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EE230-Spring09 HW1 (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 e2 0.025 eV; 1 2 0.005 eV CT *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) ex aex eiEt / h Assuming is (i) completely imaginary(=i ) or (ii) completely real .