Semiconductor Device Fundamentals Solutions Manual Robert Pierret
Semiconductor Fundamentals - PKUdsec.pku.edu.cn/~tlu/apde/apde4-5.pdf · 2017-04-03 ·...
Transcript of Semiconductor Fundamentals - PKUdsec.pku.edu.cn/~tlu/apde/apde4-5.pdf · 2017-04-03 ·...
Semiconductor Fundamentals
卢朓北京大学数学科学学院
[email protected]主页 :http://dsec.pku.edu.cn/~tlu博客 :http://hi.baidu.com/motioo
IntroductionTo understand the fundamental concepts of semiconductors, one must apply modern physics to solid materials. More specifically, we are interested in semiconductor crystals. Crystals are solid materials consisting of atoms, which are placed in a highly ordered structure called a lattice. Such a structure yields a periodic potential throughout the material.
IntroductionTwo properties of crystals are of particular interest, since they are needed to calculate the current in a semiconductor. First, we need to know how many fixed and mobile charges are present in the material. Second, we need to understand the transport of the mobile carriers through the semiconductor.
IntroductionIn this chapter we start from the atomic structure of semiconductors and explain the concepts of energy band gaps, energy bands and the density of states in an energy band. We also show how the current in an almost filled band can more easily be analyzed using the concept of holes. Next, we discuss the probability that energy levels within an energy band are occupied. We will use this probability density to find the density of electrons and holes in a band.
Introduction
Two transport mechanisms will be considered. The drift of carriers in an electric field and the diffusion of carriers due to a carrier density gradient will be discussed. Recombination mechanisms and the continuity equations are then combined into the diffusion equation. Finally, we present the driftdiffusion model, which combines all the essential elements discussed in this chapter.
晶体和非晶体
● amorphous material ( 非晶体 )● crystalline material ( 晶体 )● polycrystalline material ( 多晶 )● single crystals ( 单晶 )
半导体的特征
● 导电性介于导体与绝缘体之间的材料
● 由周期表第 IV 族、 IIIV 族及 IIVI 族● 组成能带结构中,能隙宽度宽约在 1 至 2eV左右
● 可以用掺杂 (Doping) 的技术改变传导特性
能带结构
● Recall the eigenenergy of the electron of the hydrogen atom
● the electron can take only the discrete values of the energy
● what is the property of the electrons in a crystal lattice?
能级分裂
● 两个原子带的电子在原子距离很远的时候,电子具有同样的能级结构,当两个原子靠近
时,电子的能级将会改变,称为 energy splitting.
● 一个类比,两辆汽车速度一样,发动机提供的功率相同,靠近时,要保持同样的速度前者和后者需要提供的功率不同。
● 类比不是很合适。 Don't take it seriously.
能级分裂示意图
Introduction to Bloch's Theorem● Simplify Schroedinger equation (assume
that nuclei of atoms are static)● translational symmetry of a crystal● An electron in the crystal sees primarily the
Coulomb potential energy of interaction with the atomic nuclei of the crystal, the Coulomb potential energies of interaction with other electrons
Mariam González DebsCourse: ECE 745
Professor: W.N.Hitchon
Band Structure Group Schrödinger Equation: Calculations of
Energy Bandgap using Periodic Potentials
例子 : 分片常数的周期势
● Solution to the Schrödinger wave equation using the different potential energy expression:– SquareWell potential (KronigPenney
Model)
Wavefunction for nth state:
Energy for nth state
Periodic Potential EnergyKinetic Energy
Schrödinger Wave Equation
One example of periodic potentials, V(x):
nx =unx eiknx
V x = {V , axab0 , 0xa
− ℏ2
2md2
dxV x nx =Ennx
是周期函数unx
V x在一个最小周期中的取值
V xab=V x
KronigPenney Model
Bloch 由 定理, 特征函数 x 可以写成
上面给出的分片常数周期势V x Schroedinger 场中的单电子的 方程−ℏ
2
2m ' 'x V xx =Ex
V xab=V x
kx =uk xeik x
ukx 也是周期函数,周期和V x 相同
kxab=ukxabeikxab
ukx的周期性
kxab=ukx ei k xab
=kx eikab
特别的, k−b=k aeikab
如何求解上面方程的特征值问题 I给定一个能量 E −ℏ
2
2m ' 'x V xx =Ex
就成了常微分方程,更加特别的,在 有0xa
V x =0 , 这时上面的方程就成了常系数的二次方程,这时方程的求解有成熟的理论,可以通过求解特征方程 根据根的情况得出上面方程的通解。
−ℏ2
2m
2=E
我们得出特征方程的两个纯虚根 记作=±i 2mEℏ
2
=±i K
求解上面方程的特征值问题 II给定一个能量 E −ℏ
2
2m ' 'x V xx =Ex
就成了常微分方程,更加特别的,在 有−bxa
V x =V 0 , 这时上面的方程就成了常系数的二次方程,这时方程的求解有成熟的理论,可以通过求解特征方程 −ℏ2
2m
2=E −V 0
这时我们假设 ,方程有两实根 记作=±2mV 0−E
ℏ2
=±iQ
E−V 00
In the region –b<x<0 within the barrier, the solution is:
with energy:
● In the region 0<x<a in which V=0, the eigenfunction is:
with energy:
KronigPenney Model
V(x)
(a+b) b 0 a a+b x
V0
Squarewell periodic potential
E=ℏ
2 K2
2mV 0−E=
ℏ2Q2
2m
ax =AeiKxBe−iKx bx =CeQx
−De−Qx
如何把两个区间的解拼起来
上面的方法,我们得到 Schroedinger 方程分别在两个区间上的通解,我们把两个解拼起来构成方程在整个区间上的解。
首先, Schrodinger 方程的含有二次导数,因此特征函数
x 和它的导数应该处处连续,我们只需要使 x=a处的左右函数极限和左右导数的极限相等
limx a− x =limxa x limxa− 'x =limx a 'x
其次,由 Bloch 定理,我们还要求 , 存在一个波数 使得k
k−b=k aeikab
x =k x=uk x eik x
KronigPenney Model● At x=0
● At x=a
● Solving for a vanishing determinant, the following equation is obtained:
( )
( )
( )( ) ( )
iKa iKa Qb Qb ik a b
iKa iKa Qb Qb ik a b
Ae Be Ce De eiK Ae Be Q Ce De e
− − +
− − +
+ = +− = −
( )2 2
sinh( )sin( ) cosh( )cos( ) cos( ( ))2
Q KQb Ka Qb Ka k a b
QK
�� −�� + = +����
. 上面给出了上面的两个要求得到的方程 四个方程,四个未知数 A,B ,C ,D要想得到一个非零的解,必须让次线性方程组的行列式为零
函数值相等 AB=CD , 导数值相等 iK A−B=Q C−D
KronigPenney Model● Representing the potential by the periodic delta
function obtained when we pass to the limit b=0 and , the result simplifies to:
where,
V 0∞
PK
sinKacos Ka=coska
bV 0=Pℏ
2
mP is a constant极限过程保持 乘积不变 .
分片常数的势的极限称为 函数
上面的方程中的 P , , 是常数 反应电势的强度 K 是和能量相关的,在能量确定后也是常数K=2mE/ℏ2
红色的小写k Bloch 是 定理中的波数,此波数是实数才表示了一个传播的波可是不难发现,不是对于任意的K都能找到一个实数的k 使得上面的方程成立。也就是说,能量不能任意选取。
(forbidden gap) 这就引入禁带 的概念。禁带中的能量不能使得上面的方程保证能找到一个实数的k。
求极限的过程可以用 maximaV 0=P
ℏ2
m1b
Q=2mℏ V 0−E
K=2mℏ
E
K2−Q2
2KQsinhQb
spell error in i32
limit is equal to PK
KronigPenney Model● Plots for P=3/2.● Allowed values of E lie
between ±1.● For other values of E
there are either no traveling wave or no Blocklike solutions to the wave equation.
● Forbidden gaps in the energy spectrum are formed at ka=n for integer n.
10 5 0 5 102
0
2
4
65.712
1.557−
PKa
sin Ka( )⋅ cos Ka( )+
1
1−
1010− Ka
这里Pa相当于我们用的常数 P
实数 k 应使得 coska的值在−1 和 1之间PKa
sin KacosKa=coska
不是任意 Ka -1 +1取值使得方程左边的值在 和 之间
Bloch's Theorem for 1D lattice−ℏ
2
2m∂
2x ∂ x2 V x x =Ex
V x is the periodic potential energy. V xa=V xa is the lattice spacing
The wave function x has the form of x=eikxux
ux has the periodicity of the lattice.
Bloch Theorem:
Bloch's Theorem for 3D Lattice
Periodic Potential V xr =V x Wave function of the form
x =eik⋅xu x
The fact that the wavefunction takes this simpleform allows many calculations to be performed in a relatively simple fashion.
energy band
Brillouin zone−
ak≤
a
32
Reduced Zone
33
Extended zone
34
Repeated Zone
Effective mass in a bandFree particle E=
12
p2
mp : momentum, m : mass
If a minimum or maximum occurs at k=k0 for the elctron energyin a crystal band, then when k is near this minimum or maximumthe energy in the band as a function of k−k0 may be approximated with
E k≈E0k−k02
where E0 and are constants
Rewrite the energy expression into
E k≈E0ℏ
2k−k0
2
2m*
Comparison iwth the energy expressionforthe free particle leads one to the idea of an enffective mass,
m*=
ℏ2
2
1D Density of StatesWave equation in a in a infinit quantum well
−ℏ2∂2
2m∂xx =Ex
Boundary Conditions :0=L , '0= ' LThe solutions are
x=ei kx , k=n2
L, n=0,±1,±2,⋯
A factor of two is added to account for the two possible spins of each solution.
Number of states N k denotes the number of states with a wave number less than k
N k=2kL
DOS in k -space :D k=∂N k∂k
=2 L
One dimensional DOS
Density of state in energy space D E
D E=∂N E k
∂E=∂N k∂k
∂k∂E
=2L
mℏ2E
=L
2mℏE
relation between energy and wave number
E=ℏ
2k2
2m∂k∂E
=m
ℏ 2 E
3D Density of states
k=nx ,ny ,nz
L, nx ,ny ,nz=0,1,2,⋯
The solutions of 3D Schroedinger equation of 3D quantum well are
x=sink⋅x
k=∣k∣=kx2ky
2kz
2
N k denotes the number of the states with the wave number less than k
N k=218
43k3
/ L 3
=13k3 L
3
The density of states in wave number space :
D k=∂N k∂k
=k2 L 3
3D Density of states
Density of states in energy space
D E=∂N k∂E
=∂N k∂k
∂k∂E
= L 3
k2 mℏ2 E
= L 3m3/22 E
ℏ3
Relation between energy and wave vector
E=ℏ
2
2mkx
2ky
2kz
2=
ℏ2
2mk2
DOS per unit volume and per unit energy
gE =1L3
dNdE
= 1 3m3 /22E
ℏ3
Density of states of electrons in a semiconductor
● The same analysis also applies to electrons in a semiconductor. The effective mass takes into account the effect of the periodic potential on the electron. The minimum energy of the electron is the energy at the bottom of the conduction band, , so that the density of states for electrons in the conduction band is given by:
Ec
gc E=1
Volume∂N∂E
= 1 3
2m*3 /2 1ℏ
3 E−Ec , for EEc
Carrier Distribution Functions● The distribution or probability density functions describe
the probability with which one can expect particles to occupy the available energy levels in a given system.
● Of particular interest is the probability density function of electrons, called the Fermi function.
● The derivation of such probability density functions belongs in a statistical thermodynamics course.
● Carrier density = DOS times PDF
Carrier DensitiesThe density of occupied states per unit volume and energy
Electrons nE=gc E f E
Holesp E=gv E[1− f E]
gv E is the density of states in the valence band
gv E=82
h3 m*3 /2Ev−E , for E≤Ev
Planck constant h=2ℏ
Carrier Density n(E)
Calculation of Fermi Integral
Ec : bottom of the conduction band
no=∫Ec
∞
gcE f EdE
=∫Ec
∞ 82
h3 me*3 /2E−Ec
1
1eE−EF
kT
dE
The density of electrons in a semiconductor : unit(cm−3 )
Holes Ev : top of the valley band
po=∫−∞
Ev
gv E1− f E dE
=∫−∞
Ev 82
h3 mh*3 /2Ev−E
1
1eEF−E
kT
d E
Carrier density at T=0KFermi function at T = 0 K :
f E ={1, E≤EF
0, EEF
no=∫Ec
EF
gc EdE at T=0 K
no=223
2 m*
ℏ2
3/2
EF−Ec 3 /2 , for EF≥Ec
This expression can be used to approximate the carrier density in heavily degenerate semiconductors provided EF−Ec≫kT
Nondegenerate semiconductorsNondegenerate semiconductors are defined as semiconductors for which the Fermi energy is at least 3kT away from either band edge.
no≈∫Ec
∞ 82h3 me
*3 /2E−EceEF−E
kT dE=NceEF−Ec
kT
Where Nc=2 2me* kT
h2 3/2
the hole density
po≈∫−∞
Ec 82h3 mh
*3 /2Ev−EeE−EF
kT dE=NveEv−EF
kT
Where Nv=2 2mh* kT
h2 3 /2
Nc is the effective density of states in the conduction bandNv is the effective density of states in the valence band
Degenerate semiconductors A useful approximate expression applicable to degenerate semiconductors was obtained by Joyce and Dixon and is given by:
for electrons and by:
for holes.
EF−Ec
kT≈ln
no
Nc
1
8
no
Nc
− 316
−39 no
Nc2
⋯
Ev−EF
kT≈ln
po
Nv
1
8
po
Nv
− 316
−39 po
Nv2
⋯
上一页的公式有错误?网上搜索得到的公式很多给出的是上一页给出的形式
但是原始文献给出的公式却有符号的差别
给出的公式如下:J 相当于
n0
Nc
相当于EF−Ec
kBT
Intrinsic carrier density● intrinsic semiconductors are usually non
degenerate● labeling the Fermi energy of intrinsic
material as EF
Ei
ni=no∣EF=Ei=Nce
Ei−Ec /kT
pi=po∣EF=Ei=Nv e
Ev−Ei/kT
intrinsic material means no doping, so ni=pi
band gap Eg=Ec−Ev
ni=Nc Nve−Eg /2kT
Intrinsic Fermi energy● The above equations for the intrinsic electron and
hole density can be solved for the intrinsic Fermi energy, yielding:
Ei=EcEv
2
12
kT ln Nv
Nc
Ei=EcEv
2
34
kT ln mh*
me*
Intrinsic material as referencecarrier density of non-intrinsic non-degerate material
no=nieEF−E i/KT
po=nieEi−EF /KT
Fermi energy
EF=EikT lnno
ni
EF=Ei−kT lnpo
ni
Doped semiconductors● Dopants and impurities
Shallow impurities require around the thermal energy, kT, or less - to ionize.Deep impurities reuqire more than five times the thermal energy away from either band edge, are very unlikely to ionize.
Ionized donors provide free electrons in a semiconductor, which is then called ntype, while ionized acceptors provide free holes in a semiconductor, which we refer to as being a ptype semiconductor.
Dopant and impurities● Ionized donors provide free electrons in a semiconductor,
which is then called ntype, while ionized acceptors provide free holes in a semiconductor, which we refer to as being a ptype semiconductor.
● While for shallow acceptors the hole density equals the acceptor concentration
Nd+≈Nd
Na-≈Na
Analysis of nondegenerately doped semiconductors
● assumption that the semiconductor is neutral
noNa-=poNd
+
Nonequilibrium carrier densities● constant temperature is not a sufficient condition for
thermal equilibrium● quasiFermi energy ● For a nondegenerate semiconductor
n=non=niexp Fn−Ei
kT p=pop=niexp Ei−Fp
kT
Fn and Fp are quasi Fermi energy for electrons and holes, respn and p are excess electron density and hole density, resp
● Example: A piece of germanium doped with 10E16 cm3 shallow donors is illuminated with light generating 10E15 cm3 excess electrons and holes. Calculate the quasiFermi energies relative to the intrinsic energy and compare it to the Fermi energy in the absence of illumination.
● Solution: The carrier densities when illuminating the semiconductor are:
●
● and the quasiFermi energies are:
●
● In comparison, the Fermi energy in the absence of light equals
●
● which is very close to the quasiFermi energy of the majority carriers.
n=non=10161015
=1.1×1016cm−3
p=pp≈1015cm−3
Fn−Ei=kT ln nni
=0.0259×ln 1.1time1016
2×1013=163 meV
F p−Ei=−kT lnpni
=0.0259×ln1 time1015
2×1013 =−101 meV
EF−Ei=kT lnno
ni
=0.0259×ln1016
2×1013=161 meV
Carrier Transport● A motion of free carriers in a
semiconductor leads to a current.● carrier drift and carrier diffusion● mobility and the diffusion constant● Einstein relation
Carrier drift● The motion of a carrier drifting in a
semiconductor due to an applied electric field.
● we define the mobility as the velocity to field ratio
=∣v∣
∣E∣=
q
m
where is the collision time
电子受力 −Eq
动量改变量是 m∣v∣
两次碰撞之间的时间∣E∣q冲量
冲量 = 动量改变量
ExampleElectrons in undoped gallium arsenide have a mobility of
8800 cm2s/V . Calculate the average time between collisions.
Calculate the distance traveled between two collisions(also called the mean free path). Use an average velocity of 107cm/s.
The collision time, c , is obtained from:
c=nme
*
q=
0.88×0.067×9.1×10−31
1.6×10−19 =0.34 ps
The mean free path, l , equals:l=vaveragec=107
×0.34×10−12=34 nm
Conductivity and ResistivityThe current due to eletrons and holes
J=qnveqpvh=qnnpe E
The conductivity due to electrons and holes
=∣J∣
∣E∣=qnnpp
The resistivity is defined as
=1=
1qnnpp
The sheet resistance concept is used to characterize both wafers and thin doped layers, since it is typically easier to measure the sheet resistance rather than the resistivity of the material. The sheet resistance of a uniformlydoped layer with resistivity, , and thickness, t, is given by their ratio:
While the unit of the sheet resistance is Ohms, one refers to it as Ohms per square. This nomenclature comes in handy when the resistance of a rectangular piece of material with length, L, and width W must be obtained. It equals the product of the sheet resistance and the number of squares or:
where the number of squares equals the length divided by the width.
Rs=
t
R=RsLW
Carrier diffusion
Jn=qDn∇nJp=−q Dp ∇p
diffusion current
Using the definition of the diffusion constant we then obtain the following expressions which are often referred to as the Einstein relations
Dn=nkTq
=nV tDn=p
kTq
=pV t
Total currentThe total electron current is obtained by adding the current due to diffusion to the drift current, yielding:
and similarly for holes:
The total current is the sum of the electron and hole current densities multiplied with the area, A, perpendicular to the direction of the carrier flow:
Jn=qnn EqDndndx
Jp=qpp E−qDpdpdx
I total=A Jn Jp
Carrier recombination and generation
ecewww.colorado.edu/~bart/ecen3320/newbook/chapter2/ch2_8.htm
Continuity equationThe continuity equation describes a basic concept, namely that a change in carrier density over time is due to the difference between the incoming and outgoing flux of carriers plus the generation and minus the recombination.
∂nx ,t ∂t
=1q
∂ Jnx ,t
∂xGnx ,t −Rnx ,t
∂p x ,t ∂ t
=−1q
∂ Jpx ,t
∂xGp x ,t −Rpx ,t
The diffusion equationIn the quasineutral region a region containing mobile carriers, where the electric field is small the current is due to diffusion only. In addition, we can use the simple recombination model for the net recombination rate. This leads to the timedependent diffusion equations for electrons in ptype material and for holes in ntype material:
∂np x ,t
∂t=Dn
∂2npx ,t
∂x2 −np x ,t −np0
n
∂pnx ,t
∂ t=Dp
∂2pnx ,t
∂ x2 −pnx ,t −pn0
p
The driftdiffusion modelThe assumptions of the simplified driftdiffusion model are:
● Full ionization: all dopants are assumed to be ionized (shallow dopants)
● Nondegenerate: the Fermi energy is assumed to be at least 3 kT below/above the conduction/valence band edge.
● Steady state: All variables are independent of time● Constant temperature: The temperature is constant
throughout the device.
ten variables used in DD model , the charge densityn, the electron densityp , the hole density
E , the electric field , the potentialEi , the intrinsic energyFn , the electron quasi-Fermi energyFp , the hole quasi-Fermi energyJn , the electron current densityJp , the hole current density
Ten equations of the DD modelCharge density equation
=q p−nNd+−Na
-
Electric field and potential equations∂E∂ x
=
∂
dx=−E
∂Ei
dx=qE
Drift and diffusion current equations
Jn=qnn EqDndndx
Jp=qpp EqDp
dpdx
Continuity equation in steady state with SHR recombination
0=1q
∂ Jn
∂x−
np−ni2
np2nicosh Et−Ei
kT 1
0=1q
∂ Jp
∂x−
np−ni2
np2nicosh Et−Ei
kT 1
Numerical computation 可参考
● Gummel, H.K. , A selfconsistent iterative scheme for onedimensional steady state transistor calculations, Electron Devices, IEEE Transactions on, 1964,Volume: 11, Issue: 10 On page(s): 455 465
半导体器件基础
/home/tlu/apde/Lecture 3 (Semiconductor principles).pdf
Reference● Donald A. Neamen, Semiconductor
physics and devices: basic principles, 3rd ed., McGrawHill, New York, 2003.
● http://ecewww.colorado.edu/~bart/3230.htm● ecow.engr.wisc.edu/cgi
bin/get/ece/745/hitchon/pres1_ece_745a.ppt● www.mse.nsysu.edu.tw/~kyhsieh/solid2/new/042
1.ppt