- Email: [email protected]
Physics of nanometer structure devices
34
PHYSICS OF NANOMETERSTRUCTURE DEVICES C. Hamaguchi, T. Mori, T. Wada, K. Terashima, K. Taniguchi: K. M i y a t s u j i , and H. Hihara Department of Electronics Osaka U n i v e r s i t y , Suita C i t y , Osaka 565 *VLSI Labaratory, Toshiba, Kawasaki 210 Size e f f e c t of nanometer s t r u c t u r e devices are reviewed, placing main emphasis on ( i ) d r i f t v e l o c i t y overshoot, ( i i ) hot e l e c t r o n e f f e c t in short MOSFET, ( i i i ) high f i e l d t r a n s p o r t of two dimensional electrons in MOS inversion layers, ( i v ) s e l f - c o n s i s t e n t c a l c u l a t i o n s of two dimensional e l e c t r o n density in n-AlxGal_xAs/GaAs heterostructures and (v) a novel quantum well device " s i n g l e quantum well t r a n s i s t o r (SQWT)" proposed by the present authors. Experiments are performed in short n+n n+GaAs and very short channel MOSFETs. Analysis is made by using semiclassical treatment and Monte Carlo simulation. S e l f - c o n s i s t e n t c a l c u l a t i o n s are c a r r i e d out on the two-dimensional e l e c t r o n gas formed in MOS inversion layers, n-AlxGal_xAs/GaAs heterostructures and AlxGal_xAs/GaAs/AlxGal_xAS single quantum w e l l . Monte Carlo simulations of the two dimensional electrons in MOS inversion layers are found to explain the measured f i e l d dependence of the d r i f t v e l o c i t y . Self-consistent calculations provide design p r i n c i p l e of HEMT and mechanisms of the new device SQWT. The present r e s u l t s indicate that the computer simulations are very helpful to i n v e s t i g a t e the physics of the nanometer s t r u c t u r e devices and to develop new devices.
INTRODUCTION New e f f e c t s are expected to occur when the size of semiconductor devices is reduced in the region of nanometers, f o r example, d r i f t v e l o c i t y overshoot, ballistic transport, and quantum size effect. In view of device technology, these e f f e c t s are very a t t r a c t i v e because they improve high speed operation of the devices and also e n t i r e l y new devices are created of which operation principle is quite d i f f e r e n t from the e x i s t i n g devices. In t h i s paper we w i l l concern with these e f f e c t s in semiconductors, but we w i l l r e s t r i c t ourselves to the recent work carried out in our laboratory. We deal with the f o l l o w i n g topics; (i) d r i f t v e l o c i t y overshoot of hot electrons in short n n n+ GaAs, ( i i ) hot electron e f f e c t s in short MOSFETs with e f f e c t i v e channel length from 0.25 to 2.0 ~m, where experimental r e s u l t s are analyzed by two- dimensional device simulator taking into account the d r i f t v e l o c i t y overshoot e f f e c t and by Monte Carlo simulation taking into account the two dimensional electron states formed in the inversion layers, ( i i i ) selfconsistent calculations of the two dimensional e l e c t r o n density in n-AIGaAs
/GaAs heterostructures (HEMT), and ( i v ) new quantum e f f e c t device called SQWT (single quantum well t r a n s i s t o r ) of which operation p r i n c i p l e is based on the wave function control of the twodimensional e l e c t r o n gas (2DEG) in a single quantum well composed of AIGaAs/ GaAs/AIGaAs. We w i l l point out the importance of the boundary conditions of the small size devices, e s p e c i a l l y nanometer structure devices by showing experimantal results and computer simulations. We believe that our approach presented here is very helpful when we develop a new device with nanometer s t r u c t u r e s . DRIFT VELOCITY OVERSHOOT IN n+n n+ GaAs D r i f t v e l o c i t y overshoot e f f e c t was first pointed out by Ruch [ I ] to occur in a very short channel device at high electric f i e l d s and the maximum d r i f t v e l o c i t y reaches several times the bulk drift velocity. The e f f e c t r e f l e c t s the f a c t that the energy r e l a x a t i o n time is longer than the momentum r e l a x a t i o n time and electrons move very f a s t in a very
35 short period a f t e r a high e l e c t r i c f i e l d application in which the electrons are accelerated by the f i e l d without r a i s i n g the electron temperature. Such an e f f e c t is observable in a very short channel device, f o r example, GaAs with i t s channel length less than 1 pm. The drift v e l o c i t y overshoot improves the high speed operation of semiconductor devices. When the channel length is reduced to the e l e c t r o n mean free path or less, the electrons traverse the current channel without any s c a t t e r i n g events and " b a l l i s t i c t r a n s p o r t " occurs. The e f f e c t was pointed out by Shur and Eastman [2] and several experiments have been reported [3-5] but these observations are not enough to convince us of the existence of the b a l l i s t i c motion in semiconductors. They drew the b a l l i s t i c motion from the c u r r e n t - v o l t a g e characteristic of J ~ V3/2which is obtained by assuming a s i m p l i f i e d boundary condition. The present authors, however, pointed out that space charge current a r i s i n g from the device s t r u c t u r e results in such a c u r r e n t - v o l t a g e charact e r i s t i c even i f the d r i f t and d i f f u s i o n of the electrons are assumed [ 6 ] . Holden and Debney [7] carried out calculations of. the c u r r e n t - v o l t a g e c h a r a c t e r i s t i c s of n+n n+ GaAs b a l l i s t i c diodes by taking into account the boundary conditions and found that no unique power law is obtained. Their r e s u l t s suggest that i t is not possible to identify ballistic motion from measurments of current and voltage alone. From these r e s u l t s we believe that the b a l l i s t i c motion has not been observed in semiconductors at the present stage. We proposed a novel method to study d r i f t v e l o c i t y overshoot e f f e c t in short semiconductors in which measured current vs. voltage c h a r a c t e r i s t i c s are analyzed by taking into account the boundary conditions properly [ 6 , 8 ] . The samples used in the present work are Ipm channel length n + n n + GaAs grown on semii n s u l a t i n g GaAs. The n +and n regions were obtained by ion implantation and ohmic contacts were made on the n+ regions. The e f f e c t i v e channel length is estimated to be about 0.75 pm due to the thermal d i f f u s i o n of Si donors from the n+ region to the n region. Currentvoltage c h a r a c t e r i s t i c s were measured by using 200ns pulse voltages, which show a saturation at voltages higher than 2V as shown in Fig. I , where nornalized
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I@ I0 z NORMALIZED VOLTAGE eV/kT Fig. 1 Normalized current-voltage characteristics in short n+ n n+GaAs at 30OK; c i r c l e s are measured, ( I ) curves calculated with constant mobility, and (2) with f i e l d dependent m o b i l i t y . current j = JXd/NQkT is p l o t t e d as a function of normallzed voltage eV/kT. We have to note that negative d i f f e r ential c o n d u c t i v i t y was not observed in the samples used in the present work. When we assume uniform e l e c t r i c f i e l d in the sample, the e l e c t r i c f i e l d corresponds to lOkV/cm f o r l V and thus the present observation indicates an e x i s t ence of the d r i f t v e l o c i t y overshoot. As pointed out above, the boundary condition is very important f o r the specimens with very short channel length and the uniform f i e l d assumption is no longer v a l i d . This is confirmed by solving Poisson equation and current continuity equation iteratively by taking into account the boundary condition [ 8 ] . :
e
~(N(x)
-
Nd(X))
(I)
and J = e~(x)N(x) dEF~XJ' ~
dx
(2)
where @(x) is the p o t e n t i a l , ~=Kc0 is the d i e l e c t r i c constant, Nd (x) is the donor density, EF(X) is the quasi-Fermi potential and the other symbols have their usual meanings. Equation (2) includes the d r i f t and d i f f u s i o n current by assuming Boltzmann s t a t i s t i c s and Einstein relation. The r e s u l t s are shown in Fig. 2, where the normalized values of donor density no: Nd(x)/N0, electron density n = N(x)/No , and potential u = e ¢ ( x ) / k T are p l o t t e d
36
~5
5
,o.f nO Z
%4
>-
o,2
U
l-J b.
rr C) 0
t~jlO'
'
'
0'.5
'
'
1,0
NORMALIZED DISTANCE X / L
q
Fig. 3 velocity i0 °O
O15 NORMALIZED DISTANCE X / /
Fig. 2 Spatial d i s t r i b u t i o n of normalized p o t e n t i a l u, donor density no and electron density at eV/kT=75 in GaAs. as a function of normalized distance y = x/L f o r eV/kT = 75. We used No f o r the donor density at the center of the specimen with the channel length L. We assumed Gaussian d i s t r i b u t i o n f o r the donors. In order to take i n t o account the hot electron e f f e c t we assumed the following relation f o r the inverse mobility po/P = (I + ~E2 + BE~ + yE~) I / 2
(3)
where P0 is the ohmic m o b i l i t y . In eq. (3) the term 1 + mE2 dominates at low e l e c t r i c f i e l d s and the other higher power terms are included to account f o r the i n t e r v a l l e y t r a n s f e r of hot electrons. By adjusting the parameters m, B, and y, we obtain a c u r r e n t - v o l t a g e r e l a t i o n s h i p which coincides with the experimental curve. An example is shown in Fig. 1 by the s o l i d curve. When we know the s p a t i a l d i s t r i b u t i o n of the electric field, the d r i f t v e l o c i t y is calculated as a function of distance from eq. (3). The r e s u l t is shown in Fig. 3, where we f i n d t h a t the electron d r i f t v e l o c i t y reaches about 3XIOv cm/s and thus there e x i s t s a d r i f t velocity overshoot. As mentioned e a r l i e r , the overshoot e f f e c t appears only in a short distance from the cathode at high e l e c t r i c f i e l d . The above treatment used in the present analysis is based on the assumption t h a t the electrons are q u a s i - s t a t i c and thus the s p a t i a l d i s t r i b u t i o n of the drift v e l o c i t y i s given by the s p a t i a l
Spatial d i s t r i b u t i o n of d r i f t in short n+ n n+ GaAs at 30OK.
d i s t r i b u t i o n of the e l e c t r i c f i e l d . In other words, the present treatment does not include time e v o l u t i o n of the d r i f t velocity. It is very i n t e r e s t i n g t h a t the p o t e n t i a l d i s t r i b u t i o n in the specimen e x h i b i t s a minimum near the cathode as seen in Fig. 2. We c a r r i e d out Monte Carlo simultion of hot electrons assuming a potential distribution s i m i l a r to that shown in Fig. 2. F i r s t we show the r e s u l t s when the p o t e n t i a l distribution is approximated by a t r i a n g u l a r shape with p o t e n t i a l minimum of O.03V at 0.05. 0 . I 0 , 0.15 and 0.20~m from the cathode. The r e s u l t s are shown in Fig. 4, where the applied voltage is IV. We f i n d in Fig. 4 that the d r i f t v e l o c i t y increases sharply beyond the p o s i t i o n of p o t e n t i a l minimum. When we change the minimum value of the potential from 0.01 to O.05V and f i x the p o s i t i o n at O.20pm, the d r i f t v e l o c i t y d i s t r i b u t i o n s are given by the curves shown in Fig. 5, where we f i n d no 10
%
d 4 >3
~2 i
I
DISTANCE
I
I
I
[micron]
Fig. 4 D r i f t v e l o c i t y as a function of p o s i t i o n in short GaAs, where t r i a n g u l a r d i s t r i b u t i o n of p o t e n t i a l is assumed.
37 18
% o.osv o.otv
5 o.o~v o ~
>
3
g, 0 o
0.2
0.4
DISTANCE
0.6 0.8 [micron]
Fig. 5 Drift velocity as a function of position Sn short GaAs, where triangu}ar distribution of potential is assumed.
~
o
y
j
~6
~z
0
1
DISTANCE [micron]
Fig. 6 Monte Carlo simulation of the D r i f t v e l o c i t y in short n+n n+GaAs with a given potential d i s t r i b u t i o n . significant difference in the three curves. F~omthese results we find t~at the simplified simulation explains qual i tati vely the experimentally More refined treatment is shown in Fig. 6, where the potential distribution is obtained by solving eqs. (l)-(3) for IV application and electron d r i f t velocity is obtained by Monte Carlo simulation. The overshoot is found to occur near the center of the sample but the detailed
Among them the hot electron effect plays the most important role in the FET characteristics. For example, impact ionization near the drain contact and hot electron injection into the oxide. However, the behavior of hot electrons in short MOSFET is not known well. Two approaches may be possible; one is the classical two-dimensional device simulation as done by Mock [9] and the other is quantum mechanical treatment in which scatterings of the two-dimensional electrons formed in the inversion layer is properly taken into account. As far as our knowledge is concerned, the former is usually used, in which twodimensional quantization is disregarded and the t h r e e dimensional electron motion is assumed. In this paper we will present experimental results of very short MOSFET and analysis by using two-dimensional device simulator, where three dimensional motion of electrons z~z~.~mc~ '.~e~l,~,p~.~.~ ~ h ~ of two-dimensional quantization on the layers by
using Monte Carlo simulation.
at room temperature. Results for 0.75 and O.?5~mMU~FETs are shown Yn FYq. and Fig. 8, respective]y. We see in Fig. 8 that the drain current increases at higher drain voltages. Such a behavior is explained in terms of electron impact in the region near the drain contact. Present analysis was made by using the method reported by Mock, where the cross section of the device is divided into small lattices, and static potential, electron density and current a~e o~tained by so~ing the ~oisson equation and an equation of electron current continuity, each with specified analysis are given by Mock [9] and we will not repeat i t here. In the present analysis we take into account the hot electron effect by introducing the following relation for the field dependence of the electron d r i f t velocity.
features a r e ~ot "~n gooc~ ~greement ~'~t}~ the quasi-static analysis.
HOT ELECTRONEFFECT IN SHORTMOSFET As the channel length is reduced in the range of submicrons, the short channel effects become very important.
where U~ E and v s are the low field mobility, electric field and saturation drift velocity of the electrons. We proceed fitting to obtain a good agreement between the computed results and the experimental curves by adjusting the saturation velocity. The dashed curves in Figs. 7 and 8 are calculated
38 30
l
l
I
I
I
I
we
MOS O. 7 5 ( m i c r o n ) ....
E
EXPERIMENT Rs-200 (ohm)
SIMULATION V S = l OE4 ( m l g e ¢ )
....
20
with V(z) = -e@(z) and d2~(z) e dz" = ~[(Na- Nd) + Z i N i l ~ i 12]
Vg=10(V) V 9 - 8 (V)
n, -'~ U
solve the following equations. ~2 d 2 [ - 2m3 dz 2 + V ( z ) ] ~ i ( z ) = E i ~ i ( z ) ( 5 )
10
,~" ~ ' * ,~,, ..,, _.._._._._._._ _. _._.. . . . . .
. -" "
,~" 2
3
4
Vg=4 (V)
Vg=2(V)
~ •., 1
5
where we use the same n o t a t i o n used by Stern [ I I ] and the e l e c t r o n energy is given by
6
E -2-m-*-~2 (k~ + k;) VOLTAGE
I
I
I
I
I
MOS O. 2 5 ( m l c r o n )
• ,,,
EXPERIHENT
.....
2O
SIMULATION Vs-ZOE4(m/sec)
hi • U
~.%~.
•
•
I
V9=4 (V)
...
•
. • . •" ,/ 4 "........ .--')
2
3
(7)
V9-2
The e l e c t r o n d e n s i t y N i in the subband E i is given by nvimdikT EF - Ei Ni = ~2 F [ k~ ] (8)
I
Rs-|SO(ohm) E
+ Ei.
IV]
Fig. 7 Drain c u r r e n t vs. d r a i n v o l t a g e c h a r a c t e r i s t i c s in O.75pm MOSFET.
30
(6)
d~> h
where F(x) = I n [ l + e x p ( x ) ] . We calcul a t e d the subband e n e r g i e s , Fermi energy and e l e c t r o n d e n s i t y in the subbands. An example of the s e l f - c o n s i s t e n t calculation is presented in Fig. 9, where square o f the wave f u n c t i o n s and subband energies ( h o r i z o n t a l l i n e s ) are shown f o r Si (111) surface at 30OK.
(V) T
4
I
I
5
6
T
.8 @
VOLTAGE
Iv]
Fig. 8 Drain c u r r e n t vs. d r a i n voltage c h a r a c t e r i s t i c s in 0.25 pm MOSFET. curves, where the best f i t t i n g is obtained by using v s = IXlO 7 cm/s f o r 0.75pm and 2XlOTcm/s t o r 0.25pm samples. Noting t h a t the s a t u r a t i o n v e l o c i t y in a bulk s i l i c o n is about 7XlO 6 cm/s, the obtained values o f v s i n d i c a t e t h a t the d r i f t v e l o c i t y overshoot e x i s t s in s h o r t MOSFET. The s a t u r a t i o n velocity was found to increase w i t h decreasing the channel length f o r three MOSFETs w i t h 0.25, 0.50 and O.75pm e f f e c t i v e channel length. Such a f e a t u r e is c o n s i s t e n t w i t h the Monte Carlo s i m u l a t i o n , where the overshoot becomes l a r g e r f o r s h o r t e r samples and at higher e l e c t r i c f i e l d s . In the above treatment we assumed that the e l e c t r o n motion is three dimensional. However, i t is well known t h a t the e l e c t r o n s in the i n v e r s i o n l a y e r are confined in a very narrow region near the surface and two dimensional q u a n t i z a t i o n occurs [ I 0 , I I ] . In o r d e r to obtain the e l e c t r o n s t a t e s we have to solve Schr~dinger equation and Poisson equation self-consistently. Following the treatment o f Stern [ I I ] ,
> J E
.B
v(z) :
(i 0)
FZ .4 W 0
O..2
e
18 DISTRNCE
28 FROM
38 SURFRCE
48
58
Z[nm]
Fig. 9 S e l f - c o n s i s t e n t r e s u l t s o f two dimensional e l e c t r o n states in Si ( I I I ) surface at 30OK. Ninv=l.OXlOiTcm-~ In the present a n a l y s i s we assumed t h a t the subband energies are u n a f f e c t e d by the e l e c t r i c field. We c a l c u l a t e d the d r i f t velocity and the e l e c t r o n d e n s i t y in the subbands as a function of electric f i e l d by using Monte Carlo simulation. We deal w i t h the 2DEG formed in the (I00) surface o f Si and the e l e c t r o n d e n s i t y is assumed t o be I.OXIO 13 cm-2 . Since almost a l l the e l e c t r o n s populate in the three subbands Eo, E i and E~, we take i n t o account olny the three subbands. The subband energies are found to be Eo= 208.9, El= 293.6 and E~ = 290.7meV, and t h e i r
39 average penetrationoof the electrons is 12.3, 29.8 and 28.7A, respectively. In the present simulation we included the scattering processes due to screened ionized impurity, surface roughness, acoustic phonon and intervalley phonon scatterings. Since the deformation potentials for the two-dimensional electrons in MOS inversion layer are not known, we used bulk values [12]. The intervalley phonon energies are determined f r o m magnetophonon resonance experiments by Hamaguchi, Hirose and Shimomae [13]. We picked up f- and gtype phonons which are expected to be important. The phonon temperatures used in the present calculations are 750K (g) and 630K (f) for the zero order scattering and 134K (g) and 230K (f) for the f i r s t order scattering [12], which are a l i t t l e different from our magnetophonon measurements [13]. The deformation potentials for the intervalley scattering are chosen to be Do = 9.0XlOZ°eV/m and DI = 28.0eV for the zero and f i r s t order scattering, respectively, for both g- and f-scattering processes. |'
v
i
!
m
•
v
Id
i
i
i
i
"G
L-I.5OFm
E
°290K
Ic~
•
I
I
I
!
nc
i
~ i i i
~
I
I|11
I
I
I
I
Ill=
ao~
F
ic:P
(Vlcm)
Fig. II Drift velocity as a function of electric field measured in 1.51Jm MOSFET at 77 and 290K. I.O
!
i
'
'
w
~0.5 U
01~ 0
I
,~
*
i
|
I
vo" F
i
•
I
I0 F
4
I0
5
(V/cm)
Fig. 12 Electron density vs. electric I • field in the subbands Eo, EI and E0 In Si (IOO) surface obtained by Monte Carlo simulation.
I
vo' (V/cm)
Fig. lO Drift velocity of 2D electrons in MOS (lO0) surface calculated by Monte Carlo simulation at 3OOK. The results of Monte Carlo simulation are shown in Fig. lO, where we find a weak saturation at higher electric fields. The feature is very similar to the observed results which are shown in Fig. 11 for 1.5pm MOSFET. Note that the d r i f t velocity calculated from the Monte Carlo simulation is slightly higher than the experimental value, where the experimental values are obtained by using gradual channel approximation and the estimated electron density is about
l.OXlO 13 cm-2. The difference of the drift velocities is reduced i f we increase the surface roughness scattering. I t should be note here that the d r i f t velocity in bulk Si shows a strong saturation and thus the observed results are not explained. We present the electron populations in the three subbands as a function of electric field in Fig. 12, where we find a decrease of electrons in the ground subband Eo and increase in the higher subband E# with h i g h e r density of states. Detailes of the analysis are reported elsewhere [14].
40 SELF-CONSISTENT CALCULATIONS OF 2DELECTRON DENSITY IN AlxGal_xAs/GaAs Recent progress in molecular beam epitaxy technique has made possible to fabricate A1GaAs/GaAs h e t e r o j u n c t i o n devices with high e l e c t r o n m o b i l i t i e s at low temperatures [15-17]. Among them HEMT (high e l e c t r o n m o b i l i t y t r a n s i s t o r ) is known to be a t t r a c t i v e in view of high speed devices. In HEMT the 2DEG formed in a narrow region of GaAs at the i n t e r f a c e moves very f a s t because the electrons are supplied from the n-AIGaAs and are free from the impurity s c a t t e r ing. In the sense of device f a b r i c a t i o n i t is very important to estimate the e l e c t r o n density in GaAs when doping density of AIGaAs is given. As f a r as our knowledge is concerned, the 2D e l e c t r o n density is estimated by using t r i a n g u l a r p o t e n t i a l approximation [1819] or by using v a r i a t i o n a l approximation [20], where they neglect the penet r a t i o n of the e l e c t r o n wave functions into AIGaAs layer. We report here f o r the f i r s t time s e l f - c o n s i s t e n t calculations of the 2D e l e c t r o n density in AIGaAs/GaAs heterostructures as functions of the doping density in nA1 Ga As and the thickness of the undoped AlxGal_xAs f o r x = 0.17 to 0.33 at lattice temperatures 4.2 and 77K. The c a l c u l a t i o n s were carried out by solving Schrodinger equation and Poisson equation s e l f - c o n s i s t e n t l y [21, 22] by taking into account f i v e subbands at T = 4.2K and ten subbands at T = 77K, if necessary, but the subbands of which energies exceed the b a r r i e r height are disregarded. We assume that the b a r r i e r height is given by AV = X in eV and the i o n i z a t i o n enegy of donors in n-AIGaAs is taken to be 50, I00 or 150meV. 0.5
0.4
0.3
~.2
0.
I
0.0 -50
E}
100 200 DISTANCE
300
400
500
the wave functions of the three subbands and the p o t e n t i a l energy (conduction band edge) are p l o t t e d together with the subband energies (shown by bars) and Fermi energy (longer bar). Note that e l e c t r o n wave functions penetrate into the undoped AIGaAs layer of which thickness is 50A. I O *II
a) x - 0 . 3 3 b] x - 0.30
o)~b~)
~
c)x - 025
/
~
I,...... I 0 I*
10 I? Dopln 9
Density
of
I 0 I*
AI=Go,.=As
( c m "s)
Fig. 14 2D e l e c t r o n density in GaAs vs. donor density in n-AIGaAs at T = 4.2K. I 0 ~= a) x - 0.33
~w Q
b) x
•
0.30
c) x
•
o.2s
a
)
~
)
~
..~'~/.f:~f
I 0"
w
,~ IO"
IO I? Doplng
Denslty
of
AI,Gai..As
IO ' I (c m'3)
Fig. ]5 2D e l e c t r o n density in GaAs vs. donor density in n-AlGaAs at T =77K. Figure 14 shows the e l e c t r o n sheet density in GaAs calculated as a function of doping density of AlxGal_xAs f o r x = 0.33, 0.30, 0.25 and 0.17 at 4.2K,_where spacer thickness of AIGaAs is 60X and donor i o n i z a t i o n energy is lOOmeV. The r e s u l t s at 77K are shown in Fig. 15. The present r e s u l t s give a very important information about the doping density and the thickness of the nAlxGal_xAs required to get a desired e l e c t r o n density in the GaAs layer and thus a design p r i n c i p l e of HEMT.
[A]
Fig.13 S e l f - c o n s i s t e n t r e s u l t s f o r GaAs/ AlxGal_xAs (x=0.3) f o r Ns = 5XlOlZcm-.z A typical example of the s e l f consistent r e s u l t s is shown in Fig. 13 f o r e l e c t r o n density 5.0XlO17cm -2, where
SINGLE QUANTUMWELL TRANSlSTOR WITH AIGaAs/GaAs/AIGaAs HETEROSTRUCTURES The
authors
proposed
new devices
41 called SQWT (single quantum well transistor) of which operation principle is based on the wave function control in a single quantum well and pointed out that a h i g h electron mobility will be achieved at low temperatures in a modulation doped quantum well where the half region of GaAs-well is doped by donors [22]. Confinement of 2DEG in a quantum well has been proposed by Sakaki [23] in connection with velocity-modulation transistor (VMT) in which channel conductance is modulated by changing the populations of 2DEG in two heterointerfaces. I t has been pointed out that the source and drain contacts are different between the SQWTand VMT [22]. In this paper we present physical mechanism of the modulation doped SQWT by showing a confinement of 2DEG in the un-doped region of GaAs-well by applying an external voltage. T h i s mechanism suggests a possibility to fabricate high electron mobility transistor of AIGaAs/ GaAs/Al GaAs structures. Gale. pf/f/~
Drain f//l/~
~/f/f/ff[//]
IOOA /' omGo:~$(non-doped)
.;~-:~!-~~]
' "" ~........ ',-2PEG
/ ._
f ...........
/ /
/
250~,
GoAs t non- doped ) a-~A;- . . . . . . . . . . . . . . . . .
[ SI- doped
~
.
250A
~No:2 X ' ° ' ' ~ )
--2op
Alo3GOo.7 As (non-doped)
IOOA
GoAs Sub~mte bock-gote
Fig. 16 Cross-sectional view of SQWT. In order to illustrate the basic feature of the modulation doped SQWTwe present a typical example of the selfconsistent calculations of the 2DEG in a quantum well A1 .Ga~ .As/ GaAs/Al GaI xAs (x=0.3) shown In . , ~ Fig. ,-~ 16, where xthe half region of GaAs layer is modulation doped with donors of 2.0XlO~cm - 2and A~GaAs is un-doped. The GaAs-we~l is 500A and AIGaAs barriers are lOOA, where the donors in the GaAs-well are assumed to be completely ionized and t h u s the electron sheet density is given by Ns = 5.0X10~2cm-2. In the calculations we assumed that the external voltage is applied between the un-doped AIGaAs layers of lOOA thick for simplicity. The self-consistent results are obtained by solving eqs. (5) and (6) by putting V(z) : -e@(z) + ZW[u(-z) + u ( z - d ) ]
(9)
where u(z) is the step function. For simplicity we assumed that the effective mass and the dielectric constant are given by 0.068m and 12.9 in AIGaAs and GaAs. We obtained self-consistent results of the subband energy Ei , subband electron density N~ and the wave function ~(z) with ano without an external voltage Vg. A~Gm-.As (x-03)
GO~
AI,Go,~ (x,O.3)I
0.4
m 0.2
-I00
.
0
I00
I¢J'
200
300
400
500
600
DISTANCE (~)
Fig. 17 Self-consistent results of 2D electron states in SQWTat Vg = O. Figure 17 shows the results for the gate voltage Vg = 0 at T = 4.2K, where the conduction band-edge V(z), subband energy Ei , Fermi energy EF, and square of the wave function are shown. Note that the donors are distributed in_the region of the GaAs-well from 250X to 500A as shown in Fig. 16. We assumed that the conduction band discontinuity is O.3eV. We obtained Eo = 13.1, El =23.8, E2 = 33.1, E3 =46.6 and EF =27.2meV. I t is very interesting to point out that the electrons populate in the subbands Eo (4.0XlO12cm-2) and El (O.97XlO12cm-2) and therefore they are not localized in the doped region of GaAs-well. In this case the current is mainly carried by the electrons in the subband Eo and a small part of the current consists of electron d r i f t in the subband El. When we apply a gate voltage of O.4V, the self-consistent calculations lead to the results shown in Fig. 18 where we obtain Eo = 56.0, El = 82.3, E2 = lOl.8, E~ = 123.2, and EF =73.6meV. We find in Fig. 18 that the- electrons are confined in the subband Eo or in the region 0 ~20oA of the GaAs-well because Ep < E~ . These results suggest us to f~bricate a novel FET "modulation doped SQWT". In such a device the drain current is carried by electrons confined in the non-doped region of the GaAs-well by applying an external voltage and the electrons move with a very high mobility because the electrons in this region do
42 0.6 A
.<
AI,GoJ.,As (x,03}
I00
GaAs z 0
0.4
E~
Iu
Z
u-50 Q
02
z F,<
o - ioo
Eo'~/k~/~ o
\7~.Z too
200
\ / " ~ . ~ ~o
400
. ~o
600
-02
0 EXTERNAL
DISTANCE (~)
02 VOLTAGE
0.4 (V)
Fig. 18 S e l f - c o n s i s t e n t r e s u l t s of 2D e l e c t r o n states in SQWT at Vg = 0.4V.
Fig.20 Electron density No and N1in the subbands E o and E1as a function of Vg.
not s u f f e r from the impurity s c a t t e r i n g . Therefore we can f a b r i c a t e a high speed FET using modulation doped single quantum well in which the e l e c t r o n t r a n s p o r t is quite s i m i l a r to HEMT. In Fig. 19 we present the subband energies E0, E~ , E2, E3 , and Fermi energy E~ as a function of the external gate voltage where we f i n d that the Fermi energy crosses the f i r s t e x c i t e d state E1at about V~ = 0.32V and thus the electrons populat~ only in the ground subband Eo at a gate voltage exceeding 0.32V as shown in Fig. 20. Asymmetric features of the subband energies and the e l e c t r o n populations with respect to the external voltage are due to the modulat i o n doping in the GaAs-well. These res u l t s i n d i c a t e that the threshold v o l t age of the FET c h a r a c t e r i s t i c s is e a s i l y c o n t r o l l e d by the doping p r o f i l e of the GaAs-well. We carried out s i m i l a r calc u l a t i o n s on the modulation doped SQWT with 500A GaAs-well of which h a l f region is doped by donors Nd. Our c a l c u l a t i o n s show that the gate voltage required to pupulate a l l the electrons in the ground subband Eo and in the non-doped region of the well is O.2V f o r N~= I.OXIO ~7 and O.7V f o r Nd = 4XlO17cm~3 at T = 4.2K.
In conclusion we found that a modulation doped SQWT is very promising to achieve a high speed FET l i k e HEMT. Our s e l f - c o n s i s t e n t c a l c u l a t i o n s show that the electrons distributed in the whole region of the GaAs-well are e a s i l y confined in the non-doped region of the GaAs by applying an external gate voltage between the AIGaAs b a r r i e r s . Because of the asymmetric nature of the gate voltage dependence i t is found that the threshold voltage VT is e a s i l y cont r o l l e d by changing the doping p r o f i l e and doping density in the GaAs-well. It is shown from the s e l f - c o n s i s t e n t calculations that we can achieve a hiqh electron mobility in a modulation doped SQWT proposed in our paper [22].
0.2
ACKNOWLEDGEMENTS One of the authors (C. H) is very thankful f o r the kind support of The Mitsubishi Foundation to the present work. We acknowledge the support of a Grant-in-Aid for Scientific Research from the M i n i s t r y of Education, Science and Culture. The authors (C. H and T. M) express t h e i r thanks to F u j i t s u Laboratories L t d . , f o r the kind supply of the short channel GaAs samples used in the present work.
E,
>O.I n,. b~ Z
REFERENCES O -O.2
J
i
0
i
i
0.2
EXTERNAL V O L T A G E
Fig.19
i
0.4 (V)
Subband energy Ei vs. Vg in SQWT
I . J.G. Ruch: IEEE Trans. Electron Dev. ED-19 (1972) 652. 2. M. S. Shur and L. F. Eastmen: IEEE Trans. Electron Dev. ED-26 (1979) 1677.
43 3. R. Zuleeg: IEEE Electron Dev. Lett. EDL-I (1980) 234. 4. L. F. Eastman, R. Stall, D. Woodard, N. Dandekar, C. E. C. Wood, M.S. Shur and K. Boad: Electron. Lett. ]_6_ (1980) 524 5. M. A. Hollis, L. F. Eastman and C. E. Wood: Electron. Lett. 18 (1982) 570. 6. C. Hamaguchi, K. Terashima, A. Shibatomi and O. Ryuzan: Proc. 16th Int. Conf. Semicond. Phys. (1982, Montpellier), Physica llTB & !]8B (1983) 238. 7. A. J. Holden and B. T. Debney: Electron. Lett. 18 (1982) 558. 8. T. Mori, C. Hamaguchi, A. Shibatomi and O. Ryuzan: Japn. J. Appl. Phys. 23 (1984) 212. 9. M. S. Mock: Solid-State Electronics, (1973) 601. lO. F, Stern and W. E. Howard: Phys. Rev. 163 (1967) 816. I I . F. Stern: Phys. Rev. B5 (1972) 4891. 12. D. K. Ferry: Phys. Rev. Bl___4(1976) 5364. 13. C. Hamaguchi, Y. Hirose and K. Shimomae: Jpn. J. Appl. Phys. 22 (1983) Suppl. 22-_____3,p. 190. 14. K. Terashima: Thesis, Osaka Univ. (1983) unpublished. 15. R. Dingle, H. L. Stormer, A. C. Gossard and W. Wiegman: Appl. Phys. Lett._~_3 (1978) 665. 16. T. Mimura, S. Hiyamizu, T. F u j i i , and K. Nambu: Jpn.Apl1. Phys. 19 (1980) L225. 17. S. Hiyamizu, J. Saito, K. Nambu and T. Ishikawa: Jpn. J. Appl. Phys. 22 (1983) L609 and references therein. 18. K. Lee and M. Shur: J. Appl. Phys. 54 (1983) 2093. 19. T. C. Hsieh, K. Hess and J. J. Coleman: Solid-State Electronics 26 (1983) I173. 20. F. Stern: Appl. Phys. Lett. 43 (1983) 974. 21. T. Ando: J. Phys. Soc. Jpn. 5_] (1982) 3893 & 3900. 22. C. Hamaguchi, K. Miyatsuji and H. Hihara: Jpn. J. Appl. Phys. 2__33 (1984) No.3. 23. H. Sakaki: Jpn. J. Appl. Phys. 2_! (1982) L381.
PHYSICS OF NANOMETERSTRUCTURE DEVICES C. Hamaguchi, T. Mori, T. Wada, K. Terashima, K. Taniguchi: K. M i y a t s u j i , and H. Hihara Department of Electronics Osaka U n i v e r s i t y , Suita C i t y , Osaka 565 *VLSI Labaratory, Toshiba, Kawasaki 210 Size e f f e c t of nanometer s t r u c t u r e devices are reviewed, placing main emphasis on ( i ) d r i f t v e l o c i t y overshoot, ( i i ) hot e l e c t r o n e f f e c t in short MOSFET, ( i i i ) high f i e l d t r a n s p o r t of two dimensional electrons in MOS inversion layers, ( i v ) s e l f - c o n s i s t e n t c a l c u l a t i o n s of two dimensional e l e c t r o n density in n-AlxGal_xAs/GaAs heterostructures and (v) a novel quantum well device " s i n g l e quantum well t r a n s i s t o r (SQWT)" proposed by the present authors. Experiments are performed in short n+n n+GaAs and very short channel MOSFETs. Analysis is made by using semiclassical treatment and Monte Carlo simulation. S e l f - c o n s i s t e n t c a l c u l a t i o n s are c a r r i e d out on the two-dimensional e l e c t r o n gas formed in MOS inversion layers, n-AlxGal_xAs/GaAs heterostructures and AlxGal_xAs/GaAs/AlxGal_xAS single quantum w e l l . Monte Carlo simulations of the two dimensional electrons in MOS inversion layers are found to explain the measured f i e l d dependence of the d r i f t v e l o c i t y . Self-consistent calculations provide design p r i n c i p l e of HEMT and mechanisms of the new device SQWT. The present r e s u l t s indicate that the computer simulations are very helpful to i n v e s t i g a t e the physics of the nanometer s t r u c t u r e devices and to develop new devices.
INTRODUCTION New e f f e c t s are expected to occur when the size of semiconductor devices is reduced in the region of nanometers, f o r example, d r i f t v e l o c i t y overshoot, ballistic transport, and quantum size effect. In view of device technology, these e f f e c t s are very a t t r a c t i v e because they improve high speed operation of the devices and also e n t i r e l y new devices are created of which operation principle is quite d i f f e r e n t from the e x i s t i n g devices. In t h i s paper we w i l l concern with these e f f e c t s in semiconductors, but we w i l l r e s t r i c t ourselves to the recent work carried out in our laboratory. We deal with the f o l l o w i n g topics; (i) d r i f t v e l o c i t y overshoot of hot electrons in short n n n+ GaAs, ( i i ) hot electron e f f e c t s in short MOSFETs with e f f e c t i v e channel length from 0.25 to 2.0 ~m, where experimental r e s u l t s are analyzed by two- dimensional device simulator taking into account the d r i f t v e l o c i t y overshoot e f f e c t and by Monte Carlo simulation taking into account the two dimensional electron states formed in the inversion layers, ( i i i ) selfconsistent calculations of the two dimensional e l e c t r o n density in n-AIGaAs
/GaAs heterostructures (HEMT), and ( i v ) new quantum e f f e c t device called SQWT (single quantum well t r a n s i s t o r ) of which operation p r i n c i p l e is based on the wave function control of the twodimensional e l e c t r o n gas (2DEG) in a single quantum well composed of AIGaAs/ GaAs/AIGaAs. We w i l l point out the importance of the boundary conditions of the small size devices, e s p e c i a l l y nanometer structure devices by showing experimantal results and computer simulations. We believe that our approach presented here is very helpful when we develop a new device with nanometer s t r u c t u r e s . DRIFT VELOCITY OVERSHOOT IN n+n n+ GaAs D r i f t v e l o c i t y overshoot e f f e c t was first pointed out by Ruch [ I ] to occur in a very short channel device at high electric f i e l d s and the maximum d r i f t v e l o c i t y reaches several times the bulk drift velocity. The e f f e c t r e f l e c t s the f a c t that the energy r e l a x a t i o n time is longer than the momentum r e l a x a t i o n time and electrons move very f a s t in a very
35 short period a f t e r a high e l e c t r i c f i e l d application in which the electrons are accelerated by the f i e l d without r a i s i n g the electron temperature. Such an e f f e c t is observable in a very short channel device, f o r example, GaAs with i t s channel length less than 1 pm. The drift v e l o c i t y overshoot improves the high speed operation of semiconductor devices. When the channel length is reduced to the e l e c t r o n mean free path or less, the electrons traverse the current channel without any s c a t t e r i n g events and " b a l l i s t i c t r a n s p o r t " occurs. The e f f e c t was pointed out by Shur and Eastman [2] and several experiments have been reported [3-5] but these observations are not enough to convince us of the existence of the b a l l i s t i c motion in semiconductors. They drew the b a l l i s t i c motion from the c u r r e n t - v o l t a g e characteristic of J ~ V3/2which is obtained by assuming a s i m p l i f i e d boundary condition. The present authors, however, pointed out that space charge current a r i s i n g from the device s t r u c t u r e results in such a c u r r e n t - v o l t a g e charact e r i s t i c even i f the d r i f t and d i f f u s i o n of the electrons are assumed [ 6 ] . Holden and Debney [7] carried out calculations of. the c u r r e n t - v o l t a g e c h a r a c t e r i s t i c s of n+n n+ GaAs b a l l i s t i c diodes by taking into account the boundary conditions and found that no unique power law is obtained. Their r e s u l t s suggest that i t is not possible to identify ballistic motion from measurments of current and voltage alone. From these r e s u l t s we believe that the b a l l i s t i c motion has not been observed in semiconductors at the present stage. We proposed a novel method to study d r i f t v e l o c i t y overshoot e f f e c t in short semiconductors in which measured current vs. voltage c h a r a c t e r i s t i c s are analyzed by taking into account the boundary conditions properly [ 6 , 8 ] . The samples used in the present work are Ipm channel length n + n n + GaAs grown on semii n s u l a t i n g GaAs. The n +and n regions were obtained by ion implantation and ohmic contacts were made on the n+ regions. The e f f e c t i v e channel length is estimated to be about 0.75 pm due to the thermal d i f f u s i o n of Si donors from the n+ region to the n region. Currentvoltage c h a r a c t e r i s t i c s were measured by using 200ns pulse voltages, which show a saturation at voltages higher than 2V as shown in Fig. I , where nornalized
,
.
.
.
.
.
.
.
.
.
, , , ,
•F--
,
,
/;/
Z bJ ,°'°"
~0~10 t
n W J o7,o ~ , i l l
.
.
.
.
.
.
.
.
.
.
.
.
I@ I0 z NORMALIZED VOLTAGE eV/kT Fig. 1 Normalized current-voltage characteristics in short n+ n n+GaAs at 30OK; c i r c l e s are measured, ( I ) curves calculated with constant mobility, and (2) with f i e l d dependent m o b i l i t y . current j = JXd/NQkT is p l o t t e d as a function of normallzed voltage eV/kT. We have to note that negative d i f f e r ential c o n d u c t i v i t y was not observed in the samples used in the present work. When we assume uniform e l e c t r i c f i e l d in the sample, the e l e c t r i c f i e l d corresponds to lOkV/cm f o r l V and thus the present observation indicates an e x i s t ence of the d r i f t v e l o c i t y overshoot. As pointed out above, the boundary condition is very important f o r the specimens with very short channel length and the uniform f i e l d assumption is no longer v a l i d . This is confirmed by solving Poisson equation and current continuity equation iteratively by taking into account the boundary condition [ 8 ] . :
e
~(N(x)
-
Nd(X))
(I)
and J = e~(x)N(x) dEF~XJ' ~
dx
(2)
where @(x) is the p o t e n t i a l , ~=Kc0 is the d i e l e c t r i c constant, Nd (x) is the donor density, EF(X) is the quasi-Fermi potential and the other symbols have their usual meanings. Equation (2) includes the d r i f t and d i f f u s i o n current by assuming Boltzmann s t a t i s t i c s and Einstein relation. The r e s u l t s are shown in Fig. 2, where the normalized values of donor density no: Nd(x)/N0, electron density n = N(x)/No , and potential u = e ¢ ( x ) / k T are p l o t t e d
36
~5
5
,o.f nO Z
%4
>-
o,2
U
l-J b.
rr C) 0
t~jlO'
'
'
0'.5
'
'
1,0
NORMALIZED DISTANCE X / L
q
Fig. 3 velocity i0 °O
O15 NORMALIZED DISTANCE X / /
Fig. 2 Spatial d i s t r i b u t i o n of normalized p o t e n t i a l u, donor density no and electron density at eV/kT=75 in GaAs. as a function of normalized distance y = x/L f o r eV/kT = 75. We used No f o r the donor density at the center of the specimen with the channel length L. We assumed Gaussian d i s t r i b u t i o n f o r the donors. In order to take i n t o account the hot electron e f f e c t we assumed the following relation f o r the inverse mobility po/P = (I + ~E2 + BE~ + yE~) I / 2
(3)
where P0 is the ohmic m o b i l i t y . In eq. (3) the term 1 + mE2 dominates at low e l e c t r i c f i e l d s and the other higher power terms are included to account f o r the i n t e r v a l l e y t r a n s f e r of hot electrons. By adjusting the parameters m, B, and y, we obtain a c u r r e n t - v o l t a g e r e l a t i o n s h i p which coincides with the experimental curve. An example is shown in Fig. 1 by the s o l i d curve. When we know the s p a t i a l d i s t r i b u t i o n of the electric field, the d r i f t v e l o c i t y is calculated as a function of distance from eq. (3). The r e s u l t is shown in Fig. 3, where we f i n d t h a t the electron d r i f t v e l o c i t y reaches about 3XIOv cm/s and thus there e x i s t s a d r i f t velocity overshoot. As mentioned e a r l i e r , the overshoot e f f e c t appears only in a short distance from the cathode at high e l e c t r i c f i e l d . The above treatment used in the present analysis is based on the assumption t h a t the electrons are q u a s i - s t a t i c and thus the s p a t i a l d i s t r i b u t i o n of the drift v e l o c i t y i s given by the s p a t i a l
Spatial d i s t r i b u t i o n of d r i f t in short n+ n n+ GaAs at 30OK.
d i s t r i b u t i o n of the e l e c t r i c f i e l d . In other words, the present treatment does not include time e v o l u t i o n of the d r i f t velocity. It is very i n t e r e s t i n g t h a t the p o t e n t i a l d i s t r i b u t i o n in the specimen e x h i b i t s a minimum near the cathode as seen in Fig. 2. We c a r r i e d out Monte Carlo simultion of hot electrons assuming a potential distribution s i m i l a r to that shown in Fig. 2. F i r s t we show the r e s u l t s when the p o t e n t i a l distribution is approximated by a t r i a n g u l a r shape with p o t e n t i a l minimum of O.03V at 0.05. 0 . I 0 , 0.15 and 0.20~m from the cathode. The r e s u l t s are shown in Fig. 4, where the applied voltage is IV. We f i n d in Fig. 4 that the d r i f t v e l o c i t y increases sharply beyond the p o s i t i o n of p o t e n t i a l minimum. When we change the minimum value of the potential from 0.01 to O.05V and f i x the p o s i t i o n at O.20pm, the d r i f t v e l o c i t y d i s t r i b u t i o n s are given by the curves shown in Fig. 5, where we f i n d no 10
%
d 4 >3
~2 i
I
DISTANCE
I
I
I
[micron]
Fig. 4 D r i f t v e l o c i t y as a function of p o s i t i o n in short GaAs, where t r i a n g u l a r d i s t r i b u t i o n of p o t e n t i a l is assumed.
37 18
% o.osv o.otv
5 o.o~v o ~
>
3
g, 0 o
0.2
0.4
DISTANCE
0.6 0.8 [micron]
Fig. 5 Drift velocity as a function of position Sn short GaAs, where triangu}ar distribution of potential is assumed.
~
o
y
j
~6
~z
0
1
DISTANCE [micron]
Fig. 6 Monte Carlo simulation of the D r i f t v e l o c i t y in short n+n n+GaAs with a given potential d i s t r i b u t i o n . significant difference in the three curves. F~omthese results we find t~at the simplified simulation explains qual i tati vely the experimentally More refined treatment is shown in Fig. 6, where the potential distribution is obtained by solving eqs. (l)-(3) for IV application and electron d r i f t velocity is obtained by Monte Carlo simulation. The overshoot is found to occur near the center of the sample but the detailed
Among them the hot electron effect plays the most important role in the FET characteristics. For example, impact ionization near the drain contact and hot electron injection into the oxide. However, the behavior of hot electrons in short MOSFET is not known well. Two approaches may be possible; one is the classical two-dimensional device simulation as done by Mock [9] and the other is quantum mechanical treatment in which scatterings of the two-dimensional electrons formed in the inversion layer is properly taken into account. As far as our knowledge is concerned, the former is usually used, in which twodimensional quantization is disregarded and the t h r e e dimensional electron motion is assumed. In this paper we will present experimental results of very short MOSFET and analysis by using two-dimensional device simulator, where three dimensional motion of electrons z~z~.~mc~ '.~e~l,~,p~.~.~ ~ h ~ of two-dimensional quantization on the layers by
using Monte Carlo simulation.
at room temperature. Results for 0.75 and O.?5~mMU~FETs are shown Yn FYq. and Fig. 8, respective]y. We see in Fig. 8 that the drain current increases at higher drain voltages. Such a behavior is explained in terms of electron impact in the region near the drain contact. Present analysis was made by using the method reported by Mock, where the cross section of the device is divided into small lattices, and static potential, electron density and current a~e o~tained by so~ing the ~oisson equation and an equation of electron current continuity, each with specified analysis are given by Mock [9] and we will not repeat i t here. In the present analysis we take into account the hot electron effect by introducing the following relation for the field dependence of the electron d r i f t velocity.
features a r e ~ot "~n gooc~ ~greement ~'~t}~ the quasi-static analysis.
HOT ELECTRONEFFECT IN SHORTMOSFET As the channel length is reduced in the range of submicrons, the short channel effects become very important.
where U~ E and v s are the low field mobility, electric field and saturation drift velocity of the electrons. We proceed fitting to obtain a good agreement between the computed results and the experimental curves by adjusting the saturation velocity. The dashed curves in Figs. 7 and 8 are calculated
38 30
l
l
I
I
I
I
we
MOS O. 7 5 ( m i c r o n ) ....
E
EXPERIMENT Rs-200 (ohm)
SIMULATION V S = l OE4 ( m l g e ¢ )
....
20
with V(z) = -e@(z) and d2~(z) e dz" = ~[(Na- Nd) + Z i N i l ~ i 12]
Vg=10(V) V 9 - 8 (V)
n, -'~ U
solve the following equations. ~2 d 2 [ - 2m3 dz 2 + V ( z ) ] ~ i ( z ) = E i ~ i ( z ) ( 5 )
10
,~" ~ ' * ,~,, ..,, _.._._._._._._ _. _._.. . . . . .
. -" "
,~" 2
3
4
Vg=4 (V)
Vg=2(V)
~ •., 1
5
where we use the same n o t a t i o n used by Stern [ I I ] and the e l e c t r o n energy is given by
6
E -2-m-*-~2 (k~ + k;) VOLTAGE
I
I
I
I
I
MOS O. 2 5 ( m l c r o n )
• ,,,
EXPERIHENT
.....
2O
SIMULATION Vs-ZOE4(m/sec)
hi • U
~.%~.
•
•
I
V9=4 (V)
...
•
. • . •" ,/ 4 "........ .--')
2
3
(7)
V9-2
The e l e c t r o n d e n s i t y N i in the subband E i is given by nvimdikT EF - Ei Ni = ~2 F [ k~ ] (8)
I
Rs-|SO(ohm) E
+ Ei.
IV]
Fig. 7 Drain c u r r e n t vs. d r a i n v o l t a g e c h a r a c t e r i s t i c s in O.75pm MOSFET.
30
(6)
d~> h
where F(x) = I n [ l + e x p ( x ) ] . We calcul a t e d the subband e n e r g i e s , Fermi energy and e l e c t r o n d e n s i t y in the subbands. An example of the s e l f - c o n s i s t e n t calculation is presented in Fig. 9, where square o f the wave f u n c t i o n s and subband energies ( h o r i z o n t a l l i n e s ) are shown f o r Si (111) surface at 30OK.
(V) T
4
I
I
5
6
T
.8 @
VOLTAGE
Iv]
Fig. 8 Drain c u r r e n t vs. d r a i n voltage c h a r a c t e r i s t i c s in 0.25 pm MOSFET. curves, where the best f i t t i n g is obtained by using v s = IXlO 7 cm/s f o r 0.75pm and 2XlOTcm/s t o r 0.25pm samples. Noting t h a t the s a t u r a t i o n v e l o c i t y in a bulk s i l i c o n is about 7XlO 6 cm/s, the obtained values o f v s i n d i c a t e t h a t the d r i f t v e l o c i t y overshoot e x i s t s in s h o r t MOSFET. The s a t u r a t i o n velocity was found to increase w i t h decreasing the channel length f o r three MOSFETs w i t h 0.25, 0.50 and O.75pm e f f e c t i v e channel length. Such a f e a t u r e is c o n s i s t e n t w i t h the Monte Carlo s i m u l a t i o n , where the overshoot becomes l a r g e r f o r s h o r t e r samples and at higher e l e c t r i c f i e l d s . In the above treatment we assumed that the e l e c t r o n motion is three dimensional. However, i t is well known t h a t the e l e c t r o n s in the i n v e r s i o n l a y e r are confined in a very narrow region near the surface and two dimensional q u a n t i z a t i o n occurs [ I 0 , I I ] . In o r d e r to obtain the e l e c t r o n s t a t e s we have to solve Schr~dinger equation and Poisson equation self-consistently. Following the treatment o f Stern [ I I ] ,
> J E
.B
v(z) :
(i 0)
FZ .4 W 0
O..2
e
18 DISTRNCE
28 FROM
38 SURFRCE
48
58
Z[nm]
Fig. 9 S e l f - c o n s i s t e n t r e s u l t s o f two dimensional e l e c t r o n states in Si ( I I I ) surface at 30OK. Ninv=l.OXlOiTcm-~ In the present a n a l y s i s we assumed t h a t the subband energies are u n a f f e c t e d by the e l e c t r i c field. We c a l c u l a t e d the d r i f t velocity and the e l e c t r o n d e n s i t y in the subbands as a function of electric f i e l d by using Monte Carlo simulation. We deal w i t h the 2DEG formed in the (I00) surface o f Si and the e l e c t r o n d e n s i t y is assumed t o be I.OXIO 13 cm-2 . Since almost a l l the e l e c t r o n s populate in the three subbands Eo, E i and E~, we take i n t o account olny the three subbands. The subband energies are found to be Eo= 208.9, El= 293.6 and E~ = 290.7meV, and t h e i r
39 average penetrationoof the electrons is 12.3, 29.8 and 28.7A, respectively. In the present simulation we included the scattering processes due to screened ionized impurity, surface roughness, acoustic phonon and intervalley phonon scatterings. Since the deformation potentials for the two-dimensional electrons in MOS inversion layer are not known, we used bulk values [12]. The intervalley phonon energies are determined f r o m magnetophonon resonance experiments by Hamaguchi, Hirose and Shimomae [13]. We picked up f- and gtype phonons which are expected to be important. The phonon temperatures used in the present calculations are 750K (g) and 630K (f) for the zero order scattering and 134K (g) and 230K (f) for the f i r s t order scattering [12], which are a l i t t l e different from our magnetophonon measurements [13]. The deformation potentials for the intervalley scattering are chosen to be Do = 9.0XlOZ°eV/m and DI = 28.0eV for the zero and f i r s t order scattering, respectively, for both g- and f-scattering processes. |'
v
i
!
m
•
v
Id
i
i
i
i
"G
L-I.5OFm
E
°290K
Ic~
•
I
I
I
!
nc
i
~ i i i
~
I
I|11
I
I
I
I
Ill=
ao~
F
ic:P
(Vlcm)
Fig. II Drift velocity as a function of electric field measured in 1.51Jm MOSFET at 77 and 290K. I.O
!
i
'
'
w
~0.5 U
01~ 0
I
,~
*
i
|
I
vo" F
i
•
I
I0 F
4
I0
5
(V/cm)
Fig. 12 Electron density vs. electric I • field in the subbands Eo, EI and E0 In Si (IOO) surface obtained by Monte Carlo simulation.
I
vo' (V/cm)
Fig. lO Drift velocity of 2D electrons in MOS (lO0) surface calculated by Monte Carlo simulation at 3OOK. The results of Monte Carlo simulation are shown in Fig. lO, where we find a weak saturation at higher electric fields. The feature is very similar to the observed results which are shown in Fig. 11 for 1.5pm MOSFET. Note that the d r i f t velocity calculated from the Monte Carlo simulation is slightly higher than the experimental value, where the experimental values are obtained by using gradual channel approximation and the estimated electron density is about
l.OXlO 13 cm-2. The difference of the drift velocities is reduced i f we increase the surface roughness scattering. I t should be note here that the d r i f t velocity in bulk Si shows a strong saturation and thus the observed results are not explained. We present the electron populations in the three subbands as a function of electric field in Fig. 12, where we find a decrease of electrons in the ground subband Eo and increase in the higher subband E# with h i g h e r density of states. Detailes of the analysis are reported elsewhere [14].
40 SELF-CONSISTENT CALCULATIONS OF 2DELECTRON DENSITY IN AlxGal_xAs/GaAs Recent progress in molecular beam epitaxy technique has made possible to fabricate A1GaAs/GaAs h e t e r o j u n c t i o n devices with high e l e c t r o n m o b i l i t i e s at low temperatures [15-17]. Among them HEMT (high e l e c t r o n m o b i l i t y t r a n s i s t o r ) is known to be a t t r a c t i v e in view of high speed devices. In HEMT the 2DEG formed in a narrow region of GaAs at the i n t e r f a c e moves very f a s t because the electrons are supplied from the n-AIGaAs and are free from the impurity s c a t t e r ing. In the sense of device f a b r i c a t i o n i t is very important to estimate the e l e c t r o n density in GaAs when doping density of AIGaAs is given. As f a r as our knowledge is concerned, the 2D e l e c t r o n density is estimated by using t r i a n g u l a r p o t e n t i a l approximation [1819] or by using v a r i a t i o n a l approximation [20], where they neglect the penet r a t i o n of the e l e c t r o n wave functions into AIGaAs layer. We report here f o r the f i r s t time s e l f - c o n s i s t e n t calculations of the 2D e l e c t r o n density in AIGaAs/GaAs heterostructures as functions of the doping density in nA1 Ga As and the thickness of the undoped AlxGal_xAs f o r x = 0.17 to 0.33 at lattice temperatures 4.2 and 77K. The c a l c u l a t i o n s were carried out by solving Schrodinger equation and Poisson equation s e l f - c o n s i s t e n t l y [21, 22] by taking into account f i v e subbands at T = 4.2K and ten subbands at T = 77K, if necessary, but the subbands of which energies exceed the b a r r i e r height are disregarded. We assume that the b a r r i e r height is given by AV = X in eV and the i o n i z a t i o n enegy of donors in n-AIGaAs is taken to be 50, I00 or 150meV. 0.5
0.4
0.3
~.2
0.
I
0.0 -50
E}
100 200 DISTANCE
300
400
500
the wave functions of the three subbands and the p o t e n t i a l energy (conduction band edge) are p l o t t e d together with the subband energies (shown by bars) and Fermi energy (longer bar). Note that e l e c t r o n wave functions penetrate into the undoped AIGaAs layer of which thickness is 50A. I O *II
a) x - 0 . 3 3 b] x - 0.30
o)~b~)
~
c)x - 025
/
~
I,...... I 0 I*
10 I? Dopln 9
Density
of
I 0 I*
AI=Go,.=As
( c m "s)
Fig. 14 2D e l e c t r o n density in GaAs vs. donor density in n-AIGaAs at T = 4.2K. I 0 ~= a) x - 0.33
~w Q
b) x
•
0.30
c) x
•
o.2s
a
)
~
)
~
..~'~/.f:~f
I 0"
w
,~ IO"
IO I? Doplng
Denslty
of
AI,Gai..As
IO ' I (c m'3)
Fig. ]5 2D e l e c t r o n density in GaAs vs. donor density in n-AlGaAs at T =77K. Figure 14 shows the e l e c t r o n sheet density in GaAs calculated as a function of doping density of AlxGal_xAs f o r x = 0.33, 0.30, 0.25 and 0.17 at 4.2K,_where spacer thickness of AIGaAs is 60X and donor i o n i z a t i o n energy is lOOmeV. The r e s u l t s at 77K are shown in Fig. 15. The present r e s u l t s give a very important information about the doping density and the thickness of the nAlxGal_xAs required to get a desired e l e c t r o n density in the GaAs layer and thus a design p r i n c i p l e of HEMT.
[A]
Fig.13 S e l f - c o n s i s t e n t r e s u l t s f o r GaAs/ AlxGal_xAs (x=0.3) f o r Ns = 5XlOlZcm-.z A typical example of the s e l f consistent r e s u l t s is shown in Fig. 13 f o r e l e c t r o n density 5.0XlO17cm -2, where
SINGLE QUANTUMWELL TRANSlSTOR WITH AIGaAs/GaAs/AIGaAs HETEROSTRUCTURES The
authors
proposed
new devices
41 called SQWT (single quantum well transistor) of which operation principle is based on the wave function control in a single quantum well and pointed out that a h i g h electron mobility will be achieved at low temperatures in a modulation doped quantum well where the half region of GaAs-well is doped by donors [22]. Confinement of 2DEG in a quantum well has been proposed by Sakaki [23] in connection with velocity-modulation transistor (VMT) in which channel conductance is modulated by changing the populations of 2DEG in two heterointerfaces. I t has been pointed out that the source and drain contacts are different between the SQWTand VMT [22]. In this paper we present physical mechanism of the modulation doped SQWT by showing a confinement of 2DEG in the un-doped region of GaAs-well by applying an external voltage. T h i s mechanism suggests a possibility to fabricate high electron mobility transistor of AIGaAs/ GaAs/Al GaAs structures. Gale. pf/f/~
Drain f//l/~
~/f/f/ff[//]
IOOA /' omGo:~$(non-doped)
.;~-:~!-~~]
' "" ~........ ',-2PEG
/ ._
f ...........
/ /
/
250~,
GoAs t non- doped ) a-~A;- . . . . . . . . . . . . . . . . .
[ SI- doped
~
.
250A
~No:2 X ' ° ' ' ~ )
--2op
Alo3GOo.7 As (non-doped)
IOOA
GoAs Sub~mte bock-gote
Fig. 16 Cross-sectional view of SQWT. In order to illustrate the basic feature of the modulation doped SQWTwe present a typical example of the selfconsistent calculations of the 2DEG in a quantum well A1 .Ga~ .As/ GaAs/Al GaI xAs (x=0.3) shown In . , ~ Fig. ,-~ 16, where xthe half region of GaAs layer is modulation doped with donors of 2.0XlO~cm - 2and A~GaAs is un-doped. The GaAs-we~l is 500A and AIGaAs barriers are lOOA, where the donors in the GaAs-well are assumed to be completely ionized and t h u s the electron sheet density is given by Ns = 5.0X10~2cm-2. In the calculations we assumed that the external voltage is applied between the un-doped AIGaAs layers of lOOA thick for simplicity. The self-consistent results are obtained by solving eqs. (5) and (6) by putting V(z) : -e@(z) + ZW[u(-z) + u ( z - d ) ]
(9)
where u(z) is the step function. For simplicity we assumed that the effective mass and the dielectric constant are given by 0.068m and 12.9 in AIGaAs and GaAs. We obtained self-consistent results of the subband energy Ei , subband electron density N~ and the wave function ~(z) with ano without an external voltage Vg. A~Gm-.As (x-03)
GO~
AI,Go,~ (x,O.3)I
0.4
m 0.2
-I00
.
0
I00
I¢J'
200
300
400
500
600
DISTANCE (~)
Fig. 17 Self-consistent results of 2D electron states in SQWTat Vg = O. Figure 17 shows the results for the gate voltage Vg = 0 at T = 4.2K, where the conduction band-edge V(z), subband energy Ei , Fermi energy EF, and square of the wave function are shown. Note that the donors are distributed in_the region of the GaAs-well from 250X to 500A as shown in Fig. 16. We assumed that the conduction band discontinuity is O.3eV. We obtained Eo = 13.1, El =23.8, E2 = 33.1, E3 =46.6 and EF =27.2meV. I t is very interesting to point out that the electrons populate in the subbands Eo (4.0XlO12cm-2) and El (O.97XlO12cm-2) and therefore they are not localized in the doped region of GaAs-well. In this case the current is mainly carried by the electrons in the subband Eo and a small part of the current consists of electron d r i f t in the subband El. When we apply a gate voltage of O.4V, the self-consistent calculations lead to the results shown in Fig. 18 where we obtain Eo = 56.0, El = 82.3, E2 = lOl.8, E~ = 123.2, and EF =73.6meV. We find in Fig. 18 that the- electrons are confined in the subband Eo or in the region 0 ~20oA of the GaAs-well because Ep < E~ . These results suggest us to f~bricate a novel FET "modulation doped SQWT". In such a device the drain current is carried by electrons confined in the non-doped region of the GaAs-well by applying an external voltage and the electrons move with a very high mobility because the electrons in this region do
42 0.6 A
.<
AI,GoJ.,As (x,03}
I00
GaAs z 0
0.4
E~
Iu
Z
u-50 Q
02
z F,<
o - ioo
Eo'~/k~/~ o
\7~.Z too
200
\ / " ~ . ~ ~o
400
. ~o
600
-02
0 EXTERNAL
DISTANCE (~)
02 VOLTAGE
0.4 (V)
Fig. 18 S e l f - c o n s i s t e n t r e s u l t s of 2D e l e c t r o n states in SQWT at Vg = 0.4V.
Fig.20 Electron density No and N1in the subbands E o and E1as a function of Vg.
not s u f f e r from the impurity s c a t t e r i n g . Therefore we can f a b r i c a t e a high speed FET using modulation doped single quantum well in which the e l e c t r o n t r a n s p o r t is quite s i m i l a r to HEMT. In Fig. 19 we present the subband energies E0, E~ , E2, E3 , and Fermi energy E~ as a function of the external gate voltage where we f i n d that the Fermi energy crosses the f i r s t e x c i t e d state E1at about V~ = 0.32V and thus the electrons populat~ only in the ground subband Eo at a gate voltage exceeding 0.32V as shown in Fig. 20. Asymmetric features of the subband energies and the e l e c t r o n populations with respect to the external voltage are due to the modulat i o n doping in the GaAs-well. These res u l t s i n d i c a t e that the threshold v o l t age of the FET c h a r a c t e r i s t i c s is e a s i l y c o n t r o l l e d by the doping p r o f i l e of the GaAs-well. We carried out s i m i l a r calc u l a t i o n s on the modulation doped SQWT with 500A GaAs-well of which h a l f region is doped by donors Nd. Our c a l c u l a t i o n s show that the gate voltage required to pupulate a l l the electrons in the ground subband Eo and in the non-doped region of the well is O.2V f o r N~= I.OXIO ~7 and O.7V f o r Nd = 4XlO17cm~3 at T = 4.2K.
In conclusion we found that a modulation doped SQWT is very promising to achieve a high speed FET l i k e HEMT. Our s e l f - c o n s i s t e n t c a l c u l a t i o n s show that the electrons distributed in the whole region of the GaAs-well are e a s i l y confined in the non-doped region of the GaAs by applying an external gate voltage between the AIGaAs b a r r i e r s . Because of the asymmetric nature of the gate voltage dependence i t is found that the threshold voltage VT is e a s i l y cont r o l l e d by changing the doping p r o f i l e and doping density in the GaAs-well. It is shown from the s e l f - c o n s i s t e n t calculations that we can achieve a hiqh electron mobility in a modulation doped SQWT proposed in our paper [22].
0.2
ACKNOWLEDGEMENTS One of the authors (C. H) is very thankful f o r the kind support of The Mitsubishi Foundation to the present work. We acknowledge the support of a Grant-in-Aid for Scientific Research from the M i n i s t r y of Education, Science and Culture. The authors (C. H and T. M) express t h e i r thanks to F u j i t s u Laboratories L t d . , f o r the kind supply of the short channel GaAs samples used in the present work.
E,
>O.I n,. b~ Z
REFERENCES O -O.2
J
i
0
i
i
0.2
EXTERNAL V O L T A G E
Fig.19
i
0.4 (V)
Subband energy Ei vs. Vg in SQWT
I . J.G. Ruch: IEEE Trans. Electron Dev. ED-19 (1972) 652. 2. M. S. Shur and L. F. Eastmen: IEEE Trans. Electron Dev. ED-26 (1979) 1677.
43 3. R. Zuleeg: IEEE Electron Dev. Lett. EDL-I (1980) 234. 4. L. F. Eastman, R. Stall, D. Woodard, N. Dandekar, C. E. C. Wood, M.S. Shur and K. Boad: Electron. Lett. ]_6_ (1980) 524 5. M. A. Hollis, L. F. Eastman and C. E. Wood: Electron. Lett. 18 (1982) 570. 6. C. Hamaguchi, K. Terashima, A. Shibatomi and O. Ryuzan: Proc. 16th Int. Conf. Semicond. Phys. (1982, Montpellier), Physica llTB & !]8B (1983) 238. 7. A. J. Holden and B. T. Debney: Electron. Lett. 18 (1982) 558. 8. T. Mori, C. Hamaguchi, A. Shibatomi and O. Ryuzan: Japn. J. Appl. Phys. 23 (1984) 212. 9. M. S. Mock: Solid-State Electronics, (1973) 601. lO. F, Stern and W. E. Howard: Phys. Rev. 163 (1967) 816. I I . F. Stern: Phys. Rev. B5 (1972) 4891. 12. D. K. Ferry: Phys. Rev. Bl___4(1976) 5364. 13. C. Hamaguchi, Y. Hirose and K. Shimomae: Jpn. J. Appl. Phys. 22 (1983) Suppl. 22-_____3,p. 190. 14. K. Terashima: Thesis, Osaka Univ. (1983) unpublished. 15. R. Dingle, H. L. Stormer, A. C. Gossard and W. Wiegman: Appl. Phys. Lett._~_3 (1978) 665. 16. T. Mimura, S. Hiyamizu, T. F u j i i , and K. Nambu: Jpn.Apl1. Phys. 19 (1980) L225. 17. S. Hiyamizu, J. Saito, K. Nambu and T. Ishikawa: Jpn. J. Appl. Phys. 22 (1983) L609 and references therein. 18. K. Lee and M. Shur: J. Appl. Phys. 54 (1983) 2093. 19. T. C. Hsieh, K. Hess and J. J. Coleman: Solid-State Electronics 26 (1983) I173. 20. F. Stern: Appl. Phys. Lett. 43 (1983) 974. 21. T. Ando: J. Phys. Soc. Jpn. 5_] (1982) 3893 & 3900. 22. C. Hamaguchi, K. Miyatsuji and H. Hihara: Jpn. J. Appl. Phys. 2__33 (1984) No.3. 23. H. Sakaki: Jpn. J. Appl. Phys. 2_! (1982) L381.