计算能使玻尔兹曼近似成立的最大掺杂浓度及费米能级的位置。 解：考虑T?300K时对硅进行了硼掺杂，假设玻尔兹曼近似成立的条件是EF?Ea?3kT，已知硼在硅中的电离能是Ea?Ev?0.045eV，假设本征费米能级严格等于禁带中央。在T?300K时，P型半导体的费米能级在EFi与Ea之间，所以
Ec?EvE?Ev?EF?c??Ea?Ev???EF?Ea?22Eg?N????Ea?Ev???EF?Ea??kTln?a?2?ni?EFi?EF?N1.12?0.045?3?0.0259??0.0259lna2ni0.437?0.0259lnNani
?0.437??0.437?1017?3Na?niexp??1.5?10exp?3.2?10cm????0.0259??0.0259?EFi?EF?0.437eV玻尔兹曼近似成立的最大掺杂浓度是Na?3.2?1017cm?3 费米能级高于本征费米能级EFi?EF?0.437eV。
二． 半导体中的载流子输运现象与过剩载
流子：
概念题：
30.半导体中存在两种基本的电荷输运机理，一种称谓载流子的漂移，漂移引起的载流子流动与外加电场有关；另一种电荷输运现象称谓载流子的扩散，它是由杂质浓度梯度引起的（或理解为有“扩散力”存在引起的电荷输运）。
31.给半导体施加电场，载流子的漂移速度不会无限增大，而是在散射作用下，载流子会达到平均漂移速度。半导体内主要存在
着两种散射现象：晶格散射和电离杂质散射。
32.载流子迁移率定义为载流子的平均漂移速度与所加电场的比值?p?vdpE,?n?vdnE。电子迁移率?n和空穴迁移率?p既是温度的函
数，也是电离杂质浓度的函数。
33.当所加的电场很小时，载流子的平均漂移速度与电场成线性关系；当电场强度达到104Vcm?1时，载流子的漂移速度达到饱和值
107cms?1。
34.载流子的漂移电流等于电导率与电场强度的乘积（jdrf??E）电导率与载流子浓度、迁移率成正比；电阻率是电导率的倒数。 35.载流子的扩散电流密度正比于扩散系数Dn,Dp和载流子浓度梯度。
非均匀杂质掺杂的半导体，在热平衡时，会在半导体内产生感应电场。载流子的扩散系数与迁移率的关系称谓爱因斯坦关系：
Dn?n?Dp?p?kT?Vt。 q练习题：
36. Calculate the intrinsic concentration in silicon atT?350Kand at T?400K.
The values of Nc and Nv vary as T3/2.As a first approxi -mation, neglect any variation of bandgap energy temperature. Assume that the bandgap energy of silicon is
1.12eV.the value of at T?350K is
?350?kT?0.0259??0.0302eV ??300?the value of at T?400K is
?400?kT?0.0259??0.0345eV ?300??We find for T?350K,
??E??350???1.12?22?6n?NcNvexp?g???2.8?1019??1.04?1019??exp????3.62?10cm?300??0.0302??kT?ni(350K)?1.9?1011cm?32i3For T?400K,We find
??E??400???1.12?24?6n?NcNvexp?g???2.8?1019??1.04?1019??exp?5.5?10cm??? ?300??0.0345??kT?ni(400K)?2.34?1012cm?32i337.Determine the thermal equilibrium electron and hole concentration in GaAs at T= 300K for the case when the Fermi energy level is 0.25eV above the valence-band energy Ev. Assume the bandgap energy is Eg=1.42eV.
(Ans. p0=4.5x10cm,n0=? T= 300K，Nc=4.7X10cm, Nv=7X10cm)
38.Find the intrinsic carrier concentration in silicon at (a)T=200K and at (b)T=400K < Ans.(a)8.13x10cm,(b) 2.34x10cm.ni=1.5x10cm> .
39.Consider a compensated germanium semiconductor at T=300K doped at concentration of Na=5x10cm and Nd=1x10cm.Calculate the thermal equilibrium electron and
13
-3
13
-3
12
-3
10
-3
4
-3
18
-3
14
-3
17
-3
hole
13
-3
concentrations. 13 -3 13-3 n0=1.12x10cm,ni=2.4x10cm>. N?Nd?N?Nd?213p0?a??a?n?2?10?i?22???2?1013?3.124?1013?5.124?1013cm?3n0?ni2/p0?5.76?1026/5.12?1013?1.125?1013cm?3 2?4?2.4??10226 40. Consider a compensated GeAs semiconductor at T=300K doped 16 at -3 concentration of Nd=5x10cm 15-3 and Na=2x10cm.Calculate the thermal equilibrium electron and hole 4 concentrations. -3 6 -3 16-3 n0=2.16x10cm>.(ni=1.8x10cm) 41. Consider n-type Silicon at T=300K doped with phosphorus. Determine the doping concentration such that Ed-EF=4.6kT (Asn.Nd=6.52x10cm).(Ec-Ed=0.045eV) Ec?EvE?EvE?Ev?EF?c?Ec?Ec?EF?c?Ec222EgE?Ev?EF?c?Ec?Ed?Ed???Ed?EF???Ec?Ed?22 Nd?0.56?0.045?4.6?0.0259?0.39586eV?kTlnniEF?EFi?EF??0.39586??0.39586?1016?3Nd?niexp??1.5x10exp????6.52?10cm?0.0259??0.0259?16 -3 42. Calculate the position of the Fermi energy level in n-type silicon at T=300K with respect to the intrinsic energy level. The doping concentration are Nd=2x10cmand Nd=3x10cm. (Asn.EF-EFi=0.421eV). 16 -3 17 -3
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