Purdue e-Pubs Proposed Structure for Large Quantum Interference Effects

In this letter we propose and analyze a new semiconductor structure that can be fabricated by present day technology and can lead to large quantum interference effects with potential device applications.

Proposed structure for large quantum interference effects S. Datta (Received 26 November 1985;accepted for publication 16 December 1985) In this letter we propose and analyze a new semiconductor structure that can be fabricated by present day technology and can lead to large quantum interference effects with potential device applications.
Oscillations in the conductance as a function of magnetic field due to quantum interference (Aharonov-Bohm effect) have recently been observed both in metallic rings ' and in semiconductor microstructures. 2 The magnitude of the oscillations is less than 1 %, typically much less. In this letter we propose a new semiconductor structure [ Fig. 1 (a)] that should exhibit large (up to 100% in principle) oscillations in the conductance in a magnetic field or in an electric field (electrostatic Aharonov-Bohm effect). This structure can be fabricated by etching and regrowth) using present day semiconductor technology and has potential applications as a quantum interference transistor requiring extremely low gate voltages (-1 mY or less). The current through the device is determined by the quantum interference between two parallel quantum wells and can be controlled with a third terminal by changing the relative phase between the two paths. This is very different from other quantum devices like the resonant tunneling diode 4 which involves electron transport perpendicular to the quantum wells. The quantum effects in such structures arise from the interference between the multiple reflections from various interfaces.
The basic idea is as follows. The structure consists of two parallel GaAs quantum wells separated by an AIGaAs barrier. The barrier is thinner at the two ends. The parameters are chosen such that there is considerable tunneling between the wells at the ends but hardly any in the central region. The large tunneling at the ends makes the symmetric state IS ) lower in energy than the antisymmetric state IA ) as shown in Fig. I (b) and most of the electrons occupy this state if the carrier density is such that the Fermi level is located as shown in Fig. I (b). The wave function ¢(r, t) for electrons at the ends can then be written as In the central region there is little tunneling so that the two states are degenerate. Here we can write the wave function ¢(r, t) as a linear combination of the wave functions I I) and 12) in the two (isolated) channels.

¢(r,t) = [C,(x)lI) +C 2 (x)12)]expi[k y y-Etlli].
(lb) Since the electron is in a symmetric state at the left end, it will transmit nearly symmetrically into the channels so that at x=O, Let k, and k2 be the wave vectors in the x direction along channels I and 2 respectively: At the right end (x = L) the symmetric component propagates out freely while the antisymmetric component is completely reflected. Thus the transmission coefficient IT 12 for electrons from the left end to the right end is given by Using Eqs. (2) and (3), we get from Eq. (4) We are assuming that the length L is shorter than a mean free path so that both elastic and inelastic scattering can be neglected. With a mobility of 10 5 cm 2 IV s and a velocity of 10 7 cmls the mean free path is -0.4 f.lm.
The conductance G of the structure for small voltages is proportional to the transmission coefficient IT 12.4 where m* is the effective mass and W is the width of the structure in they direction. The transmission coefficient I T 12 and hence the conductance G can be controlled if we can control (k, -k 2 ). The principle described above is very similar to optical interference experiments in which a polarized beam is split in two parts, given different phase shifts and then analyzed. The functions of the two end regions of the structure shown in Fig. 1 are analogous to those of the "polarizer" and the "analyzer," respectively.
If the two channels are identical, then k, = k2 and the transmission coefficient is equal to I. If we apply a magnetic field By so that there is a vector potential A = (Byz, 0, 0), then we can show that The wave functions in the two channels are also skewed somewhat by the magnetic field which can be neglected to first order. From Eqs. (5) and (7) we get  Assuming that kl~k2 = k, we get from Eq. (9) k. -k2 = e~V Ifw, where v = fJk 1m·.

From Eqs. (5) and (lOa)
where T, = L Iv is the transit time of electrons through the channels. It is evident from Eqs. (Sa) and (11)  k. = k ,-= k + (e~V 121iv) ,  transmission coefficient of electrons can be modulated using either a magnetic field By or an electric field g' z . An important point of difference between the two cases is that with a magnetic field, T is the same for all the electrons while with an electric field T depends on k", through the transit time T,. Consequently the overall conductance modulation, which is determined by an average over all the electrons as indicated in Eq. (6), is smaller for the electrostatic Aharonov-Bohm effect compared to the magnetic one (Fig. 2). The simple description presented above is qualitatively correct but it is complicated somewhat by multiple reflections. These can be accounted for by using scatter matrices; a similar approach has been used in the past to analyze onedimensional metal rings. 5 -7 The amplitudes C s and C A of the symmetric and antisymmetric components are defined by C s = (lIJ2)(C I + C 2 ), The antisymmetric component is completely trapped between x = 0 and x = L and cannot propagate in the two end regions with enhanced tunneling. Assuming that the reflection coefficient for the antisymmetric state is -1 at either end we get from Eq. (12) { c s~ (0) } = [SII S12] {C s+ (0) }, If we assume that the symmetric component flows freely to and from the end regions without any reflection, then the transmission coefficient T is given by T= SI2 '  Figures 2(a) and 2(b) show the conductance G calculated as a function of the magnetic field By and the voltage difference tl V respectively. In these calculations we assumed L = 5000 A and d = 300 A. The temperature is assumed low enough that (-aj IJE) in Eq. (6) can be replaced by aCE -E j ); the Fermi velocity Vj was assumed to be 10 7 cml s. As discussed earlier, the conductance G goes to zero for certain values of By but is not zero for any value of tlVbecause the phase shift is not uniform for all electrons. It is interesting to note that the conductance can be decreased by 75% with a voltage of only ~ 1 mY. This effect has potential applications as a quantum interference transistor requiring extremely low gate voltages. In practice, the symmetric component will not flow freely to and from the ends but will suffer some reflections. This can be taken into account using the scatter matrix in Eq. ( 16). We find that the plots in Fig. 2 are distorted significantly when the reflections are large and the results depend on the phase of the reflection coefficient.
To summarize, in this letter we have proposed and analyzed a new semiconductor structure that can be fabricated by present day technology and should exhibit large quantum interference effects with potential device applications.
This work was supported by the National Science Foundation under grant No. ECS-83-51-036 and by the Semiconductor Research Corporation under contract No. 83-01-001.