IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. ED-33, NO. 5, MAY 1986 625
The Delta-Doped Field-Effect Transistor (GFET)
Abstruct-A field-effect transistor (FET) using a two-dimensional electron gas (2DEG) as an electron channel is fabricated from GaAs grown by mo\ecu\ar-beam epitaxy. The doping profile of the fie\&-&- fect transistor is described by the Dirac delta (6) function. The subband structure of &-doped GaAs is calculated. The characteristics of the GFET are a high concentration of the 2DEG, a high breakdown voltage of the Schottky contact, a narrow distance of the 2DEG from the gate, and a high transconductance. These properties are analyzed. Preliminary re- sults for the extrinsic transconductance and for the transit frequency are obtained from GFET's having nonoptimized structures.
H I. INTRODUCTION
IGH-SPEED field-effect transistors fabricated from selectively doped heterostructures [ 11,  are can-
didates for future high-performance integrated digital cir- cuits based on III-V compound semiconductors. The con- ducting channel in these field-effect transistors, called twa-dimensional electron gas field-effect transistor (TEG- FET), modulation-doped field-effect transistor (MOD- FET), or high electron mobility transistor (HEMT) - (61 ~ is a two-dimensional electron gas (2DEG) with a high electron mobility at low temperatures. Originally the high tramconductance of HEMT's was attributed to the high electron mobility  exceeding 50 000 cm-2/(V * s) at 77 K. Subsequently, it was found that the mobility de- creases significantly at fields higher than E = 100 V/cm [S}-llO}. Furthermore, it was shown that HEMT's have an improved performance if the width of the spacer layer is decreased [ 1 1 3, even though this reduces the mobility. These findings imply the question of whether the proper- ties of the 2DEG (such as the confinement of electrons to two dimensions and the resulting high 3D concentration of the electron gas) rather than the high low-field mo'oiliq determines the speed of short-gate HEMT's.
We have recently proposed- an alternative field-effect transistor having a 2DEG [12J, which is entirely com- posed of GaAs and which was grown by molecular-beam epitaxy (MBE). A schematic illustration of the new field- effect transistor called GFET is shown in Fig. 1. The @ET has a Dirac-delta-function-like doping profile [ 131, [ 141. This doping profile results in a V-shaped conduction band. Electrons occupy quantized energy levels in the potential well due to size quantization. The advantages of the GFET are the high concentration of the 2DEG, the high gate- breakdown voltage, the proximity of the 2DEG from the gate, and the high transconductance.
Manuscript received November 1 1 , 1985; revised January 22, 1986. The authors are with Max-Planck-Institut fur Festkorperforschung, D-
IEEE Log Number 8607886. 7000, Stuttgart-SO, Federal Republic of Germany.
Source Gale Drwn 1
Fig. 1 . Schematic illustration of a &doped GaAs field-effect transistor grown on a semi-insulating GaAs substrate.
In this work we present design rules for the GFET, ana- lyze its properties, discuss the role of electron mobility, and present first experimental data o f both direct-current output characteristics and high-frequency measurements in the GHz range ( f 2 10 GHz).
11. RESULTS AND DISCUSSION A. 6-Doping of GaAs
The ionized impurities in the &doped epitaxial GaAs layer form a V-shaped potential well, and the electron energies are quantized for motion perpendicular to the (100) growth surface. The real-space energy-band dia- gram is shown schematically in Fig. 2. We calculate the subband structure by a method described in the Appendix. Our approach uses 1) the one electron picture, 2) takes into account bandbending due to localized impurities and free carriers, 3) takes the GaAs conduction band to be a polygonal curve, 4) takes the wavefuoction to be sinusoi- dal and, 5) neglects tunneling effects. The method can be understood as a replacement of the V-shaped potential well by an infinite square well [ 151.
In Figs. 3 and 4 the numerical results are presented for the electron subband structure of a &doped GaAs layer. The subband energies Ei and the Fermi energy EF as a function of the 2D carrier concentation are shown in Fig. 3(a) for a background acceptor concentration of NA = 1 X l O I 4 ~ m - ~ . The inset illustrates the quantized energy levels in potential well. The beginning of the population of a specific subband is marked by an arrow and the num- ber of the specific subband. At an electron concentration of nZDEG. = 1 X 1013 Cm-2, seven subbands are already populated. The sixth excited subband of energy E6 starts to be populated at a carrier concentation of n 2 ~ E G = 6.5 X 1Ol2 cm-2, Le., the total number of occupied subbands is then seven.
Fig. 3(b) shows the subband energies with a residual
0018-9383/86/0500-0625$01.00 O 1986 IEEE
626 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. ED-33, NO. 5 , MAY 1986
Fig. 2 . (a) Real-space energy band diagram of a &doped GaAs layer (lot drawn to scale). The two-dimensional doping concentration is N i D 2nd the two-dimensional depletion concentration in the p- GaAs layer is LV:~. Electrons remain close to their parent ionized donor impurities and form a two-dimensional electron gas (2DEG). (b) Schematic polygollal shape of the conduction band of a delta-doped semiconductor used ;'or the calculation outlined in the Appendix.
400r - 5 1 I
3 E 300
8 200 $ 1 W 5 f 100 0 $ 1 O l 3
0 2 4 6 8 1 0 0 2 4 6 8 1 0 2D CARRIER CONCENTRATION nDEG (d2cmz;2,
Fig. 3. Subband energies Ei and Fermi energy EF versus electron concen. tration of &-doped GaAs at a background concentration of (a) N , = 1 X loL4 and (b) NA = 1 X 10l6 ~ m - ~ . The beginning of the population of subbands is indicated by arrows.' The smaller number of occupied subbands in (b) is due to the higher background acceptor concentration.
Fig. 4. Subband separation E, - EF and distance of the Fermi energy from GaAs. the lowest subband EF - Eo versus electron concentration of &-doped
background acceptor concentration of NA = 1 x 10"j ~ m - ~ . A comparison of the subband energies with those of Fig. 3(a) shows a higher subband separation for higher background acceptor concentrations. Note that all sub- band energies approximately follow a (n2DEG)2'3 rule (see also (A7) in the Appendix).
The subband separations Ei - Eo and the distance of the Fermi energy from the lowest subband EF - E, are displayed in Fig. 4 for a background acceptor concenta- tion of 5 X loi4 ~ m - ~ , The subband separation varies from some few millielectron volts among higher excited subbands to 70 meV between the lowest and the first ex- cited subband at an electron concentration of n2DEG = 1 X loi3 cm-'.
We now compare our calculated concentrations in the individual subbands ni with experimental results obtained by Zrenner et al. . The authors determined the elec- tron concentrations in the individual subbands of &doped GaAs by Shubnikov-de Haas measurements. At a 2DEG concentration of n2DEG = 4.1 X 1 0 ' ~ they found the concentration of the first and second excited subband to be n1 = 1.0 X 1OI2 cm-2 and n2 = 0.4 X 10" cm-', respectively. Higher subband populations were not re- solved in the measurements because of the long period of Shubnikov-de Haas oscillations from weakly occupied subbands. Our calculated subband concentrations for a 2D electron concentration of n2DEG = 4.1 X 1 0 ' ~ cm-' are nl = 1.1 X 10l2 cm-2 and n2 = 4.8 X 10" cm-2, and they agree well with the experimental results.
In Table I we compare the concentrations of the indi- vidual subbands ni obtained by 1) a self-consistent cal- culation (Zrenner et al. , 2) the method described in this work, and 3) the infinite well approximation (Went- zel-Kramers-Brillouin method) [ 17). The relative devia- tions of the subband concentrations Ani/NiD from the selfconsistent calculation, i.e., the most accurate method, are given in parenthesis. The number of occupied sub- bands of our calculation coincides with the one obtained by the self-consistent calculation. In contrast, the infinite well approximation yields only two occupied subbands. Moreover, the relative deviations of the subband concen- trations are more than three times as large as compared to our method.
Fig. 5 shows the real-space widths o f eight subbands at a background acceptor concentration of NA = 5 X 1014 cm-2. The inset illustrates the real-space extent zo of the lowest subband. The lowest subband has an extent of 50 to 100 A depending on the carrier c?ncentration. Higher excited subbands extend up to 300 A in real space. The. real-space widths of the electron wave functions become smaller at high 2D carrier concentrations.
In Fig. 6 the measured Hall carrier concentration is shown as a function of the nominal 2D donor concentra- tion. For donor concentrations < 8 X 10" cm-2 the mea- sured Hall concentration closely follows the donor con- centration. In contrast, for donor concentrations > loi3
the measured Hall concentration saturates. We as- sume that this saturation is due to a population of the sat-