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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 results 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, [2] 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) [3]- (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 [7] 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 properties 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.

GFET Schottky-

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 diagram 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 number 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

t--WD--

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.

200,

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 subband 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 concentation 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 excited 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. [16]. The authors determined the electron 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 individual subbands ni obtained by 1) a self-consistent calculation (Zrenner et al. [16], 2) the method described in this work, and 3) the infinite well approximation (Went- zel-Kramers-Brillouin method) [ 17). The relative deviations of the subband concentrations Ani/NiD from the selfconsistent calculation, i.e., the most accurate method, are given in parenthesis. The number of occupied subbands 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 concentrations 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 concentration. For donor concentrations < 8 X 10" cm-2 the measured Hall concentration closely follows the donor concentration. 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-

SCHUBERT et al.: DELTA-DOPED FET 627

w "'0 2 4 6 8 IO

n2DEt (lo12cm-21 2 D CARRIER CONCENTRATION

Fig. 5 . Real-space extent zi of electron wavefunctions in the V-shaped potential well of &doped GaAs as a function of electron concentration. The inset shows the spatial extend of the lowest subband zo.

TABLE I COMPARISON OF CALCULATED ELECTRON CONCENTRATIONS IN THE FOUR

LOWEST SUBBANDS ni OBTAINED BY THREE METHODS (The relative deviations from the selfconsistent calculation (An,/NgD) are

given in parentheses.)

i e l f c o n s i s t e n t

! r e n n e r e t a l .

'1 4 /

1 .1x10"

r h i s uorU

NED=4. 1 x1 0"

N =1x10l6

2.37XlO"

( 5 . 8 % )

ellite L-minima in GaAs. In fact, the Fermi level at NDD = 8.5 X 10l2 cm-2 (see Fig. 2) has already touched the bottom of the L-minima, i.e., EF 5 EL - Er, = 310 meV [18]. Electrons in the low-mobility L-minima do not significantly contribute to the mesured Hall concentation because there are simultaneously high-mobility electrons in the I?-minimum. Due to the well-known difficulties of Hall measurements occurring when several groups of electrons with quite different mobilities are present [ 191, [20], the Hall concentration will only account for the high- mobility I'-electrons. The measured Hall concentration is

D p300K

5 10 15 DONOR IACCEPTORI WNCENrRArlON ~10'2cm~21

Fig. 6. Measured Hall carrier concentrations and Hall mobilities of n- and p-type &doped GaAs at 300 and 77 K as a function of the anticipated nominal doping concentration. The inset shows the sequence of epitaxial GaAs layers grown on a semi-insulating (SI) substrate.

thus lower than the actual free-carrier concentration. A more accurate concentration can be obtained from Shub- nikov-de Haas measurements in high magnetic fields 12 11. In addition to this first explanation we have to consider a second possibility, which can also account for the saturation of the Hall concentration: At high impurity concentrations Si atoms might no longer act as donors, but form a Si lattice. However, the concentration of available group I11 atoms on the (100) face of a GaAs crystal is 6.25 X l O I 4 cm-*, the concentration of Si atoms of l O I 3 cm-* is rather small, as compared to the number of available lattice sites. Therefore, nonionized Si atoms are probably not the origin of the measured saturation of the Hall concentration depicted in Fig. 6.

B. Intrinsic Transconductance Electrons in narrow V-shaped potential wells occupy

subbands due to size quantization. Size quantization occurs if both the electron de Broglie wavelength and the mean free path of electrons is larger as compared to the spatial extent of the V-shaped potential well. The subband energies E,, and their spatial extent z , of a strictly V-shaped potential well are approximately given by [12]

n = 0 , 1, - . - . zn = (n + 1)/2[2at?/(2m*~,,)"*], n = 0, I , * .

(2) where q is the elementary change, t? is Planck's constant divided by 2a, E is the permittivity of the semiconductor, m* is the electron effective mass, and N$D is the two-dimensional doping density.

We will now perform an analysis of the transconductance of the GFET using a model originally developed by Hower and Bechtel [22]. Their model assumes a two-re- gion approximation for the velocity-field [ u ( E ) ] characteristic and a velocity saturation at the drain end of the

628 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. ED-33, NO. 5, MAY 1986

TABLE I1 PARAMETERS OF INTRINSIC HEMT AND GFET ASSUMED FOR CALCULA~'ION

OF THE TRANSCONDUCTANCE

GFET 2 0 0 0 I l X l O ' Z

HEMT 6 0 0 0 0 1x10'2

gate. Delagebeaudeuf and Linh [23] and Drummond et al.. [24] have applied the original model to HEMT's. When we slightly modify their model, we can express the mix- imum intrinsic transconductance of a GFET in terms of an analytic expression according to

where L G and WG are the gate length and width, respw- tively, p is the mobility, us is the electron saturated cllift velocity, nZDEG is the concentration of the 2DEG, anc! d is the distance of the center of the 2DEG wavefuncticm from the gate. Equation (3) does not, however, include transient transport phenomena such as velocity overshoot and ballistic transport [25]. According to (3) an optimized FET should have 1) a high 2DEG concentration, 2) thle 2DEG close to the metal Schottky contact, 3) a high 1 0 . w field mobility, as well as 4) a high saturation velocity. 1F0r short-gate-length (LG + 0), (3) reduces to the well-kn0Lv.n saturated velocity model according to

g: = w,WG/d. ( 4.) This simple equation shows that in addition to the sat la- rated velocity only the distance d of the 2DEG from Ithe Schottky gate determines the transconductance of a sho~t- gate FET. In a depletion-mode HEMT, the distance a! is typically 660 [26]. In a GFET, distance of 300 A c m easily be achieved. Fig. 2 shows the calculated maximun transconductance (according to (3)) of an intrinsic HEMT and GFET as a function of gate length using the typical parameters of Table 11.

The saturation velocity is chosen in both devices to t~e 1.5 X lo7 cmls at 77 K [27], and will be further explaine d in Section 11-C. Fig. 7 illustrates that the HEMT has a higher transconductance for L G >> 1 pm. In short-gal e FET's ( L G << 1 pm), however, the GFET exceeds tl- e performance of the HEMT in this theoretical model.

A preliminary experimental output characteristic of a GFET having a gate length of 0.5 pm and a gate width of 300 pm is shown in Fig. 8. The normalized external tramconductance of the GFET is 75 mS1mm. By further opt -. mizing both device processing and crystal growth, higher transconductances will certainly be obtained. In Fig. 9 the transconductance of a GFET as a function of the gate voli- age is shown. The maximum transconductance occurs tit a small negative gate voltage (V , = -0.3 V). Considex-

01 0.2 0.5 1.0 2.0 5.0 10.0 GATE LENGTH L ~ l p r n l

Fig. 7. Calculated transconductances of a GFET and a HEMT according to (3) . The parameters used for the depletion-mode devices are given in the figure.

--

W -0.5V

U

-1.ov 0

1.0 2.0 3.0 DRAIN ~ SOURCE WLTAGE VDS ( V I

Fig. 8. Output characteristics of a GFET with a pinchoff voltage of Vpo = -2 V. The material and device parameters are 2DEG concentration of 5 X lo'* cm-', distance of 2DEG from the gate is 300 A, gate length is 0.5 pm, gate width is 300 pm, and source-to-drain separation is 5 pm.

GATE VOLTAGE V, I V I

Fig. 9. Measured transconductance as a function of gate voltage for a GFET. At a gate voltage of V, = -0.3 V the transconductance reaches its maximum. Device parameters as in Fig. 8.

ing the decrease of conductance versus lowering of Vgs beyond -0.3 V, this observation is not yet fully understood. A similar effect has been discussed by Cappy et al. W I . C. Concentration of the 2DEG and Breakdown Voltage

Electrons originating from donors in a &doped GaAs epitaxial layer will partly be trapped in surface states, while the other part forms the 2DEG. In Fig. 10(a) the real space energy band diagram of &doped GaAs is shown schematically. The constant electric field E between the 2DEG and the Schottky gate is (the spatial extent of the wavefunction in z-direction is taken to be zero)

with 4 B being the Schottky-barrier height (&e = 213 Eg) . Consequently, the charge density transferred into surface

SCHUBERT et al.: DELTA-DOPED FET 629

states amounts to

Accordingly, the free electron concentration in the 2DEG is given by

Using the capacitor approximation, the pinchoff voltage of the GFET is given by

4d v = - n €

2DEG.

With (5)-(8) we obtain the maximum electric field _E that occurs when the electron channel is depleted from electrons (V = V,,), Le.

Consequently, the maximum donor concentration that can be used in a GFET is obtained if the maximum electric field is equal to the avalanche breakdown field

E N;D = - &. (10)

According to this equation the maximum donor concentration does thus not depend on the distance of the 2DEG from the surface of the semiconductor. With an avalanche breakdown field of = 6-7 X lo5 V/cm (at this field the ionization coefficient is CY = lo5 cm-I) the maximum two-dimensional donor concentration in a GFET is obtained to Niymax z 5 X 1Ol2 cm-2. This concentration is significantly higher than the concentration obtained in sin- gle-channel HEMT’s.

The conduction band edge of the SFET depends linearly on the spatial coordinate z as shown in Fig. 10(a). In contrast, MESFET’s (Fig. 10(b)) and HEMT’s (Fig. lO(c)) exhibit a quadratic dependence of the conduction band edge on the spatial coordinate due to their constant doping concentration below the gate. We will now show that the GFET exhibits the largest breakdown voltage of the three devices illustrated, in Fig. 10 if all three devices are assumed to have the same gate-to-channel distance. The maximum electric field in a GFET at reverse bias is given

9

by

1 d &ax = - ($B - VI- (11)

In a conventional MESFET, illustrated in Fig. 10(b), the maximum electric field is given by [28]

2 &XIX = 2 (+B - V ) (12)

where d is the depletion width of the MESFET. In a HEMT, the maximum electric field is even higher as com-

6FET ~~

I

MESFET I

HEMT I

Fig. 10. Schematic real-space energy-band diagram of (a) GFET, (b) MESFET, and (c) HEMT. The GFET has the lowest electric field between the 2DEG and the gate, resulting in a high breakdown,voltage.

6-Doped GaAs CrAu Schottky-Contact 400 Sample # 2803

-12 -10 -8 -1 -0.5 0.5 1 m T=300K,77K

SCHOTTKY DIODE VOLTAGE V LV)

Fig. 1 1 . Current-voltage characteristic of a 8-doped epitaxial layer. The diode has an ideality factor of n = 1.05, a built-in voltage of V,i = 0.73 V, and a high breakdown voltage of V, = - I 1 V.

pared to a MESFET at zero bias (see Fig. 10 for illustration). Inspection of (1 1) and (12) manifests the lowest electric field in the GFET resulting in a larger breakdown (avalanche) voltage in a GFET, as compared to a conventional MESFET or HEMT. In Fig. 11 the current-voltage characteristic of a 6-doped GaAs epitaxial layer is shown. The large breakdown voltage is indeed experimentally ob- served and has a magnitude of V, = - 11 V. From the diode characteristic in the forward direction the ideality factor (n-value) is measured to have the excellent value of n = 1.05. The built-in voltage of the diode is Vhi = 0.73 V at 300 K.

So far; we have shown that the depletion-mode GFET exhibits a high electron concentration, a large breakdown voltage, an electron channel close to the Schottky gate, and a high transconductance. We will next discuss the role of the electron mobility.

D. Role of Electron Mobility In 8-doped GaAs, electrons remain close to their parent

donor impurities in real space due to electrostatic attrac- tion. Therefore, 8-doped GaAs layers have lower mobilities than undoped GaAs. In this section we will investi-

630 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. ED-33, NO. 5, MAY 1986

Fig. 12. S-parameters of GFET. The measuring points are performed a frequencies off = 1, 2 , 3 , . . . , 10 GHz. (SI* at 2, 4, 6 , 8, 10 GHz) Device parameters as in Fig. 8.

gate the role of ionized impurity scattering in &doped epilayers and GFET's.

Fig. 6 shows typical mobilities of 6-doped GaAs epitaxial layers at 300 and 77 K. The mobilities are as high as approximately 3000 cm2/V * s. Such mobilities are common in homogeneously doped GaAs of a donor concentration of No ' = 5 X 1017 cmF3 according to Hilsums formula [29] and experimental data [27]. The mobilities in 6-doped epilayers are higher as expected from the high three-dimensional electron concentration of the 2DEG, exceeding 10'' ~ m - ~ . The enhanced mobility in G-dopedl GaAs is most probably due to the localized impurity dis- tribution (which is 8-function-like, i.e., less than a lattice constant) and the comparatively wide electron distribu- tion, which extends over several lattice constants.

The contribution of ionized impurity scattering (or ionized impurity Rutherford scattering) to the overall mobility of a homogenously nondegenerately doped semiconductor is given by the Brooks-Herring approximation [30]

(13) where kB T = $ E is the electron energy and h is Planck's constant devided by 27r. The dependence of prr - E3I2 in (1 3) shows that the mobility due to ionized impurity scattering increases with the electron energy. Consequently, ionized impurity scattering gets irrelevant at high electric fields. The relative contribution of ionized impurity scattering to the overall electron mobility p results from Mat- thiesen's rule'

I-1 = [ F w P n ) ] - l . (14)

The negligible influence of ionized impurity scattering at high electric fields was indeed confirmed by Thornber [3 11 as well as by Ruch and Fawcett [27]. The authors showed that the saturated drift velocity of electrons does not depend on the low-field mobility. If one applies (13) to de-

'Matthiesen's rule asserts that the resistivity in the presence of several distinct scattering mechanisms is simply the sum of the resistivities one would have if each alone were present.

generately doped semiconductors, the electron energy kB T has to be replaced by the Fermi energy EF. Nevertheless, impurity scattering becomes continuously less significant at high electric fields, when the energy of high-velocity electrons below the gate exceeds the Fermi energy. Con- sequently, in degenerately doped semiconductors, the be- fore-mentioned statements also apply, and this was confirmed by Yanche et al. [32] and reviewed by Chattopadhyay and Queisser [30]. We therefore expect 6- doped epitaxial layers to have the same saturated drift velocity as homogeneously doped semiconductors or selectively doped heterostmctures. The topic of saturated ve- locities in selectively doped heterostructures is, however, still under discussion [33].

The parasitic resistances of a FET between source and gate and gate and ,drain are determined by the low-field mobility of carriers, The HEMT, for example, has very low parasitic resistances due to the high carrier mobility. In a GFET the parasitic resistances are higher due to the lower mobility. However, the parasitic resistances of a GFET can be reduced drastically by the well-known tech- niques of self-aligned gate, or by a recessed gate structure [341 . E. High-Frequency Measurements

Preliminary scattering parameters of experimental GFET's with a gate length of 0.5 pm and a gate width of 300 pm have been measured from f = 1 GHz to f = 10 GHz using a network analyzer of Hewlett-Packard, series 8410. The input reflection (S, forward transmission (S2' ) , backward transmission (SI2), and output reflection (S22) are depicted in Fig. 12. The forward transmission parameter S2, shows a constant amplitude versus frequency. From the S-parameters the current gain hzl is determined as a function of frequency. The current gain is equal to unit at a frequency of fr ' = 13 GHz. Further im- provement of both device processing and crystal growth technique will presumably improve the performance of the new GFET.

111. CONCLUSION A GaAs field-effect transistor with a Dirac-delta-func-

tion-like doping profile is proposed and grown by molecular-beam epitaxy. The subband structure of &-doped GaAs is calculated. The intrinsic depletion-mode GFET is expected to. have a transconductance exceeding 500 mS/ mm due to the proximity of the electron channel to the gate. The GFET makes feasible high electron concentrations of the 2DEG up to 5 X 10l2 cmP2 and a high breakdown voltage. Current-voltage characteristics of &doped GaAs exhibit good ideality factors of n = 1.05. As preliminary external transconductance of 75 mS/mm is measured for a GFET having a gate length of 0.5 pm. Scat- tering of electrons by ionized impurities is discussed and estimated to be irrelevant at high electric fields occurring below the gate. The forward transmission S2, parameter has a nearly constant amplitude in the frequency range 1 GHz f 10 GHz.

SCHUBERT et al.: DELTA-DOPED FET

APPENDIX SUBBAND STRUCTURE OF &DOPED 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. In Fig. l(a) we show the real-space energy band diagram. The epilayer is doped with donors of two-dimensional (2D) concentration NLD. The Dirac delta function is used for description of the charge distri- bution

N&) = NiD,DS(z). (AI)

Electrons remain close to their parent donor impurities and form a two-dimensional electron gas (2DEG). The p- GaAs with a residual acceptor concentration NA is depleted on a width WD and the depleted charge amounts to

qN2D = (~EN,E,)"~. (-42)

In this equation, E is the permittivity of the semiconductor and Eg is the bandgap energy. A small fraction of electrons recombines with acceptors in the p- GaAs layer. Therefore, the electron concentration of the 2DEG amounts to

I ~ ~ D E G = NLD - 2NiD. 643)

We first describe the principle of a new method to calculate the subband structure of a V-shaped potential well with an analytical shape of the conduction and valence band. The shape of the bottom of the conduction band in the potential well is obtained from Poisson's equation if the origin of the coordinate system is taken at the potential notch as illustrated in Fig.. l(b)

dE/dz = ($/2~) NLD, valid for z > 0. (A4)

The electron de Broglie wavelength

hdB = 2~A/(2rn*E)"~ (A51

is matched to the width of the potential well according to

(n + 1) (114) Ad, = (2dq2) (l/NiD) E,

with n = 0, 1, - - (A61

and consequently the subband energies amount to

E,, = (114) 2-lI3 (n + 1)

n = 0, 1, 0 . - . 047)

The real-space extent of subbands is given by

z, = (n + 1)/2[2ah/(2m*E,)1'12]. (AS)

Band-bending effects due to free carriers are not taken into account in (A7), and the potential well is assumed to be of infinite height, i.e., strictly V-shaped. Analytic so- lutions for the strictly V-shaped potential well have also been obtained by a calculation based on the Wentzel-Kra- mers-Brillouin (WKB) approximation [ 171. The energies of subbands obtained bv the WKB amroximation are

63 1

comparable to the energies given by (A7). In Fig. l(b) we illustrate a potential well where bund bending of the conduction band due to free curriers is taken into account. This improves the calculated values of the subband energies significantly and makes them comparable to energies of self-consistent results. As a result the energies of the bottoms of the excited subbands decrease as compared to (A7). The band bending is assumed to occur at the dis- crete points 2 J 2 , where the wavefunction intersects the bottom of the conduction band. Consequently the GaAs conduction band is a polygonal curve, i.e., an analytic function. The wavefunctions are taken to have a sinu- soidal shape, as illustrated in Fig. l(b). The application of the de Broglie wavelength is strictly valid only in a constant potential such as the square well potential but not in a varying potential. Furthermore, the neglect of tunneling effects implies an infinite barrier height of the square well. Therefore, the application of the de Broglie wavelength in a V-shaped potential can be understood as the replacement of a V-shaped potential by an infinite square well of width z, .

In addition to the lowest, the excited subbands also are obtained with good accuracy by taking into account band bending due to both ionized impurities and free electrons. The numerical calculation was performed on a desk-top computer. The computing time for the results presented here is some minutes. The calculation starts with the lowest subband only slightly occupied, i.e., the Fermi energy close to the bottom of the lowest subband. The Fermi energy is then increased by an incremental value AE,, and the subband structure is calculated for the given potential. Then the Fermi energy is increased again by the incremental value. The algorithm is terminated when the charge neutrality (2) is fulfilled, and the subband structure is obtained for one specific 2D electron concentration n2DEG. Further details of the calculation have been given in 1151 where we have applied the same method to selectively doped heterostructures.

ACKNOWLEDGMENT The authors would like to thank R. Kienzler and P. H.

Roth of the Institut fur Elektrische Nachrichtentechnik, Universitat Stuttgart for useful discussions and support. Furthermore, we would like to thank G. Colqhoun of Telefunken Electronic, Heilbronn, H. U. Habermeier, and G. Mertens for assistance with device fabrication.

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632 IEEE TRANSACTIONS QN ELECTRON DEVICES, VOL. ED-33, NO. 5, MAY 1986

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Erdmann F. Schubert ("81) was born in Stutt- gart, Federal Republic of Germany, on February 8, 1956. He studied electrical engineering and solid-state physics at the Technische Universitat Stuttgart and at Oregon State University, Corval- lis. He received the Diplom-Ingenieur degree from the Technische Universitiit Stuttgart in 1981 and the Ph.D. degree from the Max-Planck-Institut fur Festkorperforschung in Stuttgart and the Tech- nische Universitat Stuttgart. His doctoral research concerned the theoretical and experimental char-

acterization of 111-V semiconductors, heterostructures, and doping superlattices grown by molecular beam epitaxy.

In 1986, he joined AT&T Bell Laboratories, Holmdel, NJ, for post- doctoral research in the field of 111-V semiconductors.

Dr. Schubert is a member of the Verein Deutscher Elektrotechniker.

111-V semiconductors.

Albreeht Fischer was born in Bonnigheim, Baden-Wurttemberg, Germany, on July 28, 1950. He received the Dip1.-Ing. (FH) in physical tech- nics with an investigation of the characteristics of titanium sublimation pumps from the Fachhoch- schule Heilbronn.

In 1974, he joined the Max-Planck-Institut fur Festkorperforschung in Stuttgart where he has been working on the design of MBE equipment and on the development of the MBE process to fabricate highly refined 111-V semiconductor structures.

Klaus Ploog was born in Klein Kampen, Schles- wig-Holstein, Germany, on September 27, 1941. He'was educated at the Universities of Kiel and Munchen, where he received a degree in chemistry and the Dr.rer.nat. degree in inorganic solid state chemistry in 1970.

In 1974, he joined the Max-Planck-hstitut fur Festkorperforschung in Stuttgart where he has es- tablished a small group working on MBE. The group has been particularly interested in using MBE technology for bandgap engineering of