《模拟集成电路的分析与设计》，Paul R. GRAY
φ 0 = V T ln N A N D n i 2 = k T q ln N A N D n i 2 φ 0 = V T ln n i 2 N A N D = q k T ln n i 2 N A N D
N A N A a t o m s / c m 3 a t o m s / c m 3
N D N D a t o m s / c m 3 a t o m s / c m 3
n i n i 300 K 3 0 0 K
如果加上外加反向偏置电压，则有，
φ 0 + V R (0) φ 0 + V R ( 0 )
结两边总电荷必须电量相等，极性相反，则有
W 1 N A = W 2 N D (1) W 1 N A = W 2 N D ( 1 )
一维泊松方程：
d 2 V d x 2 = − ρ ε = q N A ε (2) d x 2 d 2 V = − ε ρ = ε q N A ( 2 )
ρ ρ q q 1.6 × 1 0 − 19 C 1 . 6 × 1 0 − 1 9 C ε 为硅的介电常数1.04 × 1 0 − 12 F / cm 1 . 0 4 × 1 0 − 1 2 F / cm K S K S ε 0 ε 0
ε = K S ε 0 ε = K S ε 0
对(2)式积分，得：
d V d x = q N A ε x + C 1 d x d V = ε q N A x + C 1
C 1 C 1
E = − d V d x = − ( q N A ε x + C 1 ) E = − d x d V = − ( ε q N A x + C 1 )
当x = − W 1 , E = 0 x = − W 1 , E = 0 C 1 C 1
0 = q N A ε W 1 − C 1 , C 1 = q N A ε W 1 , E = − d V d x = − q N A ε ( x + W 1 ) (3) 0 = ε q N A W 1 − C 1 , C 1 = ε q N A W 1 , E = − d x d V = − ε q N A ( x + W 1 ) ( 3 )
继续积分(3)式，得到
V = q N A ε ( x 2 2 + W 1 x ) + C 2 V = ε q N A ( 2 x 2 + W 1 x ) + C 2
如果假设电势（电压）在x = − W 1 , V = 0 x = − W 1 , V = 0 C 2 C 2
V = q N A ε ( x 2 2 + W 1 x + W 1 2 2 ) (4) V = ε q N A ( 2 x 2 + W 1 x + 2 W 1 2 ) ( 4 )
从x = − W 1 x = − W 1 V = 0 V = 0 x = 0 x = 0 V 1 V 1
V 1 = q N A ε W 1 2 2 (5) V 1 = ε q N A 2 W 1 2 ( 5 )
同理，从x = 0 x = 0 V = V 1 V = V 1 x = W 2 x = W 2 V = V 1 + V 2 V = V 1 + V 2
V 2 = q N D ε W 2 2 2 (6) V 2 = ε q N D 2 W 2 2 ( 6 )
结两端的总电压为
V 1 + V 2 = q 2 ε ( N A W 1 2 + N D W 2 2 ) = φ 0 + V R (7) V 1 + V 2 = 2 ε q ( N A W 1 2 + N D W 2 2 ) = φ 0 + V R ( 7 )
联立方程
{ q 2 ε ( N A W 1 2 + N D W 2 2 ) = φ 0 + V R W 1 N A = W 2 N D ⎩ ⎪ ⎨ ⎪ ⎧ 2 ε q ( N A W 1 2 + N D W 2 2 ) = φ 0 + V R W 1 N A = W 2 N D
可以解得
W p = W 1 = 2 ε ( φ 0 + V R ) q N A ( 1 + N A N D ) W p = W 1 = q N A ( 1 + N D N A ) 2 ε ( φ 0 + V R )
W n = W 2 = 2 ε ( φ 0 + V R ) q N D ( 1 + N D N A ) W n = W 2 = q N D ( 1 + N A N D ) 2 ε ( φ 0 + V R )
W 1 + W 2 = 2 ε ( N A + N D ) ( φ 0 + V R ) q N A N D W 1 + W 2 = q N A N D 2 ε ( N A + N D ) ( φ 0 + V R )
从以上公式可以看出，耗尽区主要向轻参杂的半导体一侧扩展。
《CMOS模拟继承电路设计（第二版）》, Phillip E. Allen
等于PN结两侧任意一侧固定的电荷的量，A 是PN结的横截面积
A b r u p t J u n c t i n : Q j = ∣ A W p q N A ∣ = A W n q N D = A 2 ε q ( φ 0 + V R ) N A N D N A + N D A b r u p t J u n c t i n : Q j = ∣ A W p q N A ∣ = A W n q N D = A N A + N D 2 ε q ( φ 0 + V R ) N A N D
G r a d e d J u n c t i o n : Q j = A ( 2 ε q ( φ 0 + V R ) N A N D N A + N D ) 1 − m , m = 1 3 G r a d e d J u n c t i o n : Q j = A ( N A + N D 2 ε q ( φ 0 + V R ) N A N D ) 1 − m , m = 3 1
PN结耗尽区形成的电容称作耗尽层电容，它是由结附近没有被中和的固定电荷形成并随着外加电压变化而改变，可以利用电容的定义求出，是非线性电容
A b r u p t J u n c t i n : C j = d Q j d V R = A ε q N A N D 2 ( N A + N D ) 1 φ 0 + V R = A ε q N A N D 2 ( N A + N D ) φ 0 1 1 + V R φ 0 A b r u p t J u n c t i n : C j = d V R d Q j = A 2 ( N A + N D ) ε q N A N D φ 0 + V R 1 = A 2 ( N A + N D ) φ 0 ε q N A N D 1 + φ 0 V R 1
《Analog Integrated Circuits Design - Johns Martin》
突变结的电荷是( V R ) 0.5 ( V R ) 0 . 5 ( V R ) 0.7 ( V R ) 0 . 7
对于大信号的结电容，利用Q = C V Q = C V V R = − φ 0 V R = − φ 0 C j = ∞ C j = ∞
C j − a v e r a g e d = Q j ( V 2 ) − Q j ( V 1 ) V 2 − V 1 C j − a v e r a g e d = V 2 − V 1 Q j ( V 2 ) − Q j ( V 1 )
当正向偏置时，由于扩散电流，靠近空间电荷区的少子将增加，这样将多出来diffusion电容，正向偏偏置时总的电容为C d + C j C d + C j τ T τ T C d C d C j C j
C d = τ T I D V T C d = τ T V T I D
根据(3)式，当x = 0 x = 0
E 0 = q N A ε W 1 = 2 q ( φ 0 + V R ) N D N A ε ( N D + N A ) (4) E 0 = ε q N A W 1 = ε ( N D + N A ) 2 q ( φ 0 + V R ) N D N A ( 4 )
最大反向偏置电压V R V R E max E m a x
E max = 3 × 1 0 5 V / cm E m a x = 3 × 1 0 5 V / cm
依据式(4)，可得
E max = 2 q ( φ 0 + V R ) N D N A ε ( N D + N A ) E m a x = ε ( N D + N A ) 2 q ( φ 0 + V R ) N D N A
( φ 0 + V R ) = ε ( N D + N A ) 2 q N D N A E max 2 ( φ 0 + V R ) = 2 q N D N A ε ( N D + N A ) E m a x 2
雪崩击穿和齐纳击穿下，反向电流
i R A = M i R = ( 1 1 − ( V R B V ) n ) i R i R A = M i R = ( 1 − ( B V V R ) n 1 ) i R
这里i R i R M M n n
正向电压V D V D x = 0 x = 0 x x
p n ( 0 ) = p n 0 exp ( V D V t ) = n i 2 N D exp ( V D V t ) p n ( 0 ) = p n 0 exp ( V t V D ) = N D n i 2 exp ( V t V D )
n p ( 0 ) = n p 0 exp ( V D V t ) = n i 2 N A exp ( V D V t ) n p ( 0 ) = n p 0 exp ( V t V D ) = N A n i 2 exp ( V t V D )
流过PN结的电流正比于过剩少数载流子在x = 0 x = 0
J p ( x ) = − q D p d p n ( x ) d x ∣ x = 0 J p ( x ) = − q D p d x d p n ( x ) ∣ ∣ ∣ ∣ ∣ x = 0
这里D p D p
p n ′ ( x ) = p n ( x ) − p n 0 p n ′ ( x ) = p n ( x ) − p n 0
过剩的少数载流子从PN结开始以指数规律衰减，L p L p
p n ′ ( x ) = p n ′ ( 0 ) exp ( − x L p ) = [ p n ( 0 ) − p n 0 ] exp ( − x L P ) p n ′ ( x ) = p n ′ ( 0 ) exp ( − L p x ) = [ p n ( 0 ) − p n 0 ] exp ( − L P x )
则有p n ′ ( x ) p n ′ ( x ) J p ( x ) J p ( x )
J p ( 0 ) = q D p p n 0 L p [ exp ( V D V t ) − 1 ] J p ( 0 ) = L p q D p p n 0 [ exp ( V t V D ) − 1 ]
同样对于p型半导体中的过剩电子，有
J n ( 0 ) = q D n n p 0 L n [ exp ( V D V t ) − 1 ] J n ( 0 ) = L n q D n n p 0 [ exp ( V t V D ) − 1 ]
总的电流密度乘以PN结横截面积，电流的表达式为
i D = A J ( 0 ) = A p [ J p ( 0 ) + J n ( 0 ) ] = q A ( q D p p n 0 L p + q D n n p 0 L n ) [ exp ( V D V t − 1 ) ] = I S [ exp ( V D V t − 1 ) ] i D = A J ( 0 ) = A p [ J p ( 0 ) + J n ( 0 ) ] = q A ( L p q D p p n 0 + L n q D n n p 0 ) [ exp ( V t V D − 1 ) ] = I S [ exp ( V t V D − 1 ) ]
g m = d I d V = I s exp ( V D V t ) = I s exp ( V D V t ) 1 V t = I D V t g m = d V d I = I s exp ( V t V D ) = I s exp ( V t V D ) V t 1 = V t I D
《Precision Temperature Sensors in CMOS Technology, Michiel A.P. Pertijs》指出了非理想因素
i D = I s [ exp ( V D V t ) − 1 ] i D = I s [ exp ( V t V D ) − 1 ]
这个公式的推导，是假设no generation or recombination of electron-hole pairs takes place in the depletion region. In practice, however, some of the carriers injected across the depletion region will recombine. The resulting recombination current can be described by
i rec = I r 0 exp ( q V D 2 k T ) i rec = I r 0 exp ( 2 k T q V D )
这里，∗ I r 0 ∗ ∗ I r 0 ∗
i D + i r e c i D + i r e c
经验公式给出了一个近似结果
i D = I s [ exp ( V D n V t ) − 1 ] i D = I s [ exp ( n V t V D ) − 1 ]
这里的n 被称作non-ideality factor：对于比较大的正向偏置电压，n ≈ 1；对于比较小的正向偏置电压（此时复合电流主导），n ≈ 2；在此之间，1 ≤ n ≤ 2；
