test the physical phenomenon underlying the observed NDC is to couple the device to a high frequency electric 7eld. If we apply simultaneously a DC and sinusoidal AC electric 7eld the equation of motion changes to dk eF eF! = + sin(!t); (5) dt ˝ ˝ where ! is now the frequency of the external AC 7eld. The additional external AC 7eld leads to a sinusoidal phase change of the velocity and therefore the inMuence of the AC 7eld is also called a frequency modulation of the Bloch oscillations. The e%ect of the AC 7eld on the current voltage characteristics of a theoretical 1D superlattice was already calculated as an approximation for conventional superlattices [10] and the resulting drift velocity is given by ∞ !B + n! F1 Jd 2 eF! d vD = Jn ; F0 2˝ n=−∞ ˝! 1 + (!B + n!)2
30GHz-Absorption
42
40
theoretical calculations
38
36 0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
Voltage [V]
Fig. 4. The static I –V -traces taken while the device is coupled to a 30 GHz external electric 7eld show a clear suppression and shift of the peak current. The inset shows theoretical solution from (6) for increasing input power.
deed quench the current peak but it would not cause a shift of its position. A higher sample temperature, on the other hand, would increase the scattering rate and could therefore also cause a shift in the position of the current peak. Temperature-dependent measurements of the I –V -traces of the sample show that with increasing device temperature the current through the sample instead increases due to leakage across the bulk superlattice. Therefore the measurements in Fig. 4 give clear evidence to Bloch oscillations as the underlying physical explanation of the observed NDC. References [1] L. Esaki, R. Tsu, IBM J. Res. Dev. 14 (1970) 61. [2] J. Feldmann, K. Leo, J. Shah, D.A.B. Miller, J.E. Cunningham, S. Schmitt-Rink, T. Meier, G. von Plessen, A. Schulze, P. Thomas, Phys. Rev. B 46 (1992) 7252. [3] K. Leo, P.H. Bolivar, F. Brggemann, R. Schwedler, K. Khler, Solid State Commun. 84 (1992) 943. [4] L. Pfeifer, K.W. West, H.L. Stormer, J.P. Eisenstein, K.W. Baldwin, D. Gershoni, J. Spector, Appl. Phys. Lett. 56 (1990) 1697. [5] H.L. Stormer, L.N. Pfei%er, K.W. Baldwin, K.W. West, J. Spector, Appl. Phys. Lett. 58 (1991) 726. [6] R.A. Deutschmann, A. Lorke, W. Wegscheider, M. Bichler, G. Abstreiter, Physica E 6 (2000) 561. [7] M. Rother, AQUILA, Walter Schottky Institut, Technische UniversitHat MHunchen, 1999. [8] A. Wacker, A.-P. Jauho, Phys. Rev. Lett. 80 (1998) 369. [9] H. Sakaki, K. Wagatsuma, J. Hamasaki, S. Saito, Thin Solid Films 36 (1976) 497. [10] A.A. Ignatov, Y.A. Romanov, Phys. Stat. Sol. (b) 73 (1976) 327.
Physica E 22 (2004) 733 – 736 www.elsevier.com/locate/physe
Atomically precise modulated two-dimensional electron gas exhibiting stable negative di%erential resistance T. Feila;∗ , B. Riederb , W. Wegscheidera , J. Kellerb , M. Bichlerc , D. Schuhc , G. Abstreiterc a Institut
fur Experimentalphysik, Universitat Regensburg, Regenburg 93040, Germany fur Theoretische Physik, Universitat Regensburg, Regensburg 93040, Germany c Walter Schottky Institut, TU M unchen, Am Coulombwall, Garching 85748, Germany
b Institut
Abstract The electron transport through a short period modulated two-dimensional electron system exhibits clear negative di%erential conductance (NDC). The modulation parameters place the NDC peak clearly into the validity range of miniband conduction and it is therefore attributed to Bloch oscillations of electrons in the lowest miniband. In contrast to conventional superlattices the NDC in our device is stable for low carrier concentrations due to the reduced dimensionality of our system. With increasing electron concentration instabilities occur in the I –V -traces at the onset of NDC. Numerical simulations con7rm this transition and a stability criterion depending on the system parameters is given. Further evidence for stable Bloch oscillations is given by coupling the device to an external high frequency 7eld. For 7xed frequency a clear suppression and shift of the peak current are observed with increasing intensity of the radiation. Both facts are predicted by the semiclassical theory when Bloch oscillations are frequency modulated by an external AC electric 7eld. ? 2003 Elsevier B.V. All rights reserved. PACS: 73.21.Cd; 73.63.Hs Keywords: Cleaved-edge-overgrowth; Superlattice; Bloch oscillations; Charge domain stability
1. Introduction The observability of Bloch oscillations in superlattices predicted by Esaki and Tsu [1] led to extensive studies of electron dynamics in superlattices over the last 30 years. Although the existence of Bloch oscillations was proven by optical excitation experiments [2,3] the formation of charge domains has inhibited the investigation of electrically-driven oscillators based ∗
Corresponding author. E-mail address: [email protected] (T. Feil).
on Bloch oscillations. The signature of such charge domains are instabilities in the I –V -characteristics of superlattices in the negative di%erential conductance (NDC) region. Our device in contrast exhibits a parameter space in which such instabilities are absent and we believe, therefore, that it allows us to study the e%ects of electromagnetic 7eld absorption and ampli7cation through Bloch oscillations. 2. System The device we are studying is depicted in Fig. 1. It is realized with the cleaved-edge-overgrowth method [4]. In a 7rst growth step a superlattice consisting of
1386-9477/$ - see front matter ? 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2003.12.111
T. Feil et al. / Physica E 22 (2004) 733 – 736
n++ contact layer GaAs/AlGaAs superlattice (100 periods) n++ contact layer
AlGaAs barrier n++ gate contact
734
description for our device holds up to voltages of about 200 mV and crosses then over into the regime of Wannier–Stark-hopping. But for lower voltages, which we will discuss here, we can apply the semiclassical theory. 3. Miniband transport
Substrate 2DEG Fig. 1. Through the combination of two perpendicular growth steps a gate controlled 2DES modulated by a short period superlattice is realized.
alternating layers of GaAs and Al0:32 Ga0:68 As with respective thicknesses of 12 and 3 nm is sandwiched between two highly doped GaAs contact layers. In a second growth step the sample is cleaved in situ along the [1 1 0] plane and then overgrown with an Al0:32 Ga0:68 As barrier followed by an additional gate contact. The result is a two-dimensional electron system (2DES) whose density can be varied by the applied gate voltage UG and which is modulated by the superlattice of the 7rst growth step. The measurements are made with the contacts outside the superlattice in two point geometry, where the contact resistance is avoided by putting two contacts on each of the two highly doped layers. The basic design was introduced in Ref. [5] and modi7ed by replacing the modulation doping with a gate structure in Ref. [6]. The resulting electronic structure consists of subbands produced by the gate con7nement potential which are split into minibands and minigaps due to the superlattice modulation. It is calculated with a two-dimensional SchrHodinger–Poisson-solver [7]. The two lowest cosine like minibands both have a width of 3:3 meV and are separated by a gap of about 7 meV. At an energetic position of 2 meV above the second miniband a continuous spectrum begins which consists of the overlapping miniband states of the higher subbands of the gate con7nement. For the following discussion we will focus only on the lowest miniband since the 7rst minigap is large compared to the miniband width and therefore neglect tunneling between minibands. A comparison of the device parameters with [8] shows that the semiclassical miniband conduction
The semiclassical transport description through a miniband of width when an electric 7eld F is applied along a superlattice of period d starts with the equations of motion given by dk eF 1 dE = ; v= ; (1) dt ˝ ˝ dk where the dispersion is given by the tight binding approximation E(k) = (1 − cos(kd)): (2) 2 Solving the Boltzmann equation combined with (1) in case of a constant electric 7eld leads to a drift velocity of the form [9] F1 (=2kB T ) Jd !B : (3) vD = F0 (=2kB T ) 2˝ 1 + !B2 2 The parameters F0 and F1 are, respectively, the zeroth and 7rst-order Fourier coeLcients of the appropriate distribution function of the electrons at temperature T; !B = eFd=˝ is the Bloch frequency and the scattering time. The result closely resembles the early prediction of Esaki and Tsu [1] and also includes the fact that a 7nite number of electrons is distributed in the system. In our device the distribution changes from the Boltzmann form to the Fermi–Dirac one for a theoretically calculated density of about 1010 cm−2 which roughly corresponds to a gate voltage of 0:3 V. The liquid helium temperature I –V -traces of our device are shown in Fig. 2. NDC is clearly observed for all gate voltages. While the I –V -traces exhibit an instability for large carrier densities the characteristics for small densities are smooth and stable. From the peaks of the low density traces a scattering time of about 2 ps and a corresponding peak Bloch frequency of about 100 GHz are deduced. An estimation of the transition density from stable to unstable calculated from the peak current Ip = wenvD |!B =1 , where w is the width of the sample, gives a value of about 2 × 1010 cm−2 .
T. Feil et al. / Physica E 22 (2004) 733 – 736
735
220 200
SL-axis
unstable
180 160
Current [µA]
140
carrier fluctuation
UG=650mV UG=600mV
120 100
UG=550mV
80
UG=500mV
sheet-like
60
UG=450mV
40
UG=400mV UG=350mV UG=300mV
20
wire-like
0 -20
-0.1
3D
stable
-40 0.0
0.1
0.2
0.3
0.4
0.5
Voltage [V]
Fig. 2. I –V -characteristics of the modulated 2DES. For low gate voltages and therefore, smaller densities the traces are stable. Above UGate = 450 mV an instability occurs at which the device jumps between two stable points.
Conventional 3D superlattices do not exhibit such stable I –V -characteristics. Once the voltage reaches the regime of NDC carrier Muctuations in the device no longer di%use but start to grow until stable charge domains are formed. The fact of stability in our device can be traced back to the reduced dimensionality of our 2D system. Both numerical simulations with the drift–di%usion-model (DDM) and the Boltzmann equation were performed. In both cases a Muctuation was introduced in the system. Then the new resulting electric 7eld distribution was calculated with the Poisson equation. This new potential in turn gives then a new density distribution and an iterative computation of these steps leads to the time evolution of the introduced density Muctuation. The di%erence between a 3D superlattice and our system is that in the 7rst case the electrons are free particles in both perpendicular directions to the superlattice axis, while there is only one such dimension in the 2D device. Owing to the fact that the electron mobility is much higher in the free directions of the systems a density Muctuation in a conventional superlattice will have sheet like geometry. Due to the gate con7nement there is only one free direction in our system, and therefore the Muctuation will have a wire-like geometry. But the 7eld of an electrically charged wire falls of much more rapidly than that of a charged plate. The two di%erent situations are shown schematically in Fig. 3. An approximate lower limit stability criterion can be derived in the following
E=const.
E~1/r
2D
Fig. 3. The drawing shows the di%erent geometries for Muctuations in a conventional superlattice compared to the lower dimensional device.
way. We assume Muctuations proportional to ei(kz−!t) for the 7eld, density and current density added to a stationary state with j0 = en0 vD (E0 ). The DDM di%erential equation and the continuity equation then correlate the frequency ! with the device parameters. The growth or suppression of the Muctuation then depends on the sign of the imaginary part of !. The resulting dispersion of the Muctuation is given by the equation
2
−bk
! = kvD (E0 ) − i Dk + (1 − e
(E0 ) ) 0
;
(4)
where b is the width of the 2DES which is typically of the order of 10 nm. In contrast to 3D superlattices an additional term −e−bk ((E0 )=0 ) appears in the dispersion. When we assume that the Muctuation wavelength is limited by the sample length L which is typically 1 m then the bk is about 10−2 . With 1−e−bk ≈ bk for small bk we then see that the growth term is reduced by a factor of 100 compared to the 3D system and a lower limit transition density of about 109 cm−2 results. This is an order of magnitude smaller than the experimental one. This is due to the fact that we assumed beforehand that the Muctuation already started out in the form of a charged wire. But this charged wire is only formed after a short-time period in which a local Muctuation has spread out along the free carrier direction. During this initial period the electric 7eld is even more reduced having an even smaller dimensionality and therefore we can expect a higher experimental transition density.
736
T. Feil et al. / Physica E 22 (2004) 733 – 736
4. Frequency modulation of Bloch oscillations
46
(6)
where F0 and F1 are again the Fourier coeLcients of the distribution and Jn is the nth order Bessel function. For ! ¡ !B , the case we are interested in here, the peak in the I –V -traces is reduced and shifted to higher 7elds due to the external high frequency 7eld. In our experimental setup the RF frequency is guided along a rectangular waveguide into the helium bath. The waveguide is adjustably shortened at the end so that the device on which a short antenna has been mounted is located =4 away from the shortened end. While the high frequency electric 7eld is coupled to the device, static I –V -characteristics are measured. In order to avoid strong geometry dependences I –V -traces for 7xed frequency but with increasing input power are compared. The experimental result for f = 30 GHz is shown in Fig. 4 together with theoretical results from Eq. (6). Both the suppression and the shift of the current peak can be clearly seen. In order to check that the trace altering is not due to electron heating or a complete heating of the sample we need to study the inMuence of both e%ects on the I –V -traces. From Eq. (6) we directly see that a higher electron temperature a%ects the I –V -traces only through the changes in the Fourier coeLcients of the distribution function. Therefore a rising electron temperature would in-
no RF Power 0dBm Power 2dBm Power 4dBm Power 6dBm Power 8dBm Power 10dBm
increasing RF Power
44
Current [µA]
An interesting experiment in order to test the physical phenomenon underlying the observed NDC is to couple the device to a high frequency electric 7eld. If we apply simultaneously a DC and sinusoidal AC electric 7eld the equation of motion changes to dk eF eF! = + sin(!t); (5) dt ˝ ˝ where ! is now the frequency of the external AC 7eld. The additional external AC 7eld leads to a sinusoidal phase change of the velocity and therefore the inMuence of the AC 7eld is also called a frequency modulation of the Bloch oscillations. The e%ect of the AC 7eld on the current voltage characteristics of a theoretical 1D superlattice was already calculated as an approximation for conventional superlattices [10] and the resulting drift velocity is given by ∞ !B + n! F1 Jd 2 eF! d vD = Jn ; F0 2˝ n=−∞ ˝! 1 + (!B + n!)2
30GHz-Absorption
42
40
theoretical calculations
38
36 0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
Voltage [V]
Fig. 4. The static I –V -traces taken while the device is coupled to a 30 GHz external electric 7eld show a clear suppression and shift of the peak current. The inset shows theoretical solution from (6) for increasing input power.
deed quench the current peak but it would not cause a shift of its position. A higher sample temperature, on the other hand, would increase the scattering rate and could therefore also cause a shift in the position of the current peak. Temperature-dependent measurements of the I –V -traces of the sample show that with increasing device temperature the current through the sample instead increases due to leakage across the bulk superlattice. Therefore the measurements in Fig. 4 give clear evidence to Bloch oscillations as the underlying physical explanation of the observed NDC. References [1] L. Esaki, R. Tsu, IBM J. Res. Dev. 14 (1970) 61. [2] J. Feldmann, K. Leo, J. Shah, D.A.B. Miller, J.E. Cunningham, S. Schmitt-Rink, T. Meier, G. von Plessen, A. Schulze, P. Thomas, Phys. Rev. B 46 (1992) 7252. [3] K. Leo, P.H. Bolivar, F. Brggemann, R. Schwedler, K. Khler, Solid State Commun. 84 (1992) 943. [4] L. Pfeifer, K.W. West, H.L. Stormer, J.P. Eisenstein, K.W. Baldwin, D. Gershoni, J. Spector, Appl. Phys. Lett. 56 (1990) 1697. [5] H.L. Stormer, L.N. Pfei%er, K.W. Baldwin, K.W. West, J. Spector, Appl. Phys. Lett. 58 (1991) 726. [6] R.A. Deutschmann, A. Lorke, W. Wegscheider, M. Bichler, G. Abstreiter, Physica E 6 (2000) 561. [7] M. Rother, AQUILA, Walter Schottky Institut, Technische UniversitHat MHunchen, 1999. [8] A. Wacker, A.-P. Jauho, Phys. Rev. Lett. 80 (1998) 369. [9] H. Sakaki, K. Wagatsuma, J. Hamasaki, S. Saito, Thin Solid Films 36 (1976) 497. [10] A.A. Ignatov, Y.A. Romanov, Phys. Stat. Sol. (b) 73 (1976) 327.