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Unformatted text preview: CHAPTER 14 14.1 (a) MS, Schottky, and Hot Canier are just alternative names for a rectifying metal—
semiconductor contact. Hot Carrier diodes are typically small area devices. (b) If (DM = x in an ideal MS contact, the contact is borderline rectifying/ohmic independent
of the semiconductor doping. (c)...Vbi is computed differently. ...In pn junction analyses it is common practice to take V: 0 on the p—side of the junction. In MS work the semiconductor bulk is typically employed as the zero~voltage
reference point. ((1) Thermionic emission current —— majority carrier injection over the surface potential
energy barrier. (e) Since I Mo—>S(VA) = IM._)3(0) = —Is.—)M(O), the M0——>S component is obtained by
evaluating the SO—aM component at zero bias. (0 The diffusion capacitance and conductance arise from the ﬂuctuation of the minority
carriers stored in a quasineutral region adjacent to the depletion region. The minority
carrier storage is large in a forward—biased pn junction diode and leads to a signiﬁcant diffusion admittance. In an MS diode there is minimal minority carrier storage for
operational forward biases. (g) There is a minimal number of stored minority carriers to be removed in going from the
forward—bias “on” condition to the reverse—bias “off” condition. (h) It is a special circuit or fabricatedin arrangement where an MS (Schottky) diode is
connected between the base and collector of a BJT. The arrangement leads to a signiﬁcant
reduction in the turnoff time when the BJT is used as a switch. (i) Ohmic contacts are usually produced in practice by heavily doping the surface region of
the semiconductor immediately beneath the contact. The device structure is also routinely
annealed (heated in an inert atmosphere) to minimize the contact resistance. (j) “Spiking” is the nonuniform penetration of Al into Si beneath an Al—Si contact. (See
Fig. 14.11a.) , 14—1 14.2 (NOTE: In the ﬁrst printing, all of the semiconductors were erroneously identiﬁed as
ntype. Combinations B, D, and F should have been labeled as NA doped.) For a given combination, it is ﬁrst necessary to determine the nature of the contact. This
requires that we compute (1)3 using (133 = z + (EC—EF)FB E Z + 56/2 — (EF~Ei)FB I kT ln(ND/ni) ntype (EF—E‘)FB
l i —kT ln(NA/ni) ptype EF—Ei EG/z <I>s
, (eV) (6V) (EVW. n————
.25x106 0.575
‘ —0.635 0.71 Noting the semiconductor type and whether (DM > CD3 or (PM < (D3, the ideal nature of the contact is deduced by referring to Table 14.1.
V Part (a) diagram Part (b) argue
Similar to , similar
i I i H ideal A Ft 142(1))  Section 14.1 . “uni—— Exeercis 14.1 _ Section14.1 7’ Exercise 14.1 _ 7. _ Rectif in . ' . 14.203) 7 (a), Erxe. 141. 142 14.3 (a) (1);; = <I>M — z = 5.10— 4.03 = 1.07 eV
0)) (Ea — EF)FB 5 EG/2 — (EF ~ EDFB = EG/Z — kT1n(ND/ni)
 = 0.56 _ (0.0259)1n(1015/1010)
= 0.26 eV Vbi = $[CDM—(EcuEF)FB] = 1.07—0.26 = 0.81 V (CYVA = 0 under equilibrium conditions and 1/2 14 ”2
W = [mvbi] : W . = 1,03 x 104 cm
qND (1.6x1019)(1015)
—19 15 «4
(d)181mx = label = 25/19; = “6X10 X10 )(1‘03X10 ) = 1.58 x 104 V/cm
K580 (11.8)(8.85X10‘14) 14.4
The computations were performed employing Eq. (14.12) with VA = 0, Eq. (14.3), and (EC — EF)FB E EG/Z —— kT ln(ND/ni)
The resultant plot and associated MATLAB m—ﬁle are displayed below. MATLAB program script“.
%Equilibrium Depletion Width (Problem 14.4) %Initialization
clear; Close %Constants and Parameters
q=1.6e—19; e0=8¢85e—14; kT=0.0259; 14—3 EG=1.12; ni=l.0e10; KS=11.8; BH=[O.5,0.6,0.7]; %Barrier Height
ND=logspace(l4,l7); %Depletion Width Calculation ECF=EG/2kT*log(ND./ni); W0=[}: for i=lz3,
Vbi=BH (i) ECF;
W=sqrt((2*KS*eO.*Vbi)./(q.*ND));
WO=[WO;W]; end %Plot result loglog(ND,W0);
axis([1.0el4,1.0e17,5e—6,5e—4]); grid
x1abel('ND (cm3)'); ylabe1('WO (cm)')
text(2e15,2.3e—5,'BH=O.5eV')
text(2e15,6e—5,'BH=0.7eV') Womm) Nowme) 14—4 14g§ (a)/(b)/(c) A sample MATLAB program that generates MS diode energy band diagrams
(equilibrium, 300 K) is included on the instructor’s disk as m—ﬁle P_14_05 In The
program can generate both 11 and p type Si diagrams plus n and p type GaAs diagrams. Sample plots are displayed below SILICON MS DIODE
w "1 SILICON MS DIODE 1.5 1.5
1 1
0.5 05
5‘ 9
3 3,
.1? S?
E o g o
E. .77
u} u'.
—0.5 415
.1 «1
1 .5 4.5
.J.._._. 4.4.... J __.l 4 .4.._J l—. “I...
4 G.5 0 0.5 1 «i O 1 2
Distance from MS interface (cm) Distance 1mm MS interface (on) x 105
(Salts MS DIODE GaAs MS DIODE
:— v ”1' r 1 u
2 2
1‘5 15
l 1
g 0.5 g 0.5
3 3;
E o E o
if: if:
m 0.5 u} as
.1 4
1.5 1 .5
.2 2
l— l awn! ‘ l I g A L 1. __ .L. I 4L,
6 4 2 0 2 4 6 B 10 1 05 0 0‘5 1 L5
Distance from MS interface (an) x 10‘ Distance from MS interface (cm) x 10‘ 14—5 14.§
Substituting Eq. (14.17) into Eq. (14.16) gives ll *2 %in *
[S.)M .. qA(i7_t_I:T’n_n_) e(EF—Ec)/kT] DX e_(mn/2kT)v% dvx h3 00 00 2 :1:
qA(4_“kT3m__§ )e(EF~Ec)/kr 0x eon/2mm? dvx
h 1)min where the second form of the above equation is obtained by interchanging the limits on the
integral and changing variables from DX to ~vx. Next evaluating the integral yields 1 ”X e—(mKIZvag dvx = _. {k_1:V»(mn*/2kmg °° : (1%) e4mg/2mvgm
m“ vmin mn 1)min  Thus, noting from Eq. (14.14) that 0,an = (2q/mn*)(Vbi~VA), we obtain 2
[5.6M = qA(____~47‘k% m5) e(EFEc)/kT eq(Vbi—VA)/kT
h3
But
qui/kT = (DB/kT + (EF—Ec)/kT
and 2 2 :5: * 2
4A m4“ T m“ = Aiﬂ Mum/C T2 = 1431’sz leading to the conclusion ISO—>M = Aﬂ*TZe—<I>B/kTquA/kf 14~6 14,7 (a) With positive current flow as deﬁned in Fig. 14.3(a), the shortcircuit photocurrent is
negative. (Note that EFM = E135 in the energy band diagram because the device is short—
circuited. However, both F N and F p deviate from Eps near the M—S interface.) Eps (Short—circuit diagram) (b) A forward bias must be developed under open—circuit conditions so that the negative:
going photocurrent is precisely balanced by a positive— going thermionic emission current. (Opencircuit diagram) (c) Paralleling the approach presented in Subsection 9.2.1, the photOcurrent (1;) will be equal to —q times the electronhole pairs photogenerated per second in the volume
A(W+Lp), or
[L = ~ qA(W + LP)GL (d) The [~V sketch should be similar to one of the G0 ¢ 0 curves in Fig. 9.3; ie, a
constant value is subtracted from the dark I—V characteristic to obtain the light~on
characteristic. Consistent with the part (a) and (b) answers, I < 0 if the device is short
circuited (VA 2 0) and V > 0 if the device is open circuited (I = 0). I 147 143 When the series resistance cannot be ignored, Eq. (14.24) assumes the modified form I = 13(e4VJ/kT 1)
where
V3 = VA;~IRS 0r VA_= Vﬁ+IRS The [—V relationships here are totally analogous to the highcurrent pn junction
relationships presented in Subsection 6.2.4. In performing computations, it is convenient
to ﬁrst choose a value for V5, compute I, and then compute VA. The requested ILV
characteristics illustrating the effect of the series resistance are reproduced below. 0 0.1 0.2 03 0.4 0.5
VA (volts) MATLAB program script“.
%Effect of RS on MS diode I—V Characteristics %Initialization
clear; close %Constants and Parameters
kT=0.0259; 14«8 Is=1.0e—8;
Rs=[0,0.1,1.0,10];
VJ=linspace(0,0.6); %Calculate I versus VA
I=Is . * (exp (VJ/kT) ~1); ' VA=; for i=1z4,
VA=[VA;VJ+I.*RS(1)};
end %Plot result semilogy(VA,I,'w‘); axis([0,0.6,1.0e9,l.0e—l]); grid xlabel('VA (volts)'); ylabel(‘I (amps)')
text(0.34,5.0e—2,'RS=O'); text(0.4l,2.0e2,'RS=1 ohm‘)
text(0.4l,4.0e—3,'Rs=10 ohms') 14.9
For a p+n junction."
2
I4>IVD and employing Eqs. (14.24/ 14.25), [TE = [8(quA/kT_ 1) = Aﬂ*Tze<DB/kT(quA/kT ~1) Thus noting
91>. 2 4/122 z «/____<kT/Q>#v
LP rp Tp
2
(IA D_p i q (1:77qu i
IDIFF : 10 2 LP ND : ’5? ND
[TE [8 Aﬂ*T26 —<I>B/kT ﬂfkTZe JDB/kT (1.6x10'19{W]1/2(&29
(106) 1016 (140)(300)2e{OM/(0.0259) = 5:05 x 10‘7 149 14,1!)
(a)/(b) The computational results for parts (a) and (b) and the associated MAILAB m—ﬁle are included after the part (c) comments. The primary relationships employed in the
computations were: Forpart (a)... 48s! “2
ND}; 2 (NDB in eV)
41th£0 1/2
lgsl = ”(IND W = [quD (Vbir'VA)]
K380 Vbi = Ell[(1)3 — (Ec~EF)FBl (Ec  EF)FB E EG/2  (EF — EDFB = EG/Z — kT ln(ND/ni) For part (b)... ..___..__’s(VA) = _____e""B(VA)/” = etemommvni/kr .._ etA<I>B<VA>—A<I>B<on/kr
Is(VA=0) ecbmm/kr (c) The A<I>B(eV) versus VA plot shows that the change in ND}; tends to be quite small,
only 0.047 eV corresponding to a AVA = SUV in the given calculation. However, because
IS depends exponentially on the barrier height, the effect of a small AQB change on IS is
very signiﬁcant.— IS changes by more than a factor of 6 over the examined voltage range! § Deana in barrier height (W)
a
b
a is normalized to In at VAo cage 1
 .45 4o .35 ‘30 <25 <20 ~15 <10 «5 0
VA (volts) 1410 MATLAB program script...
%Schottky Barrier Lowering Computation %Initialization
clear; close %Constants and Parameters q=1.6e—19: e0=8.85e—14; kT=0.0259; KSzll.8; EG=1.12: ni=l.0e10; ND=1.0e16; BH=O.72; %BH=barrier height in eV %Computation of ABH
VA;1inspace(~50,0); ECF=EG/2—kT*log (ND/Hi) ; Vbi=BH—ECF;
ES=sqrt((2*q*ND)/(Ks*e0).*(Vbi—VA));
ABH=sqrt ( (q. *ES) ./ (4*pi*KS*eO) );
plot(VA,ABH); grid xlabel(‘VA (volts)'); ylabel(‘Decrease in barrier height (eV)‘);
pause %Computation of Is/Is(0)
‘“Isn=exp((ABH—ABH(100))./kT); plot(VA,Isn); grid
x1abel('VA (volts)‘); ylabel(‘Is normalized.to Is at VAFO'); 14—11 14.11 If not explicitly given in the problem statement, the device area (A = 1.5 X 10"3 cm2) may
be obtained from Exercise 14.4. Effectin g the ﬁt employing the MATLAB program listed
below, one ﬁnds: Fit Results Exercise 14.4
M) = 9.62X1015/cm3 ND 5 9.7x1015/cm3
Vbi = 0.613 V Vbi a 0.6V
<I>B = 0.816ev (DB 5 0.8eV The fit results obviously compare very favorably with the approximate results obtained in
Exercise 14.4. MATLAB program script... % Determination of Vbi, ND, and B8 of MS diode
% employing Pl4.ll C—V data %Initialization
clear; close
format compact; format short e %Input data...Y=l/CJ2 VA= [1.09,2.08,3.07,4.06,5.05,6.04,7.03,8.02,9.01,10};
Y=l.0e21*[0.953,1.494,2.035,2.579,3°125,3.673,4.217,
4.763, 5., 320, S. 890]: %Fit p=polyfit(VA,Y,1) ND=2./ (1.6e—19*1l.8*8.85e14* (1.5e~«3) A2* (—p (1) ))
Vbi=—p(2)/p(l) %Barrier Height Computation
EG=1.12; ni=l.0e10; kT=0.0259;
ECF=EG/2nkT*log(ND/ni);
BH=Vbi+ECF %1/CJ2 vs. VA plot (not required) plot (VA,Y, '+i) axis([—11,2,0,1.1*max(Y)l); grid xlabel(‘VA (volts)‘); ylabel(‘1/CJ“2 (1/F“2)‘) 14 12 14.12 In general the development of relationships for the electrostatic variables in a linearly
graded MS diode closely parallels the uniformly doped analysis in Subsection 14.2.1. The
results obtained are analogous to the linearly graded pn junction relationships established in
Subsection 5.2.5. (a) With ND(x) = ax for x Z O, invoking the depletion approXimation yields —d—§— = J2— :—: qa x OSxSW
dx K380 K380
and
0 W
as = 4“ x'dx'
K
ax) SE“ x
or Turning to the electrostatic potential, we can write 0’"  8(X) — q“ (W2_x2) dx 2Ks£0
and
0 W
1 (JV = q“ I (W2_x'2) dx’
V(x) ZKSEO x
or V(x) = ~ 411“— (2W3 — 3W2x + x3) 6K380 Finally, V = —(ng  VA) at x = 0, and therefore 1413 (b) Paralleling the development for the linearly graded pn junction in Subsection 592.5, the
Eq. (14.3) expression for Vbi must be modified to read Vbi = ﬂdh;  (Ec— EF)lx=Wo] where W0 is the depletion width when VA 2 0. Since approximate charge neutrality applies
for x > W0, it follows that ,AFWO : nie[(EFEi)lx=Wol/kT g ND(x=W0) = 0%
OI“ aW )
ni (EC—E12)! We 5 EG/2 — kT1n( Thus, to determine Vbi, one must simultaneously solve the following two equations
employing numerical techniques. U3
W0 = [—S—M 8“ Vin]
qa
Vbi = Lbs ~ E0/2 + kT1n(aW0)]
q ”i
(C) C} = K8594 = KSSOA 1/3
K
{3 (1380 (Vm—Vo] 1414 ...
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 Spring '07
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