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Unformatted text preview: Solutions to Chapter 2 Problems
‘Fundamentals of IIIV Devices, by W. Liu 1. (a) For the (M AlwgagﬁAs/ (m QaAs heterojunction:
According toExample 13, NC for A10.35Ga0,65As is 3.72 X 1017 cm'3. n/Nc =
2/3.72 :4 0.54. From Fig. 1—21,'we locate 0.54 from the y—axis. The corresponding x—axis value is — 0.4. Therefore, Ef EC = 4 0.4 X 0.0258 =  0.01 eV. Therefore, QDN = 0.01 eV. 
For GaAs. NV = 4.7 X 1018 cm‘3. p/N‘. = 5 /4.7 =1.06. From Fig. 121, we locate 1.06
from the y—axis. The corresponding x—axis value is 0.4. Therefore, Ev  Ef: 0.4 X 0.0258 '
=/0.01 eV. Therefore, (1),, = — 0.01 eV. From Eq. (2—14), $1,, = 1.668 eV; 0N0: 1.6115 V;
4),,0: 0.0564 V; XNO = 0.104 um; Xpo = 4.15 x 10’3 um. Band Energy (eV) 2 I _
0.05 0 0.05 0.1 0.15 0.2
Position (pm) We shall use the following parameters for InGaAs: Eg = 0.75 eV, M, = 6 x 1013 cm'3; 6, , = 14; and for InP: E8 = 1.35 eV, NC = 5.8 x 1017 cm‘3; E, = 12.6. The valence band discontinuity is 0.37 eV. For the InP, n/Nc = 2/5.8 = 0.345. From Fig. 121, we locate
0.345 from the y—axis. The corresponding xaxis value is "0.9. Therefore, Ef— EC =  0.9 x 0.0258 =  0.023 eV. Therefore, GDN = 0.023 eV. For the InGaAs, p/NV = 5 / 6 = 0.83.
From Fig. 121, we locate 0.83from the yaxis. The corresponding xaxis value is 0.2. Therefore, Ev  Ef: 0.2 x 0.0258 = 0.005 eV. Therefore, 45p = — 0.005 eV. From Eq. (2
14), 0w =. 0.962 V; ¢No= 0.9285 V; ¢p0= 0.0335 V; XNO = 0.0805 mm; X,» = 3.22 x 10'3
pm.  = ,. ,, Band Energy (6V) POsition (pm) 0)) .6 WM ‘ i 
‘ For the AlGaAs, n c = 1 /3.72 = 2.7. From Fig. 121, we locate 2.7 from the y axis. The corresponding xaxis value is 2. Therefore, Ef~ EC = 2 x 0.0258 = 0.0516 eV. Therefore, CDN = ~ 0.0516 eV. For GaAs, p/Nv = 100/ 4.7 = 21.3. From Fig. 121, we
IQCate 21.3 from the y—axis. The corresponding xaxis value is 9.2. Therefore, Ev  Ef: 9.2 X 0.0258 x 0.237 eV. Therefore, q, = — 0.237 eV. From Eq. (214), $1": = 1.9566 V;
¢No= 1.9388 V; 01,0 = 0.01788 V; XNO = 0.0509 gum; Xpo = 5.09 x 10'4 um. Band Energy (eV) —0.02 ’ _ 0 0.02 . 0.04 0.06 0.08
Position (urn) Wmﬁagiqmmgm  i
For the 112?, n/Nc = 1015. = 1.72. From Fig. 121, we locate 1.72 from the yaxis. The cerresponding xaxis value is 1.2., Therefore, Ef— EC = 1.2 x 0.0258 = 0.031 eV. Therefore, 451v =  0.031 eV. For the InGaAs, p/Nv =100 / 6 = 16.7. From Fig. 121, we
locate 16.7 from the y—axis. The corresponding xaxis value is 7.8. Therefore, Ev  Ef: 7.8 x 0.0258 = 0.20 eV. Therefore, ch =  0.20 eV. From Eq. (2.14), (151,; = 1.211 V; 210 ¢N0= 1.20 V; (FPO = 0.0108 V; XNO = 0.0409 Mm; XNO = 4.09 X 10'4 um. Band Energy (CV) l 2  0.020.01 0 0.01 0.02 0.03 0.04 0.05 0.06
Position (pm) 2. (a) Em the (M AleaMAs/ 12! Gags hetergjunction: Although pi = n;2 holds only for non—degenerate semiconductors, we will continue to use
it for all samples. According to Example l2, n, for Alo,35Gao.65As is 3.6 X 102 cm'3. . Also, n, for GaAs is 1.79 x 106 cmi3. At x = XNO, n = 2 x 1017 cm3 and p = niZ/n =
(3.6 x 102)2/(2 x 1017) = 6.5 X 10'13 cm'3. Atx = 0*, we use Eq. (217) to find n,
knowing that «pm: 1.6115 V. So, It at x = 0+ is equal to 2 x 1017 x7exp( — '
1.6115/00258) _= 1.5 X 10'10 cm’3. Further, we use Eq. (2—18) to find p at x = 0+, which ' is equal to 6.5 X 10'13 X exp(1.6115/0.0258) = 8.7 X 1017 cm'3. Now, let us find the pside. At x =  Xpo, p = 5 X 1018 cm‘3 and n = nizlp = (1.79 x
105)2/(5 x 1013) = 6.2 X 10'7 cm‘3, At x = 0‘, we (use Eq. (219) to find p, knowing that
¢po= 0.0564 V. So, p at x = 0‘ is equal to 5 X 1013 X exp(  0.0564/0.0258) = 5.6 X 1017
cm'3. Finally, ‘we are ready to ﬁnd n at x = 0' . We use n = ni2/p = (1.79'x 106)2/(5.6 x ‘ 1017) = 5.7 x 10‘6 cm'3.
We tabulate the results: ‘ ,
‘ nlx=0+ Plx=0+ nlxrXNO plx=XNoni (AlGaAS) px=0— nx=0 Px=Xpo nlx=‘Xpo ni (GaAS)
(cm'3) 1.5E10 8.7E17 2El7 6.5E13 3.6E2 , 5.6E17 5.7E6 5E18 6.2E7 1.79E6
loglo 9.8 17.94 17.3} 12.1 2.56 17.75 5.24 18.7 —6.2 6.25 5n  ' [0 (carrier concentration)
hole 420 gm electron “1 10
,l
 — —  —   —
L . _ _ _ _ _ _ _ _ . _ .. .
I
0 I
,l
#
electron ‘  10:.
______—+—*———————————_———. ‘ X
Xpo o I . No
F91 Lhe 1M lnﬂ (m IanaMAs heterojunction: ‘ .n, for InP is 1.05 X 107 cm'3. Also, 11, for InGaAs is 6.31 X 1011 em'3. At x = XNO,
n = 2 x 1.017 cm'3 andp = nizln = 5.5 X 10“4 cm'3. At x = 0*, we use Eq. (217) to find n,
knowing that we: 0.9285 V. So, n at x = 0+ is equal to 2 x 1017 x exp(  '
0.9285/0.0258) = 4.7 x 101 cm’3. Further, we use Eq. (218) to find p at x = 0+, which is .
7 equal to 5.5 X 10"4 X exp(0.9285/0.0258) = 2.3 X 1012 cm‘3. Now, let us find the pside. At x =  Xpo, p = 5 x 1018 cm3 and n = ni2/p = (6.31 x
10“)2/(5 x 1013) ‘ 8.0x 104 cm'3. Atx = 0', we use Eq. (219) to ﬁnd p, knowing that
' ¢po= ¢b, — m, = 0.0335 V. So, p at x = o— is equal to 5 x 1018 x exp( — 0.0335/0.0258) =,
1.4 x 1018 cm‘3. Finally, we are ready to find 72 atx = 0‘ . We use n = nizlp = (6.31 x 1011)2/(1.4:x 1018)='2.8 x 105 cm3.
We tabulate the results: ‘ '
. nlx=0+ plx=0+ nlx=XNo Plx=XNo ni (InP) Plx=o nlx=o px=Xpo nlxéXponi (InGaA:
(cm‘3) 4.7E1 2.3E12 2E17 5.5E4 1.1E7 1.4E18 2.8E5 5E18 8E4 6.3E11
loglo 1.67. 12.4 17.3 3.25 7.0 18.1 5.4 18.7 4.9 11.8 n '. lo (carrier concentration)
hole , 4 gm 20 electron n— i ‘10I electron Xp0 0
(b) For the (121 1319356 _' As/ 12) GaAs hetegojunctjon:
J96: . . ’ n, for Alo.3sGao,65As is 3.6 X 102 cm‘3. Also, ni for GaAs is 1.79 X 106 cm‘3. At x =
XNO, =1x 1018 cm'3 and p = nizln = (3.6 x102)2/(1X 1018) = 1.3 x 1013 cm3. At x =
0*, we use Eq. (217) to ﬁnd n, knowing that (two: 1.939 V. So, n at x = 0+ is equal to 1
X 1018 X exp(  1.939/0.0258) = 2.3 X 10'15 cm‘3. Further, we use'Eq. (2—18) to findp at
x = 0* is nizln = (3.6 X 102)2/(2.3 X 1015) = 5.6 X 1019 cm‘3. ' Now, let us find the p—side. At x =  Xpo, p = 1 X 1020 cm'3 and n = ni2/p = (1.79 X
106)2/(1 X 1020) = 3.2 x 10'8 cm’3. At x = 0', we use Eq. '(2—19) to find p, knowing that
¢po= 0.01778 V. So, p at x = 0‘ is equal to 1 X 1020 X exp( — 0.01778/0.0258) = 5.02 X
1019 cm'3. Finally, we are ready to find n atx = 0’ . We use n = niZ/p = (1.79 X 106)2/(5.02 X 1019) = 6.38 X 10‘8 cm'3. We tabulate the results: nlx=0+ Plx=0+ nlx=XNo Px=XNoni (AlGaAS) Plx=0— nx=0 .Px=Xpo nx=Xpo ni (GaAs) »
' (cm'3) 2.3E—15 5.6E19 1E18 1.3E13 3.6E2 5.02E19 6.38E8 1E20 3.2E8 1.79E6
loglo l4.6 19.75 18 12.9 2.56 19.7 —7.2 20 7.49 6.25 l(3  electron 3. (a) For the (m AlmgagééAs/ 12) GaAs heterojgnctign: Let's calculate for Va = 0.5 V first. From the solutions of Problem 1a, we see that (DN = 0.01 eV and dip = — 0.01 eV. Valp is calculated to be 0.0177 V. VaN = Va  Vap = 0.4823
V, The builtin voltages at the tWO sides‘of the junction, which are critical to the construction of the band diagram, are the following. (1),, = «ppo  Vap = 0.0413 V; ¢N = ¢N0
'  VaN = 1.127 V. The depletion thicknesses, according to Eqs. (241) and (242) are, Xpr = 3.47 x to? cm‘and XN = 8.68 x 106 cm. Band Energy (eV) l .5 ,
0.05 r 0 0.05 0.1 015 Position (pm) Let's calculate for Va ==  5 V. Va? is calculated to be  0.177 V. VaN = Va  Vap = 4.823
V. The builbin voltages at the two sides of the junction, which are critical to the construction of the band diagram, are the following. ¢p = ope  Vap = 0.236 V; (M = ¢No 
VaN = 6.432 V. The depletion thicknesses, according to Eqs. (2—41) and (242) are, Xp = 8.30 x 107 cm and XN = 2.74 x 105 cm. I IS N
.1
J . ,1
o r—  ~15 2 ~ *1
5? r “g
6 L . V  Ev L
' ,8; E ___1__ _L I I _.___l
O.l 0 0.1 0.2 0.3 0.4. ‘ * , Position (pm)
The Fermi levels are nearly on top of the valence band in GaAs and the conduction band in
AlGaAs. ‘ For the gm Igﬂ (Q) InQﬂQgMAs hetergjunction:
' Let's calculate for Va = 0.5 V first. From the solutions of Problem 121, we see that CDN = 0.023 eV and (15;, =  0.005 eV. Vap is calculated to be 0.01737 V. VdN = Va  Vap =
0.4826 V. The builtin voltages at the two sides of the junction, which are critical to the construction of the band diagram, are the following. (1),, = ¢p0 — Vap = 0.01605 V; (M =
(two  VaN = 0.4459 V. The depletion thicknesses, according to Eqs. (241) and (242)
are, Xp = 2.23 x ‘10’7 cm and XN = 5.57 x 10'6 cm. ' A ' Band Energy (6‘!) I 1.5
0.05 0 0.05 , 0.1 0.15 Position (um)
Let's calculate for Va = — 5 V. Vap is calculated to be  0.1737 V. VaN = Va  Vap = 4.826
V. The builtin voltages at the two sides of the junction, which are critical to the construction of the band diagram, are the following. «7),, = (ppo  Vap = 0.207 V; (M = ¢N0  ‘
VaN = 5.755 V. The depletion thicknesses, according to Eqs. (241) and (242) are, Xp = 9M: 8.01 x 10‘7 cm and XN = 2.00 X 10'5 cm. 0
LI! ._ [m1
>
U)
iii: m
L_~2L: Band lincrgy (CV) ~0.02 0 0.02 0.04 0.06 0.08 0.1
Position (um) (b) For the (m AlmQQQﬁAS/ gm GaAs heteroju notion: Let's calculate for Va = 0.5 V first. From the solutions of Problem la, we see that (DN = 0.057 eV and (Pp =  0.237 eV. Vap is calculated to be 0.00454 V. VaN = Va  Vap =
0.4955 V. The built—in voltages at the two sides of the junction, which are critical to the construction of the band diagram, are the following. 4),, = 45150 _ Vap = 0.01324 V; ¢N =
_ ¢N0  VaN = 1;4433 V. The depletion thicknesses, according to Eqs. (241) and (242) are, Xp = 4.39 x 10'7 cm and XN = 4.39 x 10?6 cm. Band Energy (eV) 2 . . V .
0.02 0 002 ' 0.04 0.06 0.08
Position (um) Let's calculate for Va =  5 V. 'Vap is calculated to be ~ 0.0454‘V. VaN = Va  Vap = 4.955
V.‘ The builtin voltages at'the two sides of the junCtion, which are critical to the cOnstruction of the band diagram, are the following. «pp = $1” — Vap = 0.0632 V; ¢N = ¢N0
 V'aN = 6.893 V. The depletion thicknesses, according to Eqs. (241) and (242) are, Xp = 9.60 X 10‘8 cm and XNE 9.60 x 10'6 cm. at] Band Energy (eV) Position (pm) For the (M 132/ 12) IanMAs heterojunction:
' Let's calculate for Va = 0.5 V first. From the solution/sof Problem 1a, we see that @N = —0.031 eV and Q, =  0.20 eV. Vap is calculated to be 0.00446 V. VaN = Va  Val, =
0.4955V. The built‘in voltages at the two sides of the junction, which are critical to the construction of the band diagram, are the following. ¢,, = 0P0  Vap = 0.00634 V; (M =
$100  VaN =~0.7047 V. The depletion thicknesses, according to Eqs. (241) and (2—42)
are, Xp = 3.13 x 10‘7 cm and XN = 3.13 x 10‘6 cm. Band Energy (eV) .5 v "
.002 001 0 0.01 0.02 0.03 0.04 0.05 Position (um)
Let's calculate for Va 2  5 V. Val, is calculated to be  0.0446 V. VaN = Va  Vap = 4.955
' , V. The built—in voltages at the two sides of the junction, which are critical to the construction of the band diagram, are the following. «0,, = ¢p0 — Vap = 0.0554 V; ¢N = 0N0
 VaN= 6.156 V. The depletion thicknesses, according to Eqs. (241) and (242) are, X}, = 9.26 x 10'8 cm and XN = 9.26 X 10'6 cm. 1013 V 9 8 >. 2 E.“ » U 1 :5 ‘3 ;» 1 _. g ‘4 ; ': 5Q Z
5 i
._6 7 _L_ ‘ __:I
~0.05 0 0.05  0.1 0.15 Position (um) 4. Because the'doping profile of this problem is unmodified from that of Example 26,‘
the Poisson potential is still that given in Eq. (2—86). However, we the new aluminum
composition grading given in the problem, the energy gap potential will be modified from that of Eq. (299).' In particular, let us consider the region 0 S x S Xgmding. With ﬁmax
translated tolAEanax, we obtain: ’ ' ’Q‘Pg : Esp + AEcmax ‘ c..max '—l"‘( ngrading ' x2 ) 2
X grading
2e 2
V; ESP + AECJDKX ' Xgrading ' x2 )
where
26 s
Xgrading = AEC max
qud Now, in this same region (0 S x .<_ Xgmding), (ppoisson was given in Eq. (286). We sum
up qwrois‘son and mag. and get: ‘
. 2 I 2N p}
Ec(x )= q Nd( XNo  x )2  q¢~o+Egp+ AEC,max .__q d ( xgzradingx2) 265 S I
. 2N
= X1310 ‘ _2XN0x + x2 ' ngrading + x2 )' Q¢No + Egp + AEc,max
es . ' It is clear that the x2 dependence does not vanish. (NOTE: in Eq. (2100), which is the
linear conduction band proﬁle from using the correct parabolic grading, the term inside the parantheses would be,
(XNO " x? ' (Xgrading ' x)2 V: X160 ' ngrading ‘ 2(Xgrading 4 XNOV
We see that the x2 term disappears. 5. From Example 25, NC for Alo,3sGao,65As is 3.72 x 1017 cm'3 and Nv for GaAs is
v4.7«x 1018 cm‘3. From Fig. 121, we ﬁnd that for Nd/Nc = 0.267, Ef— EC = 1.2'kT =  0.031 eV and Na/Nv = 21.3, E,  Ef: 8.9kT = 0.23 eV. Hence, (DN = 0.031 eV and (Pp
=  0.23 eV. AEC max = 1247an = 0.244 eV. ¢N0, taken to be (phi, is calculated from
Eq. (214) to be 1.424 + 0.244  0.031 + 0.23 = 1.867 V. In order for the parabolic 1‘1 grading scheme to work, the grading distance needs to be that given in Eq. (296):  F—‘—"25 213.188.85 x1014
Xgrading 2, AEC max = ~ ' 1.6 x 10~19~_1 x 1017
The aluminum gnole fraction should be varied from 0 at the junction interface to 0.35 at 0.244 = 6.0 x 106 cm. Xgmdmg = 600 A in a manner given by Eq. (297):
. = .35— .5 ———X—u——2.
ﬁx) 0 03(1600A) I This way, the conduction band profile is given by Eq. (2—100). We plot the result for Va =
l and 1.2 V: _ I  ' 1.4' ‘ ‘13 (eV) 1.2 E 0.9 0.8 ' 9
0 200 400 600 800 1000 1200 Position away from the Junction, x (A) 6. This is an abrupt heterojunction because the ideality differs significantly from unity. A i
graded heterojunction would have an ideality factor very clOse to 1. According to Eq. (2
109), this ideality factor is simply I + K. K is thus equal to 0.2. If we neglect the
differences in the dielectric constants of the two heteromaterials, we establish from Eq. (2 110) that: N
. e a a
Note that, as discussed in the textbook, this ratio is obtained with the assumption that the
thermionic emission dominates the current flow. When tunneling is an significant part of .
the overall current conduction, then the ideality factor can. approach values like 1.2 although the Na greatly exceeds'Nd by, for example, 2 order of magnitude. 7. (21) During the reverse bias, the total current (consisting of the ideal pn junction diode
current and the generation current in the spacecharge region) is the sum of Eqs. (2128) and (2157). Therefore, we write; ,
DP "'2 Dn "‘2 [ (an ) ] qADni
exp H, 1 27 (XN+Xp) .. Dp ":2 Bu "1‘2 ) qADnt‘
I qAD(Lde+LnNa 21. (XN+Xpl _ The term due to spacecharge generation is the same whether the more heavily doped side is
the ntype or ptype. Therefore, we shall make the decision based entirely from the first
term; Rev'vrite the first term, which is the reverse saturation current (10), we obtain: 'uzo ' ' ' a 15), (15,21 = 0.16 eV. Now for the right hand side whose doping is 1 x 1017 cm'3. This deping level is heavy enough that we need to use the JoyceDixon approximation. n/NC =
1/4.7 = 0.213. From Fig. 121, we locate 0.213 from the yaxis. The corresponding x axis value is  1.5. Therefore, Ef— EC =  1.5 x 0.0258 = ~ 0.039 eV. Therefore, 49,12 =
0.039 eV. '
$1,, = (15,,1 CD”; = 0.16 — 0.039 = 0.121 eV. 11. (a) NC for Si is 2.8 X 1019 cm'3. For the n—GaAs layer doped at 5 X 1016 cm'3, we find Ef EC =_ 0.0258 x ln(5/2800) =  0.16 eV. Therefore, from Eq. (215), 4),,0 = 0.16
/eV. The band diagram is similar to Fig. 2—21: / Ec 0.16 eV (b) When the electron hole pairs are generated within the depletion region, the holes are
swept across the junction and move toward the emitter side. The electrons are repelled by
the potential barrier. Therefore, the current moves from the semiconductor toward the metal. 0.45 eV W/ O .
(c) The thermionic Current of the, junction .is given by} Eq. (2102a),
. . * V I , V ‘
’rhermionic=AvA T2“? lexP(%rl'1l=’°lexP(i—il‘ll . where AD is the area of the metalsemiconductor diode area. The metal is taken to be the
positive polarityside. Therefore, the current Ib h, Of part (b) is considered a negative '
current. Hence, the total device current is equ to:
I=Io[exp(%l:) 1] Ilight '
When I is made zero by the opencircuit condition, there is a positive voltage formed at the 13.2% junction so that the thermionic current cancels out lug/1,. This voltage is solved by forcing I
to be zero in the above equation. V
high, = Io [exp ( 3:7 ) — 1]
Therefore,
, I 
V=1‘—Iln[ "‘9’" +1
 ‘1 10 12. The equation shown in problem 12 is more accurate than Eq. (2—133). However, the
difference will be very small. 13. When there is no external bias, the total current should be zero. The reason that 'Eq.
(2158) suggests a nonzero current as bias goes to zero is the following. The equation
assumes that the generation‘rate U is given by Eq. (2—156). This generation rate, in turn,
assumes that the pn‘ junction is reverse biased such that the pn product, equal to
niZexp(an/kT), is approximately zero. In reality when the bias goes to zero, U is zero,
and the generation current is zero. . 9 ...
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 ev, Band Energy

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