Metal(n) AlGaAs-GaAs two-dimensional electron gas FET

IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. ED-29, NO. 6 ,J U N E 1982

h-parameters is studiedmore extensively.For a value of h=2. cm4/sthere are also two ranges witha linear decay of thejunction voltage. In the firstlinear range the lifetimeafter Daviesis determined to be 0.7 T . The second linear range begins at t = 4 T . In this region the true carrier lifetime again can be determined with a good approximation from the Davies formula. The influence of a reduction of the forward current on the time dependence of the junction voltage has also been examined. At a current turn-off from 100 A/cm2 and a value of h = 2 * cm4/sfortherecombinationparameters in the end regions, the same time dependencies, as before at current turn-off from 3000 A/cm2, appear qualitatively. The lifetime after Davies is found now to be 0.8 7 in the first linear decay of the junction voltage, while it is identical with the true lifetime in the second linear range. III. SUMMARY The results of our numerical investigation can be summarized as follows. The evaluation of the lineardecay of theopencircuit voltage after Davies gives the true value of the carrier lifetime in the i-region of p-i-n rectifiers, only if the recombination in the end regions is negligible. Taking into account the recombination in the end regions the Davies formula gives a

955

smaller value for the lifetime, when the first linear decay range of the open-circuit voltage is evaluated. The error in the determination of the carrier lifetime amounts to 40 percent for an initial current density of 3000 A/cmZ and h-parameters of cm4/s. Moreover, it increases with increasing current density and with increasing h-parameters. However, the true value of the carrier lifetime can be determined after Davies in the second linear range of the open-circuit voltage decay with a high accuracy. REFERENCES

[ l ] L. W. Davies, “The use of p-i-n structuresininvestigations of transient recombination from high injection levels in semiconduct o r s , ” & ~ ~ . I E E Evol. , 51, p. 1637, 1963. voltage [2] H. Schlangenotto and W. Gerlach, “On the post injection decay of p-s-n rectifiersat high injection levels,” Solid-State Electron., vol. 15, p. 393, 1972. [3] M. J . Ben Hamouda,“StationgresundtransientesVerhaltenvon pin-DiodenbeiHochinjektion,” M.S. thesis, Technische Universitgt Berlin, Germany, 1981. [4] F. Berz, R. W. Cooper,and S . Fagg, “Recombination in the end p. 293, regions of pin diodes,”Solid-stateElectron.,vol.22, 1979. [5] N. H. Fletcher, “The high current limit for semiconductor junction devices,” Proc. IRE, vol. 45, p. 862, 1957. [6] P. P. Debyeand E. M. Conwell,“Electricalproperties of n-type germanium,” Phys. Rev., vol. 93, p. 693, 1954.

Metal-(n) AIGaAs-GaAs Two-Dimensional Electron Gas FET DANIEL DELAGEBEAUDEUF

Abstract-Theoreticalcalculations have beendeveloped for a twodimensionalelectron gas FET(TEGFET)constitutedbya AIGaAs (n)-GaAs(nor p-) heterostructure in whichtheSchottkygate is deposited on the AlGaAs (n) top layer. The theory takes into account: i) the subband splitting in the two-dimensional electron gas (2-DEG); and ii) the existence of an undoped AlGaAs spacer layer which has been found to enhance the electron mobility. The sheet carrier concentration of the TEGFET has been calculated, and a simple analytical formula has been established for the charge control in large and small gate FET.

I.INTRODUCTION HE PRESENCE of an inversion or accumulation electron layer located at the interface of certain heterojunctions was predictedabouttwenty years ago byAnderson [ l ] . In

T

Manuscript received July 8, 198l;revised January 11, 1982. The authors are with Thomson-CSF, Central Research Laboratory, Domaine de Corbeville, BP, No. 10,91401, Orsay, France.

AND

N W E N T. LINH

1969, Esaki and Tsu 121 proposed a heterostructure in which ionized impurities and free electrons could be spatially separated givingrise toa reduced Coulomb scattering. In 1978, Dingle etal. [3] observed mobility enhancement in modulationdoped superlattices. More recently, various experiments concerning modulation-doped GaAs-AlGaAs single heterojunctions andsuperlattices have shownthe high electronmobilitybehavior [4] -181 of the two-dimensional gas [9] - [13] . At the same time, several papers were published showing the use of suchstructuresforFET applications.The two-dimensional electron gas FET (TEGFET) has the Schottky gate deposited on the GaAs layer [ 141, [I51 or on the AI,Ga,-,As layer [16] , [17]. Low noise TEGFET’s have been fabricated in our laboratorywiththelatterstructure:2.3-dB noise figure, with 10.3 dBassociated gain at 10 GHz [18] . It was also demonstrated that the latter structurepresents higher mobility [19]and higher FETperformance 1171 thantheformer structure.

0018-9383/82/0600-0955$00.75 0 1982 IEEE

IEEE TRANSACTIONS O N ELECTRON DEVICES, VOL. ED-29,

956

NO.

The solution for the longitudinal quantized well approximated by the formula

(&) (2 113

E,(eV) #

3

6 , JUNE 1982

energy is then

rrq E ) ‘ I 3 (n + : T I 3

rn?

where is thelongitudinal effective mass. For GaAs and considering onlythe lowerand theexcitedsubband,one obtains

A Fig. 1. AlxGal-,As(n)-GaAs

(p) heterojunction at equilibriumand isolated.

In this paper we shall consider transistors having the metalAlGaAs (n)-GaAs (n-orp-)structure. The TEGFETwith GaAs as the uppermost layer has been treated recently [20]. The theoretical approach we have in this work differs slightly fromthe previously published paper: we takeintoaccount thesubband splittingin the two-dimensional electron gas (2-DEG) and the existence of an undoped AlGaAs spacer layer whichspatially separates electronsfromtheirparent-donor impurities. We shall study successively: i) the equilibrium of a heterostructure which consists of two different semiconductors: a large gap n-type one and a small gap n-- or p--type one, such that the Anderson conditions for the existence of an electron accumulation or inversionlayer in the latter are satisfied; ii) thecontrol of the 2-DEG layersheet densityby a Schottky gate deposited on the large gap semiconductor; iii) the performances of FET structures using such materials.

All the numerical estimations will concern the case of GaAs (p-)/AlxGal -,As (n) heterostructures, the experimental data relative to the subbands splitting when the GaAs is n--type being is insufficientat this date. Nevertheless, the analytical treatment given below is valid for both n-- and p--cases.

Eo(eV) # 1.83 X

$‘I3

El (eV) # 3.23 X

&‘I3 Vim.& in

(2)

Thesplitting given by(2) is certainly exaggeratedresulting fromthe sublinearincrease of theconductionband edge energy with the variable x;keeping as a reasonable approximation the 3 power law for Eo and El versus $, we shall modify the two numerical coefficients of (2) to be in agreement with the most important published experimental results. First we’ shall establisha relationbetweentheinterface electric field &,, and the electronsheet concentration i) p--Type Case In the small gap semiconductor the electric field obeys the Poisson equation

n(x) being the bulk free electron concentration and NA, the ionizedacceptorsdensity in the small gap semiconductor. Integration between the limit of the depletion region (g1= 0) and the interface ( 8 , = & i l ) gives € 1 $il

= qns + ~ N ,w1 A

(4)

where el is the dielectric permittivity in the small gap semiconductor, ns the 2-DEG electron density, and w1 the space charge region width.For MESFET’s applications,wherea good mobility is required, NA, is chosen very low (the material is then qualified as “undoped” or “nonintentionally doped”) to reduce the impurity scatteringin the 2-DEG layer.Then the second term in the second member of (4) is negligible and we have a good approximation €lgil #

~ s .

(5)

ii) n--Type Case

The electric field g1obeys then thePoisson equation TWO-DIMENSIONAL ELECTRON LAYER The objective in this section is the calculation of the Fermilevel position in the 2-DEG, the free electron surface density where N o , is the ionized donors density in the semiconductor and the electric field at the interface, Only two subbands will 1. As previously noted, if d l is the small gap semiconductor be takenintoaccount and the first question concerns the layer width and if & ( d l ) # O (neutrality at d l ) the integration splitting of thesesubbands. We shall use asemi-empirical of (6) gives formulation derived from an approximate well model and € 1 gil = W, - ~ N D d l, (7) corrected to take into account the most important published I f N d l and d l are chosen sufficiently small we have results. Fig. 1 gives the used notations (a p-type small gap semicon€ 1 $il #qns (8) ductor is assumed). It starts from the assumption of a quasiconstant electric field & in the potential well (triangular well that is the same result than for the p--type case. From these assumptions and (2), (S), and (8) the subbands approximation). 11. TREATMENT OF

THE

O-DIMENSIONAL ASELECTRON LINH:DELAGEBEAUDEUF AND

FET

957

positions are given by the relations

where yo and y1 are adjustableparameterswhich would be given by

otherwise

yo = 2.26 X 10-l’

y1 = 4 X

(Systeme International (SI))

(10)

according to (2), (5), or (8). The treatment of the 2-DEG layer will be complete when a relation between n, and the Fermi-level position will have been established. Thedensity of states (associated witha single quantized energy level) for a two-dimensional system being a constant is

(spin degeneracy 2, valley degeneracy 1 ) it is easy to obtain the relation between E F , E o , E l ,and n,. The calculation is performed in Appendix I and gives

(1 2) which at low temperatures, reduces to

n, = D(EF - Eo)

(13)

or

n, = D(E1 - Eo) f ~ D ( E F - El)

(14) whenthe second subband is,respectively, unoccupiedor occupied. From the publishedresults performed at low temperatures by ShubnikovDe Haas or cyclotronresonance experiments [ 101 -[13], an estimation of yo and y1 can be done. In the p--case one finds

e which reduces the electron scattering in the 2-DEG by the ionized impurities remaining in the large gap semiconductor spacecharge region. We shall computethe electric displacementvector at theinterface in the large gap semiconductor by assuming thetotaldepletionapproximation in its space charge layer. The band bending being u z o , one easily obtains (see Appendix 11-A) e2&i,

=

a e x u z + q2N?e2 - qN2e.

(17 )

By examination of the Fig. 1,

Neglecting interface stateswe have, according to the Gauss law,

yo = 2.5 X lo-’’

y 1 = 3.2 X

Fig. 2. Sheet concentration of the 2-DEG. GaAs is taken p-type (NA = 1014 ~ m - and ~ ) A1 concentration in AlGaAs is 0.3. Full curves co:respond to various thicknesses e, dash curve to the case e = 0 but ignore the two-dimensional character of the electron sheet.

(SI)

(1 5) Then, from (5) (or (8)), (12), (17),and (19) we must solve

and from the measured cyclotron mass

-

D = 3.24 X lOI7 m-2 V-’ .

(16)

. \ / 2 q e z ~ ~ v +2 q2N;e2 0 -q

~ = qn, ~ e

qlkT(EF-0 - E l ) = DkT log [( 1 + e 4w(EF0-Eo) +e 11 For the n--case few results are published but the band bending being less pronounced the subband splitting is certainly smaller than for the p--case. This can explain why some authors [4] have observed the two subbands occupation at low tempera- where Eo and E l given by (9) are functions of n,. The solution ture when the small gap semiconductor is n-type and when the is obtained numerically according to the following procedure. We start from an arbitrary low value for E F (which ~ can be measured value of n, is moderate. negative); from (17), (18), and (19) n, is deduced and E o , E I 111. EQUILIBRIUM computed with (9). The deduced value for ns must also satisfy Fig. 1 shows the band diagram of the studied heterostructure (12) and, if it is not the case, E F ~is increased until (20) is in the p--type case at equilibrium and isolated from the influ- verified. ence of any external contact. Two of the presumed subbands The resultin the p--type case (using the yo and y1 values in the notch are shown but their position is only illustrative. given by (10)) x = 0.3 (A1 content in AlCaAs) and various In the calculation we shall take into account the presence near values of e is shown as a function of Nz (AlCaAs doping) at the interface of an undoped AlGaAs spacer layer of thickness: T = 300 K in Fig. 2 (an arbitrary low value for the acceptor

TRANSACTIONS IEEE 958

O N ELECTRON ED-29, DEVICES, VOL.

NO. 1982 6 , JUNE

Inthe absence of interfacestates to (19) €2

Q, = qn, = -(Vp2 d2

-

@M-

we have then, according

EF -t AE,

-t

Vc).

(25 )

But E F , which is a function of V c , is always very small as compared to the other terms and we obtain the approximate but important result,

where Voff

= @M - AE, -

vpz

(27 )

is the“off voltage”whichannihilates the 2-DEG. Having neglected EF t o obtainthese results it is obvious that (26) and (27) are insensitive to the exact positions of the subbands hencethey are valid for the two cases where the small gap semiconductor type is p- or n-. If the interface states charge Qi cannot beneglected, (26) remains valid but (27) becomes Fig. 3. AlXGal-,As (n)-GaAs (p) heterojunction in the 2-DEG control regime b y a Schottky gate.

density in GaAs has been chosen but the result is quite insensitive to this value). In this figure, is also reported the curve which corresponds to the case where the two-dimensional behavior is ignored (20). The temperature dependence of n, is small and the frequently observed decrease of n, at low temperatures must be attributed to carriers freeze-out in AlGaAs (reduction of the number ofionized impurities).

voff

= @M - AEc - Vp2 -

d2 1 Qi.

(28)

B. Threshold Voltagefor Charge Control For a given large gap semiconductor layer width, thereexists a threshold voltage Vcth separating the charge control regime from the equilibrium state. It can be obtained by equalizing the two expressions of e2 g j 2(17) and (24) for the equilibrium value EFo of the Fermi-energy. The result can be put in the form

IV. CHARGE CONTROL BY A SCHOTTKY GATE IN CONTACT WITH THE LARGEGAP SEMICONDUCTOR A . Charge Control Law Fig. 3 shows the band diagram of the heterostructure submitted to the influence of a Schottky gate in contact with the large gap semiconductor. We suppose as effective theinter- where (1 8) has also been used. penetration of the two depletion layers, that is, we have asV. FET APPLICATIONS sumed a sufficiently high reverse voltage on the Schottky gate V,(x) being the channel voltage under the gateat abscissa or a sufficiently small width of the large gap semiconductor x, and Vc the gate potential, the effective voltage for charge layer. control at x is With the notations used in the Fig. 3 , we establish in the Appendix 11-B the relation Veff ( x ) = vc - VC(X) (30) and (26) must be rewritten as where 4Nz Vp, = -(d2 - e ) 2 . 2E2 But we have by examination of Fig. 3 the relation ~2

then

= @M- Vc

+ EF -(23) AE,

(32)

I = Q,(x)Zv(x) where Z is the gate width and u(x) the electron velocity at x . The scattering limitation of the electron velocity is not well known for such a two-dimensional system. In addition to phonon scattering and intervalley electron transfer, other scattering mechanisms can occur such as electron-electron scattering [21 I , intersubband scattering, and real space electron scatter-

u = pg,

for G

< gC

(33)

u = us,

for E 2 gC

(34)

which is well known in the case of conventional FET [23], onecan give ananalyticaltreatment of theproblem. This approximation is purely formaland does not involve any assumption on the dominant scattering mechanism. At fields less than gCwe have (3 5 ) where dVc/dx = C% is the electric field. Thecurrent I being conservative andthechannel voltage V c ( x ) increasing fromsourcetodrain,theelectric field is maximum close to the drain and the velocity saturation will occur first at the drain side of the gate region. First we consider the case of very small drain voltage VD (linear regions of the I-VD characteristics). From (35) we have €2

I = pz -(VG dz

voff)

V c ( L ) - V c (0) ---___.

L

Experimental results have shownthattheheterojunction 2-DEGMESFET having theSchottkycontactdepositedon AlGaAs is a promising candidate as low noise FET’s. They also have demonstrated that subbandsplitting occurs in the 2-DEG, and mobility is enhanced if an undoped AlGaAs spacer layer is grown between AlGaAs (n) and GaAs (p-). Theoretical calculations developed in this paper, take these dataintoaccount.Thesheet carrier concentration in the 2-DEG has been deduced and simple analytical formulas have been established for the charge control in large and small gate FET’s. APPENDIXI ELECTRON SHEET CONCENTRATION IN THE 2-DEG AS A FUNCTION O F FERMI-LEVEL POSITION AND TEMPERATURE (Two SUBBANDS ASSUMED) For a two-dimensional density of states given by D between Eo and El and equal to 2 0 for energies greater than E l , we have, using, Fermi-Dirac, statistics

(36)

But, R s and R D being, respectively, the source and drain access resistances

dE

El

n,=D

l t e q ( E - EF) kT

Vc(0) = R s I V c ( L )= V D - R D I . Then from (36)-(38) (39)Applying

the easily verifiable formula

and the FET acts as a pure controllable resistor. At drain voltages such that the drain side electric field is less than &, the integration of (35)gives we obtain (1 2).

(40) The electricfield at abcissa x : & ( x )= dVc/dx is then easily obtained. Expressing the condition &(L)= g c

(41)

one obtains the saturated currentexpression

APPENDIX I1 ELECTRIC DISPLACEMENT VECTOR AT THE INTERFACE I N THE LARGE GAP SEMICONDUCTOR A . Equilibrium,Isolated Case Using the depletion approximation, the voltage V 2 ( x )in the space charge region of the semiconductor 2 obeys the Poisson equation

Taking the heterojunction interfaceas origin the conditions

For a large gate FET, (42) reduces to

V, (0) = 0

(43) and for a short gate FET, the control is linear

(z)

x=- w 2

=o

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960

:beingthe space charge layer width)must One obtains successively

(w2

be fulfilled.

For our case

N 2 ( x )= 0 ,

for -e < x

N 2 ( x ) = N 2 , for x