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Barrier height modification in Schottky MIS diodes
Volume 92A, number 6
PHYSICS LETTERS
22 November 1982
BARRIER HEIGHT MODIFICATION IN SCHOTTKY MIS DIODES A.H.M. SHOUSHA Department o f Physics, United Arab Emirates University, Al Ain, United Arab Emirates
Received 1 March 1982
Expressions for the potential barrier height of Schottky MIS diodes having gaussian doping profiles are derived and solved numerically both at thermal equilibrium and in the presence of applied voltages. The possibility of barrier height modification using a thin highly doped surface layer is discussed.
1. Introduction. In order to improve the performance characteristics of Schottky barrier diodes, it is sometimes required to modify their effective barrier heights [1,2]. A possible technique to achieve this modification is to introduce a suitable highly doped surface layer using low-energy ion implantation [ 3 - 5 ] . The ion-implanted profile is generally nonuniform and can be approximately described by a gaussian distribution [6]. The barrier height modification
achieved using this technique is mainly due to the change in the semiconductor space charge density and the image force barrier lowering. In determining the barrier height, the properties o f the thin interracial layer between the metal and the semiconductor must be considered specially when the density of surface states is relatively high [7]. In the present work, expressions for the potential barrier height of Schottky metal/insulator/semiconductor
~
S
m
T
q(Vbi- Vs)
~q Vn E Fs
Fm
q4o
Eg
1 Fig. 1. Energy band diagram of Schottky MIS diodes under forward bias. 0 031-9163/82/0000-0000/$02.75 © 1982 North-Holland
293
Volume 92A, number 6
PHYSICS LETTERS
(M1S) diodes having gaussian doping profiles are derived and solved numerically. The possibility of barrier height modification using, a thin highly doped surface layer is discussed,
22 November 1982
and using the depletion approximation, Poisson's equation can be written as
dE/dx = (q/es)[N s exp(-bZx 2) + Nb] ,
(6)
height of a forward-biased n-type Schottky MIS diode is given by (fig. t)
where E is the electric field, e s is the semiconductor permittivity, and Ns, N b, and b are profile-dependent constants. The solution of eq. (6) using the boundary condition E(x = W) = 0 can be expressed as
~Bn(V) = ~m -- [Xs + ( A . - V i ) + ~bi(V)]
E(x) = -(q/bes){½ v ~ N s [erfc(bx) - erfc(b W)]
2. Genera[formulatiOn. Th e potential barrier
= (vbi- vo ;Vn V-
(I)
where ~)m is the metal work furrction, Xs is the semiconductor e!ectron affinity, A is the voltage drop ~)cross the insulatgT layer a/thermal equilibrium, Oi is the image force barrie.r lowering, Vbi is the built-in voltage, and Vis theextecnal applied voltage which will partly, appear across the semiconductor (Is) and padly across the insulator layer (Vi). The total voltage drop across the insulator layer can be written as A Y Vi = Vsc + (5/ei)[ass(V) + Osc(V) ] ,
+ Nbb(W - x)) .
(7)
Integrating eq. (7), we get Vbi - Vs = (q/ZbZes) X {Ns[1 - e x p ( - b 2 W 2 ) ] +Nbb2W2),
(8)
where erfc is the complementary error function [= 1 - (2/V~) f~ exp(-a2)da] and W is the space charge layer width. The semiconductor space charge density is thus given by
(2) Qsc(V) = -esE(0 )
with
= (q/b){½V~Ns[1 - erfc(bl¢)] +NbbW). 1
(9)
8
Vsc= ei f ( 8 - x)p(x) d x , o
(3)
where 6 is the in.sutator thickness, e i is the insulator permittivity, Qssis the surface state charge density, Qsc is the semiconducto¢ Space charge density, and 0 is the space charge d|stributibn in the insulator layer. For a uniform dis'tril~ufion of acceptor-type surface states tocaRzed at the semiconductor surface and having a fixed d~ncty D~s, the surface state charge density is given by [71
Qss(V) = -qDsS(EgJq ,- ~0:- [4)Bn(V) + Oi(V)] + (EFs s - EFm)/q) ;
"
(4)
where q is the electronic charge, E'g is the semiconductor band gap, and EFs s is the imref describing the electron population of the surface states. The location ofEFs s is determ~e~ by the dynamics of the surface states and should be some~chere between EFm and EFs. The semiconductor space charge density is determined using Poisson's equation. For a gaussian doping profile of the form
Combining eqs. (1), (2), (4), and (9), we obtain ~bBn(V) + ~bi(V) = 7(~bm - Xs - Vsc) + (1 - 7)[Eg/q - (~0 + (EFss - EFm)/q] - (78q/bei) {½VrnNs [1 - erfc(b W)] + Nbb W)
= (q/2b2es){Ns [1 - exp(-b 2 W2)] + Nb b2 W2} +(kT/q){ln[Nc/(N s exp(-bZW 2) + Nb)]} + V,
(10)
where 7 = 1/(1 + qDssS]ei), k is Boltzmann's constant, T is the absolute temperature, and N C is the effective density of states in the semiconductor conduction band. The above transcendental equation can be solved numerically for W using the Newton-Raphson iteration technique. Once W is determined, the barrier height can be calculated using eq. (10) and the following relation for the image force barrier lowering q~i(V) = (q/2es)
t
Nd(X ) = U s exp(--:b2x 2) 4 Nb, 294
(5)
X {(Ns/2v~b)[1 - erfc(bW)] +NbW/n}I/2.
(111
3. Results and discussion. The results presented in
V o l u m e 92A, n u m b e r 6
22 November 1982
PHYSICS LETTERS 0.20 -
0.15
-
b
=
.....
A/s
106 cm-1 ~,~
D
z/f//
105 cm~t
Z"~\ //
"~ 4
= 10'~ c ~ v "
19 -3 %N
/,'
= 10
s
cm
/
# 0.10
-gi
%
CO
04
-g--
0.2
.~_L 16
10
, 18
17
10
10
Ns.cm
t 19
10 -3
0.05
0.0 70
10
21
02
0.0
10
O.t,
0.6
V , VOLTS
Fig. 2. Thermal equilibrium barrier height as a function of the gaussian doping profile parameters for two different surface
0.20
state densities. (q~m= 4.2 V, Xs = 3.4 V, q~o = 0.3 V, Eg = 1.12 eV, p = 0, 6 = 10 A, e s = 11.8 eo, e i = 3.82 eo,Nb = 2 X 1014 cm-S,Nc = 2.8 × 1019 cm-3, and T = 300 K.) 5
this work have been obtained using constants which are intended to refer to A u - S i O 2 - ( n - t y p e , single crystal) Si diodes. The results presented in fig. 2 show that the barrier height at thermal equilibrium CBno can be reduced by increasing the doping of the diode surface. A significant decrease in the barrier height is obtained when Ns/b exceeds 1012 cm - 2 . The surface state charge can also affect the barrier height ifDss is sufficiently high. In calculating the results presented in fig. 3, it is assumed that the surface states equilibrate with the semiconductor electrons (i.e., EFs s = EFs ). In this case, [~bBn(V) - qSBno] increases with increasing applied voltage, surface layer doping, and surface state density. On the other hand, when the surface states equilibrate with the metal (i.e., EFs s = EFm ), [¢Bn(V) -- ~bBno] changes slightly with diode parameters and is almost independent of the applied voltage over a wide voltage range. An approximate expression for the barrier height can be obtained when the ion-implanted surface layer is very thin such that the doping profile can be expressed approximately by Nd(X )
=
Nst6(x)
+
Nb ,
(12)
where Nst is the number of implanted atoms per unit surface area [for gaussian profiles, Nst = Ns(w'n/2 b)]
-~,'~ 0.I0
>-
/ _ ~ , ~ - I/
.,#/
16 10
~17
=
IU
Cm eV
~"~B !0
/
t9 10
Ns ~tm -3
Fig. 3. Dependence of [~Bn{V) - CBnol on the applied voltage, doping profiIe patameters, arid surface state density. (Same constants as in fig. 2 and taking EFs s = EFs. ) and 6(x) is the Dirac delta function. Following the analysis given above and making reasonable assumptions (7), we can get t~Bn(V)
= 7(¢m
-- Xs -
+ (1 - 7 ) [ E g / q
Vsc
-
q6Nst/ei)
- d20 + (EFs s - ~"Vm)/q]
-- ( q / 2 e s ) ( N s t / n ) l / 2
(13)
The barrier heights obtained using the above equation (fig. 4) have been found to be reasonably accurate as long as bW >> 1. This condition is usually satisfied when b is greater than 10 6 cm -1 . A similar analysis for diodes using p-type semiconductor shows that the barrier height can be approxi295
Volume 92A, number 6
PHYSICS LETTERS
22 November 1982
mately written as
o,8~
~s~
- - ' . ~ . "-.
l lo k
CBp(V) = 7(Eg/q + Xs - Cm + Vsc - qSNst/ei) + (1 - 7)[q50 - (EFs s - EFm)/q]
d c~ 7>
0,6
-- (q/Zes)(Nst/n)l/2 .
g " ~ o,4
Oss 1012
0,2 2 ×1011
"~
It is thus shown that the effective barrier height of Schottky MIS diodes can be controlled b y the proper choice of the highly doped surface layer parameters.
""""i~
""
1012
1013
N st 9cm-2 100
/ = >
Ill
II . ~" /
75
E.
%~1II
•
1
ill" /
/ /
iI
ii
l/I
;
'
,x%/ "
,
50
_
-%
, 25
,/
/, / iii I/
,
/ "/
?.
° > 1 " " *"
I
[
../ ii
~"
o
'
o 0,0
0,1
0,2 V ~ V OLTS
0,3
0,¢
Fig. 4. Barrier heights and their voltage dependence for different interfacial layer thicknesses and surface state densities assuming that Nd(x ) = Nst6 (x) + N b. (Same constants as in fig. 2)
296
(14)
References [1] K. Kajiyama, S. Sakata and Y. Mizushima, Proc. IEEE 62 (1974) 1287. [2] A.H.M. Shousha, J. Phys. D: Appl. Phys., to be published. [3] Y. Anand and W.J. Moroney, Proc. IEEE 59 (1971) 1182. [4] J.M. Shannon, Appl. Phys. Lett. 24 (1974) 369. [5] C. Wu, J. Appl. Phys. 51 (1980) 4919. [6] J. Mayer, L. Eriksson and J. Davis, Ion implantation in semiconductors (Academic Press, New York, 1970). [7 ] S.M. Sze, Physics of semiconductors devices (Wiley, New York, 1969).
PHYSICS LETTERS
22 November 1982
BARRIER HEIGHT MODIFICATION IN SCHOTTKY MIS DIODES A.H.M. SHOUSHA Department o f Physics, United Arab Emirates University, Al Ain, United Arab Emirates
Received 1 March 1982
Expressions for the potential barrier height of Schottky MIS diodes having gaussian doping profiles are derived and solved numerically both at thermal equilibrium and in the presence of applied voltages. The possibility of barrier height modification using a thin highly doped surface layer is discussed.
1. Introduction. In order to improve the performance characteristics of Schottky barrier diodes, it is sometimes required to modify their effective barrier heights [1,2]. A possible technique to achieve this modification is to introduce a suitable highly doped surface layer using low-energy ion implantation [ 3 - 5 ] . The ion-implanted profile is generally nonuniform and can be approximately described by a gaussian distribution [6]. The barrier height modification
achieved using this technique is mainly due to the change in the semiconductor space charge density and the image force barrier lowering. In determining the barrier height, the properties o f the thin interracial layer between the metal and the semiconductor must be considered specially when the density of surface states is relatively high [7]. In the present work, expressions for the potential barrier height of Schottky metal/insulator/semiconductor
~
S
m
T
q(Vbi- Vs)
~q Vn E Fs
Fm
q4o
Eg
1 Fig. 1. Energy band diagram of Schottky MIS diodes under forward bias. 0 031-9163/82/0000-0000/$02.75 © 1982 North-Holland
293
Volume 92A, number 6
PHYSICS LETTERS
(M1S) diodes having gaussian doping profiles are derived and solved numerically. The possibility of barrier height modification using, a thin highly doped surface layer is discussed,
22 November 1982
and using the depletion approximation, Poisson's equation can be written as
dE/dx = (q/es)[N s exp(-bZx 2) + Nb] ,
(6)
height of a forward-biased n-type Schottky MIS diode is given by (fig. t)
where E is the electric field, e s is the semiconductor permittivity, and Ns, N b, and b are profile-dependent constants. The solution of eq. (6) using the boundary condition E(x = W) = 0 can be expressed as
~Bn(V) = ~m -- [Xs + ( A . - V i ) + ~bi(V)]
E(x) = -(q/bes){½ v ~ N s [erfc(bx) - erfc(b W)]
2. Genera[formulatiOn. Th e potential barrier
= (vbi- vo ;Vn V-
(I)
where ~)m is the metal work furrction, Xs is the semiconductor e!ectron affinity, A is the voltage drop ~)cross the insulatgT layer a/thermal equilibrium, Oi is the image force barrie.r lowering, Vbi is the built-in voltage, and Vis theextecnal applied voltage which will partly, appear across the semiconductor (Is) and padly across the insulator layer (Vi). The total voltage drop across the insulator layer can be written as A Y Vi = Vsc + (5/ei)[ass(V) + Osc(V) ] ,
+ Nbb(W - x)) .
(7)
Integrating eq. (7), we get Vbi - Vs = (q/ZbZes) X {Ns[1 - e x p ( - b 2 W 2 ) ] +Nbb2W2),
(8)
where erfc is the complementary error function [= 1 - (2/V~) f~ exp(-a2)da] and W is the space charge layer width. The semiconductor space charge density is thus given by
(2) Qsc(V) = -esE(0 )
with
= (q/b){½V~Ns[1 - erfc(bl¢)] +NbbW). 1
(9)
8
Vsc= ei f ( 8 - x)p(x) d x , o
(3)
where 6 is the in.sutator thickness, e i is the insulator permittivity, Qssis the surface state charge density, Qsc is the semiconducto¢ Space charge density, and 0 is the space charge d|stributibn in the insulator layer. For a uniform dis'tril~ufion of acceptor-type surface states tocaRzed at the semiconductor surface and having a fixed d~ncty D~s, the surface state charge density is given by [71
Qss(V) = -qDsS(EgJq ,- ~0:- [4)Bn(V) + Oi(V)] + (EFs s - EFm)/q) ;
"
(4)
where q is the electronic charge, E'g is the semiconductor band gap, and EFs s is the imref describing the electron population of the surface states. The location ofEFs s is determ~e~ by the dynamics of the surface states and should be some~chere between EFm and EFs. The semiconductor space charge density is determined using Poisson's equation. For a gaussian doping profile of the form
Combining eqs. (1), (2), (4), and (9), we obtain ~bBn(V) + ~bi(V) = 7(~bm - Xs - Vsc) + (1 - 7)[Eg/q - (~0 + (EFss - EFm)/q] - (78q/bei) {½VrnNs [1 - erfc(b W)] + Nbb W)
= (q/2b2es){Ns [1 - exp(-b 2 W2)] + Nb b2 W2} +(kT/q){ln[Nc/(N s exp(-bZW 2) + Nb)]} + V,
(10)
where 7 = 1/(1 + qDssS]ei), k is Boltzmann's constant, T is the absolute temperature, and N C is the effective density of states in the semiconductor conduction band. The above transcendental equation can be solved numerically for W using the Newton-Raphson iteration technique. Once W is determined, the barrier height can be calculated using eq. (10) and the following relation for the image force barrier lowering q~i(V) = (q/2es)
t
Nd(X ) = U s exp(--:b2x 2) 4 Nb, 294
(5)
X {(Ns/2v~b)[1 - erfc(bW)] +NbW/n}I/2.
(111
3. Results and discussion. The results presented in
V o l u m e 92A, n u m b e r 6
22 November 1982
PHYSICS LETTERS 0.20 -
0.15
-
b
=
.....
A/s
106 cm-1 ~,~
D
z/f//
105 cm~t
Z"~\ //
"~ 4
= 10'~ c ~ v "
19 -3 %N
/,'
= 10
s
cm
/
# 0.10
-gi
%
CO
04
-g--
0.2
.~_L 16
10
, 18
17
10
10
Ns.cm
t 19
10 -3
0.05
0.0 70
10
21
02
0.0
10
O.t,
0.6
V , VOLTS
Fig. 2. Thermal equilibrium barrier height as a function of the gaussian doping profile parameters for two different surface
0.20
state densities. (q~m= 4.2 V, Xs = 3.4 V, q~o = 0.3 V, Eg = 1.12 eV, p = 0, 6 = 10 A, e s = 11.8 eo, e i = 3.82 eo,Nb = 2 X 1014 cm-S,Nc = 2.8 × 1019 cm-3, and T = 300 K.) 5
this work have been obtained using constants which are intended to refer to A u - S i O 2 - ( n - t y p e , single crystal) Si diodes. The results presented in fig. 2 show that the barrier height at thermal equilibrium CBno can be reduced by increasing the doping of the diode surface. A significant decrease in the barrier height is obtained when Ns/b exceeds 1012 cm - 2 . The surface state charge can also affect the barrier height ifDss is sufficiently high. In calculating the results presented in fig. 3, it is assumed that the surface states equilibrate with the semiconductor electrons (i.e., EFs s = EFs ). In this case, [~bBn(V) - qSBno] increases with increasing applied voltage, surface layer doping, and surface state density. On the other hand, when the surface states equilibrate with the metal (i.e., EFs s = EFm ), [¢Bn(V) -- ~bBno] changes slightly with diode parameters and is almost independent of the applied voltage over a wide voltage range. An approximate expression for the barrier height can be obtained when the ion-implanted surface layer is very thin such that the doping profile can be expressed approximately by Nd(X )
=
Nst6(x)
+
Nb ,
(12)
where Nst is the number of implanted atoms per unit surface area [for gaussian profiles, Nst = Ns(w'n/2 b)]
-~,'~ 0.I0
>-
/ _ ~ , ~ - I/
.,#/
16 10
~17
=
IU
Cm eV
~"~B !0
/
t9 10
Ns ~tm -3
Fig. 3. Dependence of [~Bn{V) - CBnol on the applied voltage, doping profiIe patameters, arid surface state density. (Same constants as in fig. 2 and taking EFs s = EFs. ) and 6(x) is the Dirac delta function. Following the analysis given above and making reasonable assumptions (7), we can get t~Bn(V)
= 7(¢m
-- Xs -
+ (1 - 7 ) [ E g / q
Vsc
-
q6Nst/ei)
- d20 + (EFs s - ~"Vm)/q]
-- ( q / 2 e s ) ( N s t / n ) l / 2
(13)
The barrier heights obtained using the above equation (fig. 4) have been found to be reasonably accurate as long as bW >> 1. This condition is usually satisfied when b is greater than 10 6 cm -1 . A similar analysis for diodes using p-type semiconductor shows that the barrier height can be approxi295
Volume 92A, number 6
PHYSICS LETTERS
22 November 1982
mately written as
o,8~
~s~
- - ' . ~ . "-.
l lo k
CBp(V) = 7(Eg/q + Xs - Cm + Vsc - qSNst/ei) + (1 - 7)[q50 - (EFs s - EFm)/q]
d c~ 7>
0,6
-- (q/Zes)(Nst/n)l/2 .
g " ~ o,4
Oss 1012
0,2 2 ×1011
"~
It is thus shown that the effective barrier height of Schottky MIS diodes can be controlled b y the proper choice of the highly doped surface layer parameters.
""""i~
""
1012
1013
N st 9cm-2 100
/ = >
Ill
II . ~" /
75
E.
%~1II
•
1
ill" /
/ /
iI
ii
l/I
;
'
,x%/ "
,
50
_
-%
, 25
,/
/, / iii I/
,
/ "/
?.
° > 1 " " *"
I
[
../ ii
~"
o
'
o 0,0
0,1
0,2 V ~ V OLTS
0,3
0,¢
Fig. 4. Barrier heights and their voltage dependence for different interfacial layer thicknesses and surface state densities assuming that Nd(x ) = Nst6 (x) + N b. (Same constants as in fig. 2)
296
(14)
References [1] K. Kajiyama, S. Sakata and Y. Mizushima, Proc. IEEE 62 (1974) 1287. [2] A.H.M. Shousha, J. Phys. D: Appl. Phys., to be published. [3] Y. Anand and W.J. Moroney, Proc. IEEE 59 (1971) 1182. [4] J.M. Shannon, Appl. Phys. Lett. 24 (1974) 369. [5] C. Wu, J. Appl. Phys. 51 (1980) 4919. [6] J. Mayer, L. Eriksson and J. Davis, Ion implantation in semiconductors (Academic Press, New York, 1970). [7 ] S.M. Sze, Physics of semiconductors devices (Wiley, New York, 1969).