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Negative conductance at THz frequencies in multi-well structures
So/M-State Electronics Vol. 37. Nos 4-6, pp. 1235-1238, 1994 Copyright © 1994 Elsevier Science Ltd 0038-1101(93)E0053-4 Printed in Great Britain. All rights reserved 0038-1101/94 $ 6 . 0 0 + 0 . 0 0
Pergamon
NEGATIVE CONDUCTANCE AT THz FREQUENCIES IN MULTI-WELL STRUCTURES W. S. TRUSCOTr Department of Electrical Engineering and Electronics, University of Manchester Institute of Science and Technology, PO Box 88, Manchester M60 IQD, England Abstract--An algorithm by which the exact first order a.c. response of quantum electronic devices, including tunnelling structures, may be calculated rapidly is described. Novel multi-well tunnelling structures with negative conductance maxima at 1.4 x 10t2 Hz (1.4 THz) and 14 THz at temperatures of 80 and 300 K respectively are presented. Devices of this sort are expected to operate as power sources, mixers and amplifiers between 0.5 and 20 THz. The effects of bias and departures from the ideal geometry are discussed as are the factors which indicate that such devices are realisable in practice.
Three new aspects of the response at extremely high frequencies of electronic transport structures, particularly tunnelling and other quantum devices, are presented in this paper: first, an algorithm to calculate both the terminal currents and the internal carrier displacement currents that are induced when an external a.c. potential is applied to a structure; second, a novel device structure which may operate as a power source or amplifier at frequencies in the range 0.5-20 T H z (5 x 10tl-2 x 1013Hz); and, third, experimental and theoretical considerations related to the practical production and utilisation of these devices. This work is based on exact solutions to Schrodinger's equation for systems in uniform time dependent fields[l]. The solutions are inherently selfconsistent, i.e. where one of these solutions applies, the additional time dependence caused by the a.c. field is independent of the cartier-carrier interactions. Full details of, and justification for, their use in the calculation of the a.c. response of semiconductor devices will be reported later; here the method is outlined. The response of a wave function to a time dependent field is close to the motion of a classical particle in that field, in addition there is a time-dependent phase modulation. As a specific example, A exp(ikx) is a wave function for a particle of mass m in a region of constant potential, when the potential - F x cos(cot) is added an exact wave function is:
Aexp{ik[x-~ lEt h
~
F c°s(cot) ]
d
iFx sin(cot ) hco
iF2[2cot - sin(cot )] )
~T,,~
~ (l)
Exact solutions also exist for regions in which the potential is a linear or parabolic function of distance. A semiconductor device subject to an a.c. bias may be divided into regions within each of which there are many time dependent wave functions that will be exact solutions o f Schrodinger's equation; however,
most of these will not satisfy the continuity equations at regional boundaries since the adjoining wave functions will have different time dependencies for their amplitudes and phases. For the above example, this will always occur if either F or k changes at a boundary. In a seminal paper concerned with a transit time for tunnelling which has acted as a major stimulus for this work, Bfittiker and Landauer have given a solution[2]: for any wave function with energy E that describes the device with a static potential, there is a related one for the structure subject to an additional sinusoidally time dependent field of period 2n/co which has a similar wave function at energy E together with a series of functions with time dependencies which differ from it by e x p ( - i n c o t ) where n is any integer. The latter are wave functions for the local potential with energies E + nhco selected so that their sum satisfies the continuity equations and the boundary conditions at all times. The first order response is associated solely with the components identified with exp(_+ icot ); a total of three functions in any region are sufficient to describe the linear behaviour of a device. To first order, the time dependent continuity equations at any boundary divide into two parts separating the effects of the a.c. field in adjacent regions. The algorithm used for this paper calculates the effect of the a.c. field on each region of the potential having an exact solution: with this approach the waves generated at the two boundaries of a region tend to cancel outside it. First order forms for the amplitude and spatial derivative of the wave function of eqn (1) for a boundary at x = a may be found by multiplying the function: A exp ika - -~- + - ~ sin(cot ) , by
1235
W.S. TRUSCOTT
1236 and
Iik ~k2F cos(o,t ) + ~--~ ikF sin(ogt)l -
respectively. If the a.c. potential is continuous then the original function will also be continuous at x = a and it is only the time dependence associated with the multipliers that has to be counterbalanced by wave functions at energies E + ho9; waves of the required amplitude and derivative are added at one side of the boundary with respect to the other. They propagate through the region to the second boundary at x = b where a similar set of functions describe the amplitude and derivative. For small values of co the latter at E + ho~ will almost exactly cancel those required for the boundary at x = a: the residual sum of the waves satisfying the boundary conditions at x = a and at x = b propagates into the rest of the structure and gives rise to the time dependent terminal current. The carrier displacement current in any region of the structure is found by summing the net current due to all other regions, the current associated with the initial time dependent wave function and that associated with the waves within the region generated at its boundaries. The results in this paper are from a computer programme which uses the above algorithm. This programme can analyse any structure which is composed of a sequence of regions of constant potential separated by abrupt steps in potential. There are many advantages in restricting the structures in this way: all the functions within the algorithm are analytical and any result may be checked (in simple structures) by an independent calculation; because the functions involved are either exponential or trigonometrical the calculation is much faster than it would be if the functions had to be numerically integrated; the programme is simple and has been checked and verified over a long period of time, the probability of error in these results is very small. This algorithm is readily extended to include regions with a linear or parabolic variation of potential and programmes have been developed to analyse structures including such regions: the results from these are not different in kind from those presented here. The modulation of the current emerging from a double barrier resonant tunnelling diode structure (DBRTS) calculated using the programme is consistent with results published elsewhere[3-5]: the overall response is that of a system with a single resonance for which a characteristic time, r. is defined by the ratio of Plank's constant to the half width of the transmission peak. The limitations on the frequency response intrinsic to the D B R T S are overcome in structures with two or more quasi-bound levels which have spacings corresponding to the operating frequency, f0- The algorithm has allowed the behaviour of many different structures to be investigated; the overall pattern of phase and amplitude variation is characteristic of
spatially distinct states whose coupling through an external potential is characterised by a resonant response. The argument that this resonant modulation of a unidirectional particle flux by an external field is a novel quantum effect will be given in a later paper; this is concerned with the practical implications of the effect. The pattern of the results that have been obtained allows optimised device structures to be evolved: examples of the conduction band diagram of two of these are shown in Fig. 1(a, b). The key features of these are the wider barriers on either side of the left hand well and the nearly uniform spacing in energy of the three lowest quasi-bound levels when the device is biased at the optimum operating point. The wider barriers cause the tunnelling transmission to be dominated by the quasibound state at energy E2 associated with the left hand well. There will be a significant accumulation of carriers in this well when a tunnelling current flows. The rest of the structure is designed so that the energies of two other levels, E l and E3 satisfy E 3 - E1 = 2hf0. The calculated response to an applied field when E2 = ½(El + E3) is very small because the effects of the coupling to levels 1 and 3 are equal and opposite. This is shown by Fig. 2 which illustrates for electrons with energies corresponding to the transmission window at E2, the reactive and conductive parts of the terminal current as a function of the energy of the state to which the tunnelling electrons are coupled by the a.c. field. The reactive currents are opposite in sign for coupling to the states at E l and E3, as are the conductive currents for a given modulation frequency if E2 = ½(El + E2); these two parts always vary with energy in accordance with expectations based on the K r a m e r s - K r o n i g relationships. However if E2 > ~(E1 + E3), the case shown in Fig. 2, the coupling between levels 2 and 3 occurs at lower frequencies than that between 2 and 1; it is the overlap between
E +hf
41A
41A
54 A
27A
54 A
54 A
27A 300 meV
180 A
°°vr H1 l,Ov Fig. 1. Conduction band energy diagrams of two structures exhibiting negative conductance. They are optimised for 1.4 THz (a) and 14 THz (b) respectively.
Negative conductance at THz frequencies in multi-well structures
J
susce~tancescancel t~ E2 E ~
~
~ negativeconductancesadd
eneDgy energy eneDgy
Fig. 2. An explanation of the calculated admittance frequency curves of Fig. 3. The upper curve shows the transmission peak at energy E2. The lower curves are sketches of the susceptance and conductance for any electron with energy close to E2 coupled to energies close to El or E3 by the applied a.c. field at a frequency f. the two responses which gives the large negative terminal conductance which is constant over a substantial bandwidth. Figure 3(a) and (b) show the frequency dependence of the real and imaginary parts of the admittance of the structures in Fig. l(a) and (b) respectively when bias is applied so that E2 >~(EI +E3). The quantities plotted in Fig. 3(a) are the real and imaginary components of the a.c. terminal current produced by unit fieldacross the device structure for an incident electron spectrum at the lefthand side of Fig. l(a) that is characterised by a temperature of
200[ ~SOl
~ -50j '~ -loo! -200 L
1.0
1.4 1.8 ModulationFrequency(THz)
(a)
-15
10
14 18 ModulationFrequency(THz)
(b) Fig. 3. The calculated conductance (-II-) and susceptance (-O-) (arbitrary scales) as a function of frequency for the two device structures of Fig. 1.
1237
80 K, while those plotted in Fig. 3(b) are the identical quantities for the structure of Fig. l(b) with an incident electron temperature of 300 K; in both cases an effective mass one tenth the electron mass has been assumed. The algorithm described above is used to calculate at a fixed frequency and incident electron energy the magnitude and phase of the modulation of the tunnelling current produced by the a.c. field. The observable admittance at that frequency is found by integrating the modulation weighted by the transmission probability over the electron spectrum: this integration extends over the range of incident energies that cover the transmission peaks E l , E2 and E3. For some structures and frequencies there is significant cancellation between the response of the electrons with the highest energies in the central transmission peak and those with the lowest energies. The structure shown in Fig. I(a) minimises this with barrier widths designed so that the major transmission peak at energy E2 is about half the width of those at energies E1 and E3, in which case the phase of the modulation produced by the a.c. field does not vary significantly over the transmission peak at E2. The total current flowing through any device similar to those shown in Fig. 1 is composed of two parts, first, the tunnelling current which passes through the right hand barrier into the contact and. second, the displacement current associated with the electrical polarisation of the material and carriers within the structure. The dielectric polarisation is a fixed function of the device geometry, the carrier polarisation normally adds to this, increasing the overall structural capacitance. In these devices however, the negative resistance is the result of the reduction in the density of carriers adjacent to the final barrier when the a.c. field is directed towards it; the carrier displacement within the device under optimum operating conditions is therefore inductive. These new devices therefore exhibit an effect of great importance in THz applications, namely that the capacitance of the device is reduced below its structural value by the carrier displacement currents within the active region of the device. Calculation of the combined structural and carrier capacitance is possible using the above theory but, since this effect is of the same order as the variation in the applied field over the device due to carrier accumulation, such a calculation will only be valid in a model which includes these effects. Devices using this new concept can only be realised experimentally and exploited commercially if the range of parameters such as structural width, barrier height and device bias over which the effect is exhibited covers the probable range of deviations that may occur in the normal manufacturing processes and operating conditions for such structures. The effect on the negative conductance characteristic of changing the energy offset between the barriers and the wells for the structure in Fig. l(a) by 5% is shown in Fig. 4(a), as is the effect of changing the overall width scale by 2%. It can be seen that, as a result of the
W. S. TRUSCOTT
1238
lOO
I
°
o
~
-5o! -loo
"~ 0 -150 -20O .0
1.4
1.8
Modulation Frequency (THz)
(a) 200~ ~)
1oo
~
-200 ~
1
~
~
1
1
1.4
~
1.8
Modulation Frequency (THz)
(b) Fig. 4. Variation of the conductance for the device in Fig. l(a) with (a) device geometry and (b) applied bias. The design conductance curve ( - 0 - ) is compared with those for devices with a 5% decrease in barrier height (-O-) and a 2% increase in geometrical widths (-II-), and curves are presented for forward biases which are larger (-II-) and smaller (-O-) than the optimum bias and also for a small negative bias (-Q-).
bandwidth over which the device has a constant negative conductance, devices with either of these departures from the designed structure would still perform as oscillators or amplifiers at the design frequency of 1.4 THz. Since these variations are in excess of the degree of control which can be achieved in MBE growth, normal structural variations will not restrict the performance of these devices. Figure 4(b) shows the negative resistance characteristic with three different values of forward bias, the effect of reducing the bias by a few mV over the active region of the device to the condition that E2 >~(E1 + E3) is to switch the sign of both the conductance and susceptance at the operating frequency. This device may therefore be used as a modulator. Changing the bias by a similar amount in the opposite direction splits the features in the device response associated with the coupling between E2 and E3 from that between E2 and EI giving a reduced magnitude for the negative conductance, but an increased bandwidth and a correspondingly less rapid change of susceptance with frequency. A single device may therefore be used in high gain narrow bandwidth and low gain wide bandwidth applications depending on the applied bias. The band structure diagrams shown in Fig. 1 are significantly different from those for real devices fabricated in GaAs/A1GaAs, a material system which minimises the broadening of the quasi-bound levels, subject to a d.c. bias prompting an electron flow. The
bias will cause an electric field over the whole structure; in addition the accumulation of carriers, particularly in the left hand well, will make this field vary over the active region. Although it might seem a very difficult task to produce a structure with three energy levels whose relative energies are correct within 0.5 meV, there are two factors that are a considerable assistance for structures similar to that of Fig. l(a): first, levels E1 and E3 are associated with the mixed states of right hand wells; the minimum value of E3 - E1 as the d.c. bias is varied is determined only by the barrier between these wells, and it changes slowly with this field about this condition because of the mixing. There will therefore be a useful range of applied bias over which the E3 - E1 will be close to 2hfo. Second, the position of E2 with respect to E l and E3 is controlled by the bias, a suitable value can always bring their energies into the correct relation to show negative conductance at some frequency. Because E 3 - E1 varies slowly over a certain range of applied bias, the design requirements reduce to establishing an appropriate barrier thickness between the right hand wells and choosing the widths of these wells so as to ensure that the bias at which E2 is in the correct relative position is close to that at which E 3 - E l reaches its minimum value. An important factor determining the practical applications of these devices is temperature at which they can be operated. The negative conductances shown in Fig. 3(a) and (b) have been calculated for electrons at 80 and 300 K; the structures therefore have the potential to operate at practical temperatures. The other factor which will determine the maximum operating temperature is the broadening of the quasi-bound levels by phonon scattering. Roskos et al.[6] have reported the observation of up to 15 cycles of radiation at a frequency of 1.5 THz caused by coherent electron oscillations in an optically excited double well structure at a temperature of 10 K. The basis for this phenomenon is closely related to that giving rise to the negative conductance; this observation strongly suggests that the high frequency negative conductance will be observed in suitable structures at practical temperatures as does the observation at temperatures up to 185 K of maxima in the tunnelling current-voltage curve associated with a 38 meV miniband spacing in a multi-well structure[7]. REFERENCES
1. W. S. Truscott, Phys. Ret,. Lett. 70, 1900 (1993). 2. M. Biittiker and R. Landauer, Phys Ret,. Lett. 49, 1740 (1982). 3. W. R. Frensley, Appl. Phys. Lett. 51, 448 (1987). 4. H. C. Liu, AppL Phys. Lett. 52, 453 (1988), 5. V. Kislov and A. Kamenev, AppL Phys. Lett. 59, 1500 (1991). 6. H. G. Roskos, M. C. Nuss, J. Shah, K. Leo, D. A. B. Miller, A. M. Fox, S. Schmitt-Rink and K. K6hler, Phys. Rev. Lett. 68, 2216 (1992). 7. P. C. Harness, Ph.D. thesis, Univ. of Manchester, pp. 170 177 (1992).
Pergamon
NEGATIVE CONDUCTANCE AT THz FREQUENCIES IN MULTI-WELL STRUCTURES W. S. TRUSCOTr Department of Electrical Engineering and Electronics, University of Manchester Institute of Science and Technology, PO Box 88, Manchester M60 IQD, England Abstract--An algorithm by which the exact first order a.c. response of quantum electronic devices, including tunnelling structures, may be calculated rapidly is described. Novel multi-well tunnelling structures with negative conductance maxima at 1.4 x 10t2 Hz (1.4 THz) and 14 THz at temperatures of 80 and 300 K respectively are presented. Devices of this sort are expected to operate as power sources, mixers and amplifiers between 0.5 and 20 THz. The effects of bias and departures from the ideal geometry are discussed as are the factors which indicate that such devices are realisable in practice.
Three new aspects of the response at extremely high frequencies of electronic transport structures, particularly tunnelling and other quantum devices, are presented in this paper: first, an algorithm to calculate both the terminal currents and the internal carrier displacement currents that are induced when an external a.c. potential is applied to a structure; second, a novel device structure which may operate as a power source or amplifier at frequencies in the range 0.5-20 T H z (5 x 10tl-2 x 1013Hz); and, third, experimental and theoretical considerations related to the practical production and utilisation of these devices. This work is based on exact solutions to Schrodinger's equation for systems in uniform time dependent fields[l]. The solutions are inherently selfconsistent, i.e. where one of these solutions applies, the additional time dependence caused by the a.c. field is independent of the cartier-carrier interactions. Full details of, and justification for, their use in the calculation of the a.c. response of semiconductor devices will be reported later; here the method is outlined. The response of a wave function to a time dependent field is close to the motion of a classical particle in that field, in addition there is a time-dependent phase modulation. As a specific example, A exp(ikx) is a wave function for a particle of mass m in a region of constant potential, when the potential - F x cos(cot) is added an exact wave function is:
Aexp{ik[x-~ lEt h
~
F c°s(cot) ]
d
iFx sin(cot ) hco
iF2[2cot - sin(cot )] )
~T,,~
~ (l)
Exact solutions also exist for regions in which the potential is a linear or parabolic function of distance. A semiconductor device subject to an a.c. bias may be divided into regions within each of which there are many time dependent wave functions that will be exact solutions o f Schrodinger's equation; however,
most of these will not satisfy the continuity equations at regional boundaries since the adjoining wave functions will have different time dependencies for their amplitudes and phases. For the above example, this will always occur if either F or k changes at a boundary. In a seminal paper concerned with a transit time for tunnelling which has acted as a major stimulus for this work, Bfittiker and Landauer have given a solution[2]: for any wave function with energy E that describes the device with a static potential, there is a related one for the structure subject to an additional sinusoidally time dependent field of period 2n/co which has a similar wave function at energy E together with a series of functions with time dependencies which differ from it by e x p ( - i n c o t ) where n is any integer. The latter are wave functions for the local potential with energies E + nhco selected so that their sum satisfies the continuity equations and the boundary conditions at all times. The first order response is associated solely with the components identified with exp(_+ icot ); a total of three functions in any region are sufficient to describe the linear behaviour of a device. To first order, the time dependent continuity equations at any boundary divide into two parts separating the effects of the a.c. field in adjacent regions. The algorithm used for this paper calculates the effect of the a.c. field on each region of the potential having an exact solution: with this approach the waves generated at the two boundaries of a region tend to cancel outside it. First order forms for the amplitude and spatial derivative of the wave function of eqn (1) for a boundary at x = a may be found by multiplying the function: A exp ika - -~- + - ~ sin(cot ) , by
1235
W.S. TRUSCOTT
1236 and
Iik ~k2F cos(o,t ) + ~--~ ikF sin(ogt)l -
respectively. If the a.c. potential is continuous then the original function will also be continuous at x = a and it is only the time dependence associated with the multipliers that has to be counterbalanced by wave functions at energies E + ho9; waves of the required amplitude and derivative are added at one side of the boundary with respect to the other. They propagate through the region to the second boundary at x = b where a similar set of functions describe the amplitude and derivative. For small values of co the latter at E + ho~ will almost exactly cancel those required for the boundary at x = a: the residual sum of the waves satisfying the boundary conditions at x = a and at x = b propagates into the rest of the structure and gives rise to the time dependent terminal current. The carrier displacement current in any region of the structure is found by summing the net current due to all other regions, the current associated with the initial time dependent wave function and that associated with the waves within the region generated at its boundaries. The results in this paper are from a computer programme which uses the above algorithm. This programme can analyse any structure which is composed of a sequence of regions of constant potential separated by abrupt steps in potential. There are many advantages in restricting the structures in this way: all the functions within the algorithm are analytical and any result may be checked (in simple structures) by an independent calculation; because the functions involved are either exponential or trigonometrical the calculation is much faster than it would be if the functions had to be numerically integrated; the programme is simple and has been checked and verified over a long period of time, the probability of error in these results is very small. This algorithm is readily extended to include regions with a linear or parabolic variation of potential and programmes have been developed to analyse structures including such regions: the results from these are not different in kind from those presented here. The modulation of the current emerging from a double barrier resonant tunnelling diode structure (DBRTS) calculated using the programme is consistent with results published elsewhere[3-5]: the overall response is that of a system with a single resonance for which a characteristic time, r. is defined by the ratio of Plank's constant to the half width of the transmission peak. The limitations on the frequency response intrinsic to the D B R T S are overcome in structures with two or more quasi-bound levels which have spacings corresponding to the operating frequency, f0- The algorithm has allowed the behaviour of many different structures to be investigated; the overall pattern of phase and amplitude variation is characteristic of
spatially distinct states whose coupling through an external potential is characterised by a resonant response. The argument that this resonant modulation of a unidirectional particle flux by an external field is a novel quantum effect will be given in a later paper; this is concerned with the practical implications of the effect. The pattern of the results that have been obtained allows optimised device structures to be evolved: examples of the conduction band diagram of two of these are shown in Fig. 1(a, b). The key features of these are the wider barriers on either side of the left hand well and the nearly uniform spacing in energy of the three lowest quasi-bound levels when the device is biased at the optimum operating point. The wider barriers cause the tunnelling transmission to be dominated by the quasibound state at energy E2 associated with the left hand well. There will be a significant accumulation of carriers in this well when a tunnelling current flows. The rest of the structure is designed so that the energies of two other levels, E l and E3 satisfy E 3 - E1 = 2hf0. The calculated response to an applied field when E2 = ½(El + E3) is very small because the effects of the coupling to levels 1 and 3 are equal and opposite. This is shown by Fig. 2 which illustrates for electrons with energies corresponding to the transmission window at E2, the reactive and conductive parts of the terminal current as a function of the energy of the state to which the tunnelling electrons are coupled by the a.c. field. The reactive currents are opposite in sign for coupling to the states at E l and E3, as are the conductive currents for a given modulation frequency if E2 = ½(El + E2); these two parts always vary with energy in accordance with expectations based on the K r a m e r s - K r o n i g relationships. However if E2 > ~(E1 + E3), the case shown in Fig. 2, the coupling between levels 2 and 3 occurs at lower frequencies than that between 2 and 1; it is the overlap between
E +hf
41A
41A
54 A
27A
54 A
54 A
27A 300 meV
180 A
°°vr H1 l,Ov Fig. 1. Conduction band energy diagrams of two structures exhibiting negative conductance. They are optimised for 1.4 THz (a) and 14 THz (b) respectively.
Negative conductance at THz frequencies in multi-well structures
J
susce~tancescancel t~ E2 E ~
~
~ negativeconductancesadd
eneDgy energy eneDgy
Fig. 2. An explanation of the calculated admittance frequency curves of Fig. 3. The upper curve shows the transmission peak at energy E2. The lower curves are sketches of the susceptance and conductance for any electron with energy close to E2 coupled to energies close to El or E3 by the applied a.c. field at a frequency f. the two responses which gives the large negative terminal conductance which is constant over a substantial bandwidth. Figure 3(a) and (b) show the frequency dependence of the real and imaginary parts of the admittance of the structures in Fig. l(a) and (b) respectively when bias is applied so that E2 >~(EI +E3). The quantities plotted in Fig. 3(a) are the real and imaginary components of the a.c. terminal current produced by unit fieldacross the device structure for an incident electron spectrum at the lefthand side of Fig. l(a) that is characterised by a temperature of
200[ ~SOl
~ -50j '~ -loo! -200 L
1.0
1.4 1.8 ModulationFrequency(THz)
(a)
-15
10
14 18 ModulationFrequency(THz)
(b) Fig. 3. The calculated conductance (-II-) and susceptance (-O-) (arbitrary scales) as a function of frequency for the two device structures of Fig. 1.
1237
80 K, while those plotted in Fig. 3(b) are the identical quantities for the structure of Fig. l(b) with an incident electron temperature of 300 K; in both cases an effective mass one tenth the electron mass has been assumed. The algorithm described above is used to calculate at a fixed frequency and incident electron energy the magnitude and phase of the modulation of the tunnelling current produced by the a.c. field. The observable admittance at that frequency is found by integrating the modulation weighted by the transmission probability over the electron spectrum: this integration extends over the range of incident energies that cover the transmission peaks E l , E2 and E3. For some structures and frequencies there is significant cancellation between the response of the electrons with the highest energies in the central transmission peak and those with the lowest energies. The structure shown in Fig. I(a) minimises this with barrier widths designed so that the major transmission peak at energy E2 is about half the width of those at energies E1 and E3, in which case the phase of the modulation produced by the a.c. field does not vary significantly over the transmission peak at E2. The total current flowing through any device similar to those shown in Fig. 1 is composed of two parts, first, the tunnelling current which passes through the right hand barrier into the contact and. second, the displacement current associated with the electrical polarisation of the material and carriers within the structure. The dielectric polarisation is a fixed function of the device geometry, the carrier polarisation normally adds to this, increasing the overall structural capacitance. In these devices however, the negative resistance is the result of the reduction in the density of carriers adjacent to the final barrier when the a.c. field is directed towards it; the carrier displacement within the device under optimum operating conditions is therefore inductive. These new devices therefore exhibit an effect of great importance in THz applications, namely that the capacitance of the device is reduced below its structural value by the carrier displacement currents within the active region of the device. Calculation of the combined structural and carrier capacitance is possible using the above theory but, since this effect is of the same order as the variation in the applied field over the device due to carrier accumulation, such a calculation will only be valid in a model which includes these effects. Devices using this new concept can only be realised experimentally and exploited commercially if the range of parameters such as structural width, barrier height and device bias over which the effect is exhibited covers the probable range of deviations that may occur in the normal manufacturing processes and operating conditions for such structures. The effect on the negative conductance characteristic of changing the energy offset between the barriers and the wells for the structure in Fig. l(a) by 5% is shown in Fig. 4(a), as is the effect of changing the overall width scale by 2%. It can be seen that, as a result of the
W. S. TRUSCOTT
1238
lOO
I
°
o
~
-5o! -loo
"~ 0 -150 -20O .0
1.4
1.8
Modulation Frequency (THz)
(a) 200~ ~)
1oo
~
-200 ~
1
~
~
1
1
1.4
~
1.8
Modulation Frequency (THz)
(b) Fig. 4. Variation of the conductance for the device in Fig. l(a) with (a) device geometry and (b) applied bias. The design conductance curve ( - 0 - ) is compared with those for devices with a 5% decrease in barrier height (-O-) and a 2% increase in geometrical widths (-II-), and curves are presented for forward biases which are larger (-II-) and smaller (-O-) than the optimum bias and also for a small negative bias (-Q-).
bandwidth over which the device has a constant negative conductance, devices with either of these departures from the designed structure would still perform as oscillators or amplifiers at the design frequency of 1.4 THz. Since these variations are in excess of the degree of control which can be achieved in MBE growth, normal structural variations will not restrict the performance of these devices. Figure 4(b) shows the negative resistance characteristic with three different values of forward bias, the effect of reducing the bias by a few mV over the active region of the device to the condition that E2 >~(E1 + E3) is to switch the sign of both the conductance and susceptance at the operating frequency. This device may therefore be used as a modulator. Changing the bias by a similar amount in the opposite direction splits the features in the device response associated with the coupling between E2 and E3 from that between E2 and EI giving a reduced magnitude for the negative conductance, but an increased bandwidth and a correspondingly less rapid change of susceptance with frequency. A single device may therefore be used in high gain narrow bandwidth and low gain wide bandwidth applications depending on the applied bias. The band structure diagrams shown in Fig. 1 are significantly different from those for real devices fabricated in GaAs/A1GaAs, a material system which minimises the broadening of the quasi-bound levels, subject to a d.c. bias prompting an electron flow. The
bias will cause an electric field over the whole structure; in addition the accumulation of carriers, particularly in the left hand well, will make this field vary over the active region. Although it might seem a very difficult task to produce a structure with three energy levels whose relative energies are correct within 0.5 meV, there are two factors that are a considerable assistance for structures similar to that of Fig. l(a): first, levels E1 and E3 are associated with the mixed states of right hand wells; the minimum value of E3 - E1 as the d.c. bias is varied is determined only by the barrier between these wells, and it changes slowly with this field about this condition because of the mixing. There will therefore be a useful range of applied bias over which the E3 - E1 will be close to 2hfo. Second, the position of E2 with respect to E l and E3 is controlled by the bias, a suitable value can always bring their energies into the correct relation to show negative conductance at some frequency. Because E 3 - E1 varies slowly over a certain range of applied bias, the design requirements reduce to establishing an appropriate barrier thickness between the right hand wells and choosing the widths of these wells so as to ensure that the bias at which E2 is in the correct relative position is close to that at which E 3 - E l reaches its minimum value. An important factor determining the practical applications of these devices is temperature at which they can be operated. The negative conductances shown in Fig. 3(a) and (b) have been calculated for electrons at 80 and 300 K; the structures therefore have the potential to operate at practical temperatures. The other factor which will determine the maximum operating temperature is the broadening of the quasi-bound levels by phonon scattering. Roskos et al.[6] have reported the observation of up to 15 cycles of radiation at a frequency of 1.5 THz caused by coherent electron oscillations in an optically excited double well structure at a temperature of 10 K. The basis for this phenomenon is closely related to that giving rise to the negative conductance; this observation strongly suggests that the high frequency negative conductance will be observed in suitable structures at practical temperatures as does the observation at temperatures up to 185 K of maxima in the tunnelling current-voltage curve associated with a 38 meV miniband spacing in a multi-well structure[7]. REFERENCES
1. W. S. Truscott, Phys. Ret,. Lett. 70, 1900 (1993). 2. M. Biittiker and R. Landauer, Phys Ret,. Lett. 49, 1740 (1982). 3. W. R. Frensley, Appl. Phys. Lett. 51, 448 (1987). 4. H. C. Liu, AppL Phys. Lett. 52, 453 (1988), 5. V. Kislov and A. Kamenev, AppL Phys. Lett. 59, 1500 (1991). 6. H. G. Roskos, M. C. Nuss, J. Shah, K. Leo, D. A. B. Miller, A. M. Fox, S. Schmitt-Rink and K. K6hler, Phys. Rev. Lett. 68, 2216 (1992). 7. P. C. Harness, Ph.D. thesis, Univ. of Manchester, pp. 170 177 (1992).