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Negative Differential Conductance and Threshold Voltage

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					          Negative Differential Conductance and Threshold Voltage Distribution
                  in Two-island Single-electron Tunnelling Structures

                            Gareth J. Evans1 and Hiroshi Mizuta2
    1
      Microelectronics Research Centre, Cavendish Laboratory, Cambridge CB3 OHE, UK
      2
        Hitachi Cambridge Laboratory, Cavendish Laboratory, Cambridge CB3 OHE, UK

A huge variety of experimental devices have shown Coulomb Blockade (CB) effects in systems
ranging from metal-insulator-metal systems to semiconductor dots produced by electrostatic,
geometric or dopant fluctuation confinement. In principle, CB effects allow current flow to
be manipulated on a per electron basis and has attracted attention as a possible future ULSI
technology.
    Single-island systems, which are called Single Electron Transistors (SETs), have well-
understood simple characteristics. However, the next logical extension, a two-island system,
exhibits a huge variety of complex characteristics (for example see [1]).
    This paper describes the analysis of the two-island system in Figure 1 and relates its be-
haviour to the development of its two-dimensional phase space (q) as the source-drain voltage
Vds changes. We propose a geometrical approximation called the polytope approximation to
track the changes in phase space. (A polytope is a finite volume polyhedron and for the two-
island case is a polygon.)
    As an example of the variety of behaviour possible for two-island systems, we demonstrate
a device that exhibits strong negative differential conductance (NDC) shown in Figures 2 and
3. Interestingly, this NDC is not connected to discrete quantum energy levels but due to the
charging energy considerations. Previously, Nakashima et al. have shown NDC to exist in linear
arrays of seven- and nine-island systems [2,3], while Heij et al. [4] demonstrated a two-island
system that exhibited NDC. Shin et al. [5] have also shown that NDC can exists in ring-arrays of
junctions. Our two-island system is most similar to Nakashima et al.’s device and the polytope
approximation lets us derive the following necessary condition for NDC,
                                          C1d   C2d
                                              >
                                          Cs1   Cs2
    Figure 2 demonstrates that NDC exists only when the offset-charge or gate biases are tuned
to the correct part of phase space. The offset-charge problem makes a system’s characteristics
very unpredictable. Assuming that the offset charge is uniformly distributed in the range −0.5e
to +0.5e then the threshold voltage distribution for a structure can be calculated from the poly-
tope approximation as shown in Figure 4 and the actual polytopes are shown in Figure 5, where
regions less than 1 fA are black.[6,7]

[1] G. Y. Hu and R. F. O’Connell, J. Phys.: Condens. Matter 5 7259 (1993)
[2] H. Nakashima and K. Uozumi, Jpn. J. Appl. Phys. 34 L1659 (1995)
[3] H. Nakashima and K. Uozumi, J. Vac. Sci. Tech. B 15 (4) 1411 (1997)
[4] C. P. Heij, D. C. Dixon, P. Hadley and J. E. Mooij, Appl. Phys. Lett. 74 (7) 1042 (1999)
[5] M. Shin, S. Lee, K. Park and El-H. Lee, J. Appl. Phys. 84 (5) 2974 (1998)
[6] G. J. Evans, H. Mizuta and H. Ahmed, Jpn. J. Appl. Phys. 40 5837 (2001)
[7] G. J. Evans, Transport in Silicon Nanowires, PhD Thesis (submitted 2001), University of
Cambridge
                                                                                                                                                                        0.15
                                                                                                                                       Vds (V)            0.1
                                                                                                                                       0.05
                                                  source (Vds )                                               0
                                                                                             -8
                                               Cs1




                                                                                                                           log10 Ids
                                                                                             -10

                                   1                                                         -12
                         Cs2             C12                                                 -14
                                                                                             -16
                                                                                                                              0.4
                                                          C1d                                                                           0.2
                                          2                                                                                                      q1   0
                                                                                                                                                                   -0.2

                                               C2d                                                                                                                               -0.4

                 drain (0 V)
                                                                                     Figure 2: log10 [Ids ] as a function of Vds and the offset
Figure 1: The two island circuit under                                               charge on island 1. Changing the offset charge on island
analysis. This is the most general two-                                              2 from −0.5 to 0 roughly changes the phase of the oscil-
island linear array possible.                                                        lations by π.

                                                                                                                           12
                                        Vds (V)
                       0.025     0.05         0.075     0.1       0.125       0.15
         log10 Ids




  -10
                                                                                                                           10
                                                                                              Prob. Density (arb. units)


 -12.5
                                                                                                                            8
  -15

                                                                                                                            6
 -17.5

  -20                                                                                                                       4
                                                                                                                                       0.00832


                                                                                                                                                        0.02496




                                                                                                                                                                                        0.05824


                                                                                                                                                                                                     0.07488


                                                                                                                                                                                                               0.09152
                                                                                                                                                                        0.0416
 -22.5                                                                                                                      2

  -25
                                                                                                                                                 0.02              0.04             0.06              0.08               0.1   0.12
 -27.5
                                                                                                                                                                                                  Vds (V)


Figure 3: Part of Fig. 2 in the NDC region at                                               Figure 4: Threshold voltage distribution for a
temperatures from 8.2 K (top) to 1.2 K in 1 K                                               non-NDC circuit. The line is the polytope ap-
steps. Vertical lines are predictions of the valley                                         proximation’s prediction and the dots are from a
positions and the solid lines of valley currents.                                           Monte Carlo simulation of the device.



                      0.4                                            0.4                                                                                          0.4
                      0.2                                            0.2                                                                                          0.2
                       0                                                  0                                                                                        0
                     -0.2                                         -0.2                                                                                    -0.2
                     -0.4                                         -0.4                                                                                    -0.4
                                Vds = 0.00832 V                                Vds = 0.02496 V                                                                                    Vds = 0.0416 V
                                  -0.2 0 0.2 0.4
                               -0.4                                              -0.2 0 0.2 0.4
                                                                              -0.4                                                                                                  -0.2 0 0.2 0.4
                                                                                                                                                                                 -0.4



                      0.4                                            0.4                                                                                          0.4
                      0.2                                            0.2                                                                                          0.2
                       0                                                  0                                                                                        0
                     -0.2                                         -0.2                                                                                    -0.2
                     -0.4                                         -0.4                                                                                    -0.4
                                Vds = 0.05824 V                                Vds = 0.07488 V                                                                                    Vds = 0.09152 V
                                  -0.2 0 0.2 0.4
                               -0.4                                              -0.2 0 0.2 0.4
                                                                              -0.4                                                                                                  -0.2 0 0.2 0.4
                                                                                                                                                                                 -0.4
Figure 5: The current as a function of q at the Vds corresponding to the vertical lines in Figure 4. The
polytopes are marked with solid lines and their interiors indicated with dashed lines.

				
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