Ballistic transport in semiconductor at low temperatures for low-power high-speed logic

IEEE TRANSACTIONS ON ELECTRON

DEVICES, VOL.

ED-26, NO. 1 1 , NOVEMBER 1979 1677

Ballistic Transport in Semiconductor at Low Temperatures for Low-Power High-speed Logic MICHAEL s. SHUR, MEMBER,

IEEE, AND

Abstmct-At low temperatures, a mean free path of electrons in semiconductors may exceed device dimensions. Current-voltage characteristics, potentials, electrical field, and carrier distributions are calculated for a two-terminal device undersuch conditions whenthe electron transportis ballistic. Current-voltagecharacteristics of a “ballistic” FET are analyzed using an approach similar to the Shockley model. It is shown that very high drift velocities can be obtained at low voltages leading to high speed and low power consumption in possible applications in logic circuits. For example, GaAs logic devices with characteristic dimensions about a micrometer or less at 77 K will be comparable with or better than Josephson tunneling logic gates.

I. INTRODUCTION

A

T LOW TEMPERATURES, a mean free path (X) of carriers in semiconductors may become larger than the length of a semiconductor device. For example, the electronic mobility p of high-purity GaAs at 77 K is about 150 000 cm2/V . s leading to the values of

- --

X ruth

Er

4

(3kTm*)’I2

- 1.32 pm.

Here r is the momentum relaxation time, vth is the thermal velocity, T is thelatticetemperature, m* is the electronic effective mass, q is the electronic charge, and k is the Boltzmannconstant. For the above estimate we assumed p = 150 000 c m 2 / V -s, T = 7 7 , and m* -0.068 m,, where m, is the free electronic mass.Thevalue of X is certainly much larger than the minimum device length which could be achieved by the modern submicrometer technology [ l ]. It should be also stressed that higher values of the low-field mobility may be reached in GaAs.More energetic electrons might have an even larger mean free path because of the reduced ionized impurity scattering cross section. There arealso othercompounds and alloys which could be grown on thesemi-insulating substrate and yield even longer scattering times. The gate length of GaAs MESFET’s, for example, can be made about 0.2 pm and probably less. But even for a 0.5-1pm-long device made by ordinary photolithography the electron scattering will be relatively small if h 1.32 pm. In the present paper we analyze the current-voltage characteristics of a semiconductor under such conditions. We con-

-

Manuscript received January 31, 1979;revised May 31, 1979. M. S. Shur is with Department of Electrical Engineering, University of Minnesota, Minneapolis, MN 55455. L. F. Eastman is with the School of ElectricalEngineering, Comell University, Ithaca, NY 14853.

LESTER F. E A S T ” ,

FELLOW, IEEE

sider a one-dimensional problem so that most of our results are applicable, strictly speaking, to a two-terminal device (a “ballistic” diode). We consider, however, current-voltage characteristics of a “ballistic” FET using an approach which is similar to the Shockley model for the conventional FET’s. We also use our results to speculate about the possible device implications. Ouranalysisshows that in “ballistic” GaAs devices electrons travel like in an electron beam reaching high values of drift velocity at very low voltages leading to short propagation delay and low power Consumption. Our estimates indicate that high-speed (picosecond-range) low-power logic elements canbe implemented with powerdelay products of a small fraction of a femtojoule. At this point, our work is parallel and extends the recent paper o f Rees et al. [2] who proposed a novel typeof GaAs FET operated at low temperature and predicted that such a device would allow a high switching speed at drastically reduced powerlevels.Our analytical estimates of power-delay products are in good agreement with the power-delay product obtained in computer simulation of a low-doped GaAs FET at 80 K performed by Rees et al. [ 2 ].

11. BASICEQUATIONS We consider a one-dimensional problem for an n-type semiconductor. We neglect diffusion and electron scattering. In such conditionsthe basic steady-state equations are the equation for the currentdensity j = qnv

(1)

Poisson’s equation

and the equation for theelectron velocity

Here n is the electron concentration, no is the doping density, U is the electric potential, x is the distance from the cathode (we assumed U(0) = 0), and is the permittivity. This simple approach is valid if X > L , where L is the active length of the device. If X < L , either a full Monte Carlo technique [ 3 ] ora simplified approach [ 4 ] ba.sed on Monte Carlo calculations has to be used. When X 3.43 and

(y )

Pr(fJ) = 0.1602 u(V)

112 2~

4

vel< - 0.4053

n2

(see (24) and (33)). The current-voltage characteristics j versus u for a 1-pm GaAs diode are presented in Fig. 5. The voltagerange is limited by 4

L(pm) . s ( p m 2 )

0.354 (~01)~’’

at low voltages

%U