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Three-dimensional simulation of single electron transistors

Three-dimensional simulation of single electron transistors
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  2004 4th zyxw EEE Conference on Nanotechnology Three-dimensional simulation of Single Electron Transistors G. Fieri',*, M. G. Pala and G. Iannaccone'?' Dipartimento di Ingegneria dell'hformazione, UniversitA degli Studi di zyxwvut  isa, Via Caruso 1.56122 Pisa, Italy IEIIT-CNR, Via Caruso, 1-56122 Pisa, Italy email:' zyxw AbsfnW- We present a three-dimensional (30) approach for the simulation of single electron transistor (SET) in which subregions with different types of confinement zyxwvu re present simultaneously. In particular, we have applied our model, based on the solution of the Schrodinger equation with DFT to a split gate zyxwvutsrq nd a silicon on insulator (SOI) single electron transistor (SET). The solution of the Schrodinger equation with open boundary conditions bas allowed us to compute the three-dimensional conductance in the linear response regime. Index Terms-Single Electron Transistor, PoissodSchrodinger solver, nanoelectronics device simulation. I. INTRODUCTION The improvement of fabrication techniques in recent years has made possible the realization and characteriza- tion of devices capable of exploiting the discreteness of the electron charge. Single Electron Transistors (SET) are one of this kind zyxwvutsr f devices, and they represent the final step of semiconductor electronics, exploiting the transmission of information through a single carrier. In this kind of devices, it is necessary to take into account the effects of quantum confinement, typically three dimensional around the dot, two dimensional along the wires connecting the contacts, and one dimensional in the two dimensional electron gas (2DEG) [l]. In this paper we present a code in which the Pois- sodSchrodinger equation is solved self-consistently in a zyxwvu 0 domain, by means of a multigrid algorithm and with density functional theory, local density approximation [Z]. The Poisson equation is solved over the entire 3D domain, while the solution of the Schrodinger equation is limited to the confined regions and solved in the momentum space [31. In addition, in order to compute the conduc- tance of the SET, we have developed a method based on the solution of the 3D Schrodinger equation with open boundary conditions along the propagation direction by means of the scattering matrix formalism. 11. PHYSICAL ODELS AND NUMERICAL METHOD The PoissodSchrodinger equation has been solved self- consistently, by means of a multigrid algorithm [4]. While a semiclassical approximation is assumed in the whole domain for charge concentrations, in strongly confined regions, the electron concentration is computed by solving the Schrodinger equation with density functional theory, and local density approximation [Z], [51, [61 [71. In particular, we have considered subregions with one dimensional confinement for the leads and three dimen- sional confinement for the dot. From a numerical point of view, the demand of detailed band profiles implies the use of a finely discretized grid, which in turn poses huge memory requirements for the solution of the Schrodinger equation. Our approach is able to avoid these drawbacks, solving the Schrodinger equation in the momentum space [3]  Indeed, once the eigenvalue problem is transferred in the momentum space by means of the Fast Fourier Transform (FIT), we solve a reduced eigenvalue problem and then we anti-transform the solution in the direct space. Concerning the 3D quantum region, the number of elec- trons in the confined region is fixed, so that its chemical potential can be derived in a simple way by means of Slater's formula [8]. In order to compute the quasi-equilibrium conductance between two generic regions of the structure, we have solved the 3D Schrodinger equation with open hound- ary conditions along the direction of propagation, using the conduction band previously computed, in a post- processing phase. We have then obtained the quantum conductance G at zero temperature using the Landauer-Biittiker formula [9],  [IO] that relates G with the transmission matrix t: where zyxwv   is the electron charge and h is Planck's constant. In particular, the numerical method is based on subdi- viding the propagation direction in several slices and in computing the transmission and reflection coefficients of the wave function, imposing the continuity of the wave function and of the current density at the interface between adjacent slices. 111. RESULTS AND DISCUSSIONS zyx . Split Gate SET We have applied our model to a single electron transistor defined by split gates on an AIGaAdGaAs modulation 0-7803-8536-5/04/ 20.00 02004 IEEE 337  TABLE I zyxwv OTAL OT CAPACITANCE AND GATE-DOT CAPACITANCE DERIVED doped heterostructure. The layer structure is shown in Fig. la, while the gate layout adopted in the simulation is shown in Fig. Ib. Surface states have been considered FROM SIMULATIONS AS A FUNCTION OF THE NUMBER OF ELECTRONS IN THE DOT. transition zyxwvu   zyxwv lectrons) 2 2 zy  3 4 z 9 3.84 zyxw F 1.68 aF 4.67 aF CT 105.21 aF 46.93 aF 131.95 zy F 0.713 aF 0.297 aF 0.805 aF 99.94aF 44 66 aF 125.67 aF Cj C Fig. 1 a) Layer smture of zyxwvutsrqp he simulated device b Gate layout of th simulated device. In lhe picture are shown the regions n which 1D and 3D quantum analysis have been performed. Gate 1.3.4 and 6 are the extemal zyxwvutsrqpon ates that deli ne the ODEG. while 2 and 3 are the gates that determine the number of electrons in the dot. on the ungated region at the top of the shucture using a model typically applied to metal-semiconductor inter- faces [Ill 12]. In the two-dimensional electron gas, two kinds of con- finement have been considered. As shown in Fig. lh, in the center of the ZDEG, the three dimensional Schrodinger equation has been solved, since quantum confinement is strong along all the three directions, while, in the rest of the 2DEG, the Schrodinger equation has been solved only in the vertical direction. Fig. 2 shows the chemical potential Gate Voltage (V) Fig. 2. Chemical potentials as a function of the inner gate voltlge for a number of elemns ranging from I to 4, and equivalent cvpaciwnce circuit of the SET. as a fiinction of the inner gate pair voltage, for a voltage applied to the extemal gates VG=-l.l V. From these curves it is possible to extract the values of the gate capacitance between the dot and the inner gates C,) and the total capacitance zyxw CT) efined as CT = cj cg +cm, (2) where Cj is the capacitance between the dot and each reservoir and Cm is the capacitance between the dot and the other gates. In Fig. 2 the equivalent circuit of the SET is shown. In particular [13] we have, where 4Vc and 4p are the distance between two curves corresponding to p N) and p N + 1) at fixed VG and p respectively, as shown in Fig. 2. The values of the capacitance extracted from Fig. 2 are shown in Table I. The capacitance Cj has been obtained in a similar way as 4) where VD is the drain voltage, and the remaining C capacitance can he finally obtained from (2). Cj values are quite small, meaning that in this structure, the dot is weakly coupled with the near reservoirs. E. Silicon-on-insulator SET We now focus on the silicon-on-insulator (SOI) SET fabricated and characterized at the University of Tiihingen by Augke et al. [141 (Fig. 3). The SET consist in two silicon side gates and in a silicon dot in the center of the structure that is separated by tunnel barrier from source and drain. In Fig. 3 the equivalent circuit of the SET is also shown, where Cs and CO are the dot-to-reservoir capacitances, CQ the side capacitance, and Cec s the dot to back gate capacitance. In order to estimate the device capacitances, we are primarily interested in the evaluation of the chemical potential of the dot (Fig. 4a). Since from the SEM picture it is not clear whether the central region is prismatic or rounded, we have performed simulations for both geometries. In particular, in Table 11, we show the capacitance values obtained for the two structures and the experimental measurements given in 338  TABLE zyxw I COMPARISON BETWEEN SIMULATION CAPACITANCE AND EXPERIMENTAI. DATA zyxwv   zyx 41. ‘Vback gale a) b) Fig. 3. SEM micro-gnph of the SO1 SET 1141 and the equivalent Circuit. 0.1 , , , , , , , , -1 -0.8 -0.6 -0.4 x 10 zyxwvutsrqpo   VG (v) yo 01 Fig. 4. a The chemical potential of the dot for different zyxwvut umber of electrons as VG is varied for the prismatic sl~cture. The zyxwvut eaks correspond respectively to zyxwvutsrqpon   = 1 N = 2, N = 3, N = 4. Temperature is 4.2 K. b) The coulomb blockade oscillations obtained as Vc is varied for the prismatic smcfure. Temperature is from 4.2, to 7, to 10 and to 15 K. The oscillation Deriod is 0.1 V. Ref. [I41 : it is clear that the results are very sensitive to the dot shape. In Fig. 4b conductance oscillations at different tem- peratures are plotted for the prismatic structure. Even if the conductance differs by a factor of IO and the oscillation period is approximately twice as compared to the experimental data, we can consider our conductance simulations acceptable, since we do not know the actual geometry of the dot. IV. CONCLUSIONS In this paper we have presented a 3D simulation approach particularly suitable for semiconductor single electron transistor that require a huge number of grid points. We present a code based on the solution of the CG 1.6 zyxw F 4aF 3.4 aF CD laF 6aF 16 aF Schrodinger equation with DFT by means of a multigrid algorithm, the solution of the single particle Schriidinger equation in a reduced momentum space basis, and the computation of scattering matrices for 3D structures. In the examples shown we have been able to recover all sig- nificant microscopic quantities such as charge distribution and conduction band profiles as well as capacitances and transport properties. ACKNOWLEDGMENT Financial support from IST NanoTCAD project (EC contract IST-1999-108’28) is gratefully acknowledged. Au- thors acknowledge fruitful discussions with A. Schenk and E Heinz. REFERENCES [I] A. Scholze, A. Schenk. and W. Fichtner, “Single-electron device Simulation’’ IEEE Trans. Elect. De”,, Vol. 47, No.10, pp.1811-1818, 2000.  [21 J. C. Inkson, Many body theory of solids- on nrmduction, New York Plenum, 1984.  [3] M. G. Pala and G. lannaccone. “A three-dimensional solver of the  Schr’odinger equation in momentum space for the detailed simulation  of nano~VUc1ures” londechnolo~y. o1.13, pp.369-372, 2002. [4] W. H. Press, S. A. Teukolsky. W. T. Vending and B. P. Flannery Nu- merical Recipes in Fortran 77, second edition New York Cambridge University ss. 1992. [5] H. Hohenber and W. Kohn, “Inhomogeneous Electmn Gas” Phys. Rev. .Vol. 136. oo.B8M-B871. 1964. [6] W. Kohn and ‘i. . Sham, .“Self-Consistent Equations Including Exchange and Correlation Effects” Phys. Rev, Vol. 140, pp.AI133- A1138, 1965. [7] U. von Bxlh and L. Hedin, “A Id xchange-correlation potential for the spin polarized case”. J Phys. C o1.5.pp.1629-1642, 1972. [8] J. C. Slater, “A Simplification of the Haruee-Fock Method“,Phys. Rev., Vo1.81, pp.385-390, 1951.  [9] R. Landauer, ”Spatial vwivtion of currents and i elds due to localized scatten in metallic conduction”, IBMJ Res. Dev., Vol.l, p.223, 1957. [IO] M. Buidker. “Symmetry ofelectrical conduction”. IBM 3 Res. Dev., Vo1.32. p.306, 1988. [Ill S. Sze, Physics of Semiconductor Devices. 2nd Edition. New York [I21 M. G. Pala, G. lannaccone. S. Kaiser, A. Schliemann. L. Wonchech and A. Forchel, ”ExuactiOn of parameten of surface states from experimental test smctures” Nanotechnology, Vol. 13, pp.373-377, 2002. [I31 M. Macucci, K. Hess, and G. I. lafnte, “Electronic energy spectrum and the concept of capacitance in quantum doe”, Phys. Rev. B. Vo1.48. pp.17354-17363, 1993. Cl41 R. Augke, W. Eberhardt, C. Single, F. E. Ens, D.A. Wharam. and D. P. Kem, ‘Doped silicon single electron vansiston with single island chancteristics”, Appl. Phys. Len. Vo1.76, pp.2065-2067,ZWO. I. wiley sans. 1981. 339
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