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(Simple model for ballistic nanoscale devices)
1.(a) Using the equations for N (Eq. 1.4.2) and I (Eq. 1.4.3) plot current vs. drain
voltage and number-of-electrons vs. drain voltage as shown in Fig. 1.4.6. Use
  0;  0.2 eV; Vg = 0; kT = 0.0259; CD/CT = 0.5; CG/CT = 0.5; CS/CT = 0 and
U0 
 0.025 eV; 1   2  0.005 eV
*Note that you need to self-consistently solve for these equations. Also while
writing the program, pay attention to the consistency of units.
1.(b) Now vary gate voltage from 0-0.5 Volt and plot I vs. Vd for different gate
voltages in the same plot (3-4 gate voltages within this range will suffice).
1. (c) In bulk transistors the current saturates due to the well-known pinch off
effect. Such mechanisms do not exist in ballistic transistors. However 1(b) and 1(c)
shows a saturation of the current. What is the reason for this saturation?
1.(d) Now fix the gate voltage to zero and plot I-Vd for three different U0:
U0  0; 0.025 eV; 7*0.025 eV;
plot the results in the same figure.
From the plots what can you say about the effect of self-consistent field on the
 behavior of a ballistic transistor? How is the power dissipation of the device related
to this self-consistent field?
Strength of the simple model:
Although very simple, this model is an extremely powerful tool for understanding
current flow in nanoscale devices. Replacing the Lorentzian density of states (DOS)
by the proper DOS of the material being used, it is possible to explain the behavior of
close-to-ballistic transistors. You may find it instructive to read the following paper
for a description of how it is done:
A. Rahman, J. Guo, S. Datta, and M. Lundstrom, "Theory of Ballistic Nanotransistors",
IEEE Transactions on Electron Devices, 50, pp. 1853-1864, 2003.
Later in the course we shall see how the DOS can be calculated.
In the following we shall look at two other examples that illustrate the strength of
this very simple model:
2. (Thermoelectric Effect): If the DOS around the equilibrium Fermi level  is such
that the DOS increases with increasing energy then it emulates a conduction band
whereas in the opposite case it emulates a valence band (c.f. the band s of a bulk
material). Hence by placing the Fermi level below the peak of the lorentzian we can
a valence band.
model a conduction band and by placing it above we can emulate
(a) Using the same program developed in prob. 1, plot current I as a function of
equilibrium Fermi level  for Vd=1V. Vary  from -1 to +1 eV. You should see that
the sign of the current is independent on the placement of  .
(b) Now put Vd =0; Change the temperature of contact 1 from that of contact 2 so
 Use these temperatures
that T1=1.33T2;
when you calculate f1 and f2. Due to these
 although there is no voltage
different temperatures, now some current will flow
anywhere in the device. This is the thermoelectric effect. Plot current I as a function
of  . You will see that the sign of the current is reversed depending on where you
place the  .
There was a long debate on whether current flowing through a single molecule goes
through the valence band or the conduction band. The aforementioned theory was
proposed in
Paulsson and Datta, Phys. Rev. B 67, 241403 (2003).
Subsequently the following experiment was performed
Reddy et al., Science 315, 1568 (2007)
that resolved the issue.
3. In the same program as developed in Prob 1, change 1 such that
1(E)  0.005 eV; E > 0
= 0 otherwise
Keep  2 unchanged. Such a coupling can be used to simulate a semiconductor
contact where the bandgap starts below E=0. Plot I-Vd at Vg=0; You should see that
a Negative Differential Resistance (NDR) appears only for the positive side of the
voltage bias.
The fact that a molecule sandwiched between a semiconductor and metal may show
such NDR switching was proposed :
Rakshit et. al., Nano Letters, 2004, 4 (10), pp 1803–1807.
And subsequently experimentally demonstrated:
Guisinger et al. PNAS June 21, 2005 vol. 102 no. 25 8838-8843.
Schrodinger’s Equation:
4. (E2.3) Find out the probability current density for a wavefunction
(x,t)  ex  aex eiEt / h
Assuming  is (i) completely imaginary(=i  ) or (ii) completely real  .