Quantum adiabatic transport through an array of constrictions

Physica B 165&166 (1990) 849-850 North-Holland

QUANTUM ADIABATIC TRANSPORT THROUGH AN ARRAY OF CONSTRICTIONS Michel van Eijck, Frank Hekking, Leo Kouwenhoven and Gerd Schlln Department of Applied Physics, Delft University of Technology, Lorentzwegl, 2628 CJ Delft, The Netherlands Using the concept of adiabatically transmitted edge channels we calculate the conductance of a system consisting of N constrictions in series as a function of the gate voltage. The system shows an incipient band structure, which is reflected in the behaviour of the conductance. We also study the effect of small disorder on the conductance. The results agree with recent experimental observations by Kouwenhoven et al. (1). 1. INRODUCTION Recently, Kouwenhoven et al. (1) studied the conductance of a 1 dimensional lateral array consisting of 16 constrictions in series separated by cavities. The periodic confinement is defmed by means of a split gate technique in a 2 DEG. A constant magnetic field is applied perpendicular to the array. The overall conductance of the array as a function of the gate voltage is quantized, however, it shows pronounced minima, separated by 15 resonances in accordance with the number of cavities. Experimentally (2) it has been shown that in a magnetic field transport in double point contact devices takes place in 1 dimensional edge channels. If the magnetic field is strong enough, these channels are transmitted adiabatically through both the point contacts and the cavity, which means that no scattering occurs between them. A theoretical account for this has been given by Glazman and Jonson (3). Also, in the cavity between the point contacts discrete electron states have been observed, the so called zero dimensional states (2). The experiment on the array (1) has been performed under conditions such that there is only one conducting ld channel. The effect of increasing the gate-voltage is twofold. Firstly, the transmission of each constriction increases, such that the ground states in the cavities become more strongly coupled. The ground state energies thus develop a band structure. Secondly, the effective size of each cavity increases, so that the phase <1>0 acquired by the electron during one revolution around the cavity increases. The gate voltage can thus be used to vary the phase of the electron. A magnetic field of sufficient strength ensures that no channel mixing will occur; it further introduces an Aharonov-Bohm flux through the cavities. The Fermi energy of the system also depends on the gate-voltage, which we won't take into account here.

2.TIIE MODEL The model is closely related to the work ofTsu and Esaki (4) on ld superlattices. We describe the array by N identical equally spaced constrictions in series. Some models have been developed in order to calculate the transmission and reflection amplitudes as a function of the gate voltage for a constriction in a magnetic field (5). One can describe each cavity by a potential that varies slowly in the longitudinal direction (parallel to the array); in the transverse direction it confines the electron. The Schrodinger equation connected to it then has to be solved in the adiabatic limit, which means that the longitudinal coordinate enters as a parameter in the transverse wave function: 0921-4526/90/$03.50

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x

J

'l'x;k(x)(y)

[1]

From this wave function one can calculate the phase integral over one complete period of the classical motion around the cavity (3): [2]

<1>0 = Jk(x)dx

c

The phase <1>0 will increase when the cavity becomes wider, because then the length of the contour will increase. In a magnetic field there will be an extra contribution to <1>0 due to the Aharonov-Bohm phase AB. So we can reduce the problem to the 1 dimensional model depicted in figure I, consisting of an array of N identical barriers at a constant distance with transmission and reflection amplitudes t, 1', r and r'. Each time the electron travels back and forth once between two successive barriers it acquires a phase <1>0.

2

3

4

N

Figure 1. The quantum mechanical ld N barrier system. The transmission amplitude tN of the full system can be calculated following for instance Thouless (6). It is straightforward to show that: t tN-l tN = l-r'TN-lexp(io) TN where

= ~

r+rN-l~exp(io)

.

[3]

l-r'TN-lexp(io) = tt' - rr'. Note that we do not average over the phase

<1>0 in this case since it is assumed that the phase breaking length l~ is much larger than the system length. We then can express the conductance aN of this system in terms of the total transmission amplitude tN by the well-known Landauer formula (7):

Elsevier Science Publishers B.V. (North-Holland)

850

M. van Eijck, F. W.J. Hekking, L.P. Kouwenhoven, G. Schon

[4]

We can now calculate O'N as a function of the gate voltage and the magnetic field by a suitable parametrization of the dependence of T=tt* and «Po on these quantities (1). In the following we will concelltrate however on the dependence on the phase

CPo.

3. RESULTS We now study the transmission of one conducting channel through the array as a function of the various introduced parameters. In fig. 2 we plotted the transmission TN obtained with eq. [3] as a function of «Po for T = 0.90 for various values ofN.

:r:= I

2

Figure 4. T16 as a function of cjlo. The upper curve shows the effect of one barrier with a different phase; the lower curve is obtained by giving two cavities an extra phase contribution.

PHASE/21t

It clearly shows that a band structure develops for increasing N out of the transmission maxima of the individual cavities, with bands symmetric around CPo = 2nx (integer n). Finite size effects, reflected by the N-I resonances in each band, disappear with increasing N. In fig. 3 we study the behaviour for the case N=I6 and T=0.90, 0.25 and 0.0025. The gaps become narrower with increasing T, corresponding to the gradual transition from a Id tight binding limit (smalllj to a Id nearly free electron model (T '" 1.0).

~1J:::U=Jd

Col . . 1 J

PHASE/21t

:rrn "DOJ PHASE/21t

Figure 2. Transmission TN as a function of CPo for T=O.90 and various N.

It

Finally in fig. 4 we show some effects of a weak disorder introduced in the system. The upper curve was obtained by changing the phase of one of the barriers in the array. The resonances then pair into groups of two, reflecting the onset of breaking the array into two parts. The lower curve was obtained by introducing small disorder in the phase «Po in one of the cavities. The resulting band structure is aperiodic; in some parts the band gaps become more shallow, while the resonant structure gets a larger amplitude. However, the number of resonances decreases, again showing the effect of dividing the array in various parts.

I

T~O.25

2

Figure 3. Transmission Tl6 as a function of «Po for various T.

The obtained results agree with the experimental observations of Kouwenhoven et ai. (1), see also (8). It is also possible to perform voltage and temperature averaging; details will be published elsewhere (9).

REFERENCES (1)

(2) (3) (4) (5) (6) (7) (8) (9)

L.P. Kouwenhoven, F.W.J. Helling, B.J. van Wees, C.J.P.M Harmans, C.E. Timmering and C.T. Foxon, submitted to Phys. Rev. Lett; L.P. Kouwenhoven et al. in Localization and Confinement of Electrons in Semiconductors, ed. K. von Klitzing (Springer-Verlag Berlin, Heidelberg, New York, Tokyo). B.J. van Wees et ai. Phys. Rev. Lett. 62 (1989) 2523. L.P. Kouwenhoven et ai. Phys. Rev. B 40 (1989) 8083. L.I. Glazman and M. Jonson, Gtlteborg preprint 89-30, May 1989. R. Tsu and L. Esaki, Appl. Phys. Lett. 22 (1973) 562 H.A. Fertig and B.I. Halperin, Phys. Rev. B 36 (1987) 7969; Y. Avishai and Y.B. Band, Phys. Rev. B 40 (1989) 3429. D.J. Thouless in Physics in One Dimension, eds. J. Bernasconi and T. Schneider (Springer-Verlag Berlin Heidelberg 1981). D.S. Fisher and P.A. Lee, Phys. Rev. B 23 (1981) 6851. RJ. van Wees et ai., this volume. Frank Helling et ai., in preparation.