+
+多了一个 **interstitial**
+
+* **Frenkel defect**
+
+##### line dislocation
+
+### 1.6.2 impurities in Solids
+
+##### lattice defects
+
+*substitutional* impurities
+
+*interstital* impurities
+
+
+
+##### doping (参杂原子)
+
+* impurity diffusion
+
+ high temperature - vacancy - impurity partical move from high concentration to lower - low down the temp
+
+* ion implantation
+
+ low temp - ions be accelerated
+
+ defect: lattice displacement damage
+
+ solve: thermal anneal 热退火
+
+
+
+## 1.7 growth of semiconductor materials
+
+### 1.7.1 growth from a melt
+
+#### Czochralski Method（Growth）
+
+seed - the same material in liquid phase - pulled - bigger
+
+#### zone refining（Purify）
+
+At the solid–liquidinterface, there is a distribution of impurities between the two phases. The parameter that describes this distribution is called the **segregation coefficient**: the ratio of the
+concentration of impurities in the solid to the concentration in the liquid.
+
+r-f coil 射频线圈
+
+#### 切割
+
+flat（110）plane
+
+#### polish
+
+chemecal etch
+
+### 1.7.2 Epitaxial Growth
+
+外延生长
+
+single-crystal substrate
+
+*homoepitaxy* one material
+
+*heteroepitaxy* e.g. Growing epitaxial layers of the ternary alloy AlGaAs on a GaAs substrate
+
+#### CVD chemical vapor-phase deposition
+
+chemical reaction release material atom to deposit on the material
+
+#### Liquid-phase epitaxy
+
+melt temp lower than the semiconductor itself
+
+practiced at low temp (III-V)
+
+#### Molecular Beam Epitaxy
+
+vacuum iat temp 0f 400-800
+
+semiconductor and dopant atoms evaporated onto the surface of the substrate
+
+## 1.8 Summary
+
+**substrate** A semiconductor wafer or other material used as the starting material for further semiconductor processing, such as epitaxial growth or diffusion.
+
+**ternary semiconductor** A three-element compound semiconductor, such as aluminum gallium arsenide (AlGaAs).
+
+**elemental semiconductor** A semiconductor composed of a single species of atom, such as silicon or germanium.
+
diff --git a/docs/JY_FENG/Semiconductor Physics and Devices/2 Introduction to Quantum Mechanics.md b/docs/JY_FENG/Semiconductor Physics and Devices/2 Introduction to Quantum Mechanics.md
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+# 2 Introduction to Quantum Mechanics
+
+## 2.0 Preview
+
+In this chapter, we will:
+
+* Discuss a few basic principles of quantum mechanics that apply to semiconductor device physics.
+* State Schrodinger’s wave equation and discuss the physical meaning of the wave function.
+* Consider the application of Schrodinger’s wave equation to various potential functions to determine some of the fundamental properties of electron behavior in a crystal.
+* Apply Schrodinger’s wave equation to the one-electron atom. The result of this analysis yields the four basic quantum numbers, the concept of discrete energy bands, and the initial buildup of the periodic table.
+
+## 2.1 Principles of Quantum Mechanics
+
+### 2.1.1 Energy Quanta
+
+photoelectric effect
+$$
+E=hv\space (h=6.625\times 10^{-34}J\cdot s)
+$$
+**photon** the particle- like packet of energy
+
+**work function** The minimum energy required to remove an electron
+
+The **maximum kinetic energy** of the **photoelectron**
+$$
+T= \frac1{2}mv^2=hv-\Phi
+$$
+
+### 2.1.2 Wave -Particle Duality
+
+#### Compton effect
+
+A portion of the x-ray beam was defl ected and
+the frequency of the defl ected wave had shifted compared with the incident wave.
+
+#### Brogile wave-particle duality principle
+
+$$
+p=\frac{h}{\lambda}
+$$
+
+$\lambda$ *de Broglie wavelength*
+
+##### Davisson Germer experiment
+
+
+
+
+
+### 2.1.3 The Uncertainty Principle
+
+#### Heisenberg uncertainty principle
+
+$$
+\Delta p\Delta x\ge \hbar\\
+\Delta E\Delta t\ge \hbar
+$$
+
+to *probability density function*
+
+## 2.2 Schrodinger's Wave Equation
+
+**wave-particle**
+
+### 2.2.1 The Wave Equation
+
+one-dimensional, nonrelativistic Schrodinger's wave equation
+$$
+\frac{-\hbar}{2m}\cdot\frac{\partial^2\Psi(x,t)}{\partial x^2}+V(x)\Psi(x,t)=j\hbar\frac{\partial\Psi(x,t)}{\partial t}
+$$
+$\Psi(x,t)$ wave function
+
+$V(x)$ potential function
+$$
+\Psi(x,t)=\psi(x)\phi(t)\\
+(use\ the\ technique\ of\ seperation\ of\ variables)\\
+\frac{-\hbar}{2m}\frac1{\psi(x)}\frac{\partial^2\psi(x)}{\partial x^2}+V(x)=j\hbar\frac1{\phi(t)}\frac{\partial\phi(t)}{\partial t}
+$$
+Since the left side of Equation (2.9) is a function of position x only and the right side of the equation is a function of time t only, **each side of this equation must be equal to a constant**.
+$$
+\eta=j\hbar\cdot\frac1{\phi(t)}\cdot \frac{\partial\phi(t)}{t}\\
+\Rightarrow \phi(t)=e^{-j(\eta/\hbar)t}\\
+E=hv=\frac{h\omega}{2\pi}\Rightarrow\omega=\frac{\eta}{\hbar}=\frac{E}{\hbar}\Rightarrow\phi(t)=e^{-j\omega t}\\
+\therefore \frac{-\hbar^2}{2m}\cdot\frac1{\psi(x)}\cdot\frac{\partial^2\psi(x)}{\partial x^2}+V(x)=E\\
+$$
+
+$$
+\Rightarrow \frac{\partial^2\phi(x)}{\partial x^2}+\frac{2m}{\hbar^2}(E-V(x))\psi(x)=0
+$$
+
+
+
+### 2.2.2 Physical Meaning of the Wave Function
+
+the relation between the function and the electron
+$$
+\Psi(x,t)=\psi(x)\phi(t)=\psi(x)e^{-j(E/\hbar)t}=\psi(t)e^{-j\omega t}
+$$
+$\Psi$不代表任何实际的物理含义
+
+$\bold{\left|\Psi(x,t)\right|^2dx}$ the probability of finding the particle between x and $x + dx$ at a given time
+$$
+\begin{align}
+\left|\Psi(x,t)\right|^2&=\Psi(x,t)\cdot\Psi^*(x,t)\\
+&=[\psi(x)e^{-j(E/\hbar)t}][\psi^*(x)e^{+j(E/\hbar)t}]\\
+&=\psi(x)\psi^*(x)=\left|\psi(x)\right|^2\\
+\end{align}
+$$
+
+### 2.2.3 Boundary Conditions
+
+$$
+\int^{\infin}_{-\infin}\left|\psi(x)\right|^2 dx =1
+$$
+
+$E\ and\ V(x) are\ finite$
+
+impose postulate(都是假设)
+
+* $\psi(x)$must finite, single valued, and continuous
+
+ > If the probability density were to become infinite at some point in space, then the probability of finding the particle at this position would be certain and the uncertainty principle would be violated.
+
+* $\psi'(x)$must finite, single valued, and continuous
+
+ > If the total energy E and the potential V(x) are finite
+ > everywhere, then from **Equation (8)**, **the second derivative must be finite**, which implies that the first derivative must be continuous. The first derivative is related to the particle momentum, which must be finite and single-valued.
+
+
+
+For these cases, the first derivative will not necessarily be continuous, but the remaining boundary conditions will still hold.
+
+## 2.3 Applications of Schrodinger's Wave equation
+
+### 2.3.1 Electron in Free Space
+
+$$
+\frac{\partial^2\phi(x)}{\partial x^2}+\frac{2m}{\hbar^2}(E-V(x))\psi(x)=0
+$$
+
+If there is no force acting on the particle, then the potential function V(x) will be constant and we must have E>V(x). Assume $V(x)=0$,
+
+$(14)\Rightarrow\\$
+$$
+\frac{\partial^2\phi(x)}{\partial x^2}+\frac{2mE}{\hbar^2}\psi(x)=0\\
+$$
+
+$$
+\psi(x)=A\ \exp[\frac{jx\sqrt{2mE}}{\hbar}]+B\ \exp[\frac{-jx\sqrt{2mE}}{\hbar}]\\
+\Rightarrow \psi(x)=A\ \exp[jkx]+B\ \exp[-jkx]\\
+\small{k=\frac{\sqrt{2mE}}{\hbar}}
+$$
+
+$$
+\therefore \Psi(x,t)=A\exp[j(kx-\omega t)]+B\exp[-j(kx+\omega t)]
+$$
+
+**wave function** that a particle moving in free space is represented by a traveling wave.
+
+$A\exp[j(kx-\omega t)]$ 正行波
+
+$B\exp[-j(kx+\omega t)]$ 负行波
+
+
+
+假设B=0, 此时物质仅仅由正行波A决定
+$$
+k=\sqrt{\frac{2mE}{\hbar^2}}=\frac{p}{\hbar}\\
+E=\frac{p}{2m}此时把并未考虑相对论效应，只能说E\approx \frac{p}{2m}\\
+\small{（当动量p很小，即远小于静止质量能量m_0c^2时）}
+$$
+利用德布罗意方程
+$$
+p=\frac{h}{\lambda}\\
+k=\frac{2\pi}{\lambda}
+$$
+A free particle with a well-defined energy will also have a well-defined wavelength and momentum. But it can be found anywhere since $|\Psi(x,y)|^2=AA^*$.
+
+**wave packet**
+
+### 2.3.2 The Infinite Potential Well
+
+
+$$
+\frac{\partial^2\phi(x)}{\partial x^2}+\frac{2m}{\hbar^2}(E-V(x))\psi(x)=0\\
+$$
+**E** total energy of the particle. If E is finite, the wave function must be zero $\psi(x)=0$ in region I and III.
+
+>A particle cannot penetrate these infinite potential barriers, so the probability of finding the particle in regions I and III is zero.
+
+$$
+\frac{\partial^2\phi(x)}{\partial x^2}+\frac{2mE}{\hbar^2}\psi(x)=0\\
+$$
+
+in region II: V=0
+$$
+\psi(x)=A_1 \cos kx+A_2\sin kx\\
+k=\sqrt{\frac{2mE}{\hbar^2}}
+$$
+boundary condition:
+
+* $\psi(x=0)=\psi(x=a)=0$
+
+$$
+\therefore A_1=0\\
+and\ k=\frac{n\pi}{a}\\
+$$
+
+**Negative values of n are not considered** Negative values of n simply introduce a negative sign in the wave function and yield redundant solutions for the probability density function.
+
+而$A_2$的解可以由$\int^{\infin}_{-\infin}\left|\psi(x)\right|^2 dx =1$给出。
+
+如果波函数为实数，则
+$$
+\because \int_0^a A_2^2\sin^2 kx\ dx=1\\
+\therefore A_2=\sqrt{\frac2{a}}
+$$
+
+$$
+\psi(x)=\sqrt{\frac2{a}}\sin(\frac{n\pi x}{a})\ (n=1,2,3,\cdots)
+$$
+
+**standing wave** 驻波
+
+上述方程为驻波解，电子被束缚在这个驻波解中
+
+k的取值只能为n，因此E的取值也只能为限定值（E和k的关系式）
+
+> This result means that the energy of the particle is quantized. That is, the energy of the particle can only have particular discrete values.
+
+$$
+E=E_n=\frac{\hbar^2n^2\pi^2}{2ma^2}, where\ n=1,2,3,\cdots
+$$
+
+* We may note that as the energy increases, the probability of fi nding the particle at any given value of x becomes more uniform.
+
+
+
+### 2.3.3 The Step Potential Function
+
+
+
+We will assume that the particles are traveling in the +x direction and that they originated at $x=-\infin$.
+
+讨论一个比较特殊的案例$E
+
+transmission coefficient T
+$$
+T=\frac{v_t\cdot A_3\cdot A_3^*}{v_i\cdot A_1\cdot A_1^*}=\frac{A_3\cdot A_3^*}{A_1\cdot A_1^*}\approx 16(\frac{E}{V_0})(1-\frac{E}{V_0})exp(-2k_2a)
+$$
+
+## 2.4 Extensions of the Wave Theory to Atoms
+
+### 2.4.1 The One-Electron Atom
+
+the potential function of H
+$$
+V(r)=\frac{-e^2}{4\pi \epsilon_0 r}
+$$
+
+$$
+\grad ^2\phi(r,\theta,\phi)+\frac{2m_0}{\hbar^2}(E-V(r))\phi(r,\theta,\psi)=0
+$$
+
+$\phi$ can be written in the form of
+$$
+\phi(r,\theta,\phi)=R(r)\cdot\Theta(\theta)\cdot \Phi(\phi)
+$$
+ then, we obtain
+$$
+{\sin^2\theta\over R}\cdot\frac{\partial}{\partial r}(r^2\frac{\partial R}{\partial r})+\frac1{\Phi}\cdot\frac{\partial^2\phi}{\partial \phi^2}+\frac{\sin\theta}{\Theta}\cdot \frac{\partial}{\partial\theta}(\sin\theta\cdot \frac{\partial\Theta}{\partial\theta})+r^2\sin^2\theta\sdot\frac{2m_0}{\hbar^2}(E-V)=0
+$$
+
+$$
+\frac1{\Phi}\cdot\frac{\partial\Phi}{\partial\phi^2}=-m^2
+$$
+
+the solution is
+$$
+\phi=e^{jmp}
+$$
+m is integer, $m=0,\pm1,\pm2,\dots$
+
+we can further seperate the variables $\theta$ and $r$ and generate two additional seperation-of-variables constants l and n
+$$
+\text{the principle quantum number } n=1,2,3,\dots\\
+\text{the angular quantum number }l=n-1,n-2,n-3,\dots,0\\
+\text{the magnetic quantum number} \abs{m}=l,l-1,\dots,0
+$$
+the electron energy is,
+$$
+E_n=\frac{-m_0e^4}{(4\pi\epsilon_0)^2\cdot 2\hbar^2n^2}
+$$
diff --git a/docs/JY_FENG/Semiconductor Physics and Devices/3 Introduction to the Quantum Theory of Solids.md b/docs/JY_FENG/Semiconductor Physics and Devices/3 Introduction to the Quantum Theory of Solids.md
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+## 3.1 Allowed and Forbidden Energy Bands
+
+
+
+
+
+
+
+
+
+
+
+### 3.1.3 The $k$-Space Diagram
+
+the k comes from that all one-electron wave functions, for problems involving periodically varying potential energy functions, must be of the form $\psi(x)=u(x)e^{jkx}$
+$$
+P'\frac{\sin\alpha a}{\alpha a}+\cos \alpha a=\cos ka\\
+P'=\frac{mV_0ba}{\hbar^2}
+$$
+Equation gives the relation between the parameter $k$ total energy
+$E$ (through the parameter $\alpha$), and the potential barrier $bV_0$.
+
+We assume that $V_0=0$, then we acquire
+$$
+\cos \alpha a=\cos ka\\
+or\quad
+\alpha =k
+$$
+$\alpha$ is given by$\sqrt{\frac{2mE}{\hbar^2}}=\frac{p}{\hbar}=k$, so we obtain the relationship between $k$ and $E$
+$$
+E=\frac{k^2\hbar^2}{2m}
+$$
+
+
+Now, we consider the relation between $E$ and $k$ form Equation for the particle in the single-crystal lattice.
+
+define the left side of the Equation as
+$$
+f(\alpha a)=P'\frac{\sin\alpha a}{\alpha a}+\cos\alpha a
+$$
+to the right side, we also have,
+$$
+f(\alpha a)=\cos ka
+$$
+
+
+
+
+the right side can also be written as $\cos ka=\cos(ka+2n\pi)=\cos(ka-2n\pi)$
+
+then we gain
+
+
+
+## 3.2 Electrical Conduction In Solids
+
+### 3.2.1 The Energy Band and the Bond Model
+
+
+
+
+
+
+
+### 3.2.2 Drift Current
+
+the drift current density
+$$
+J=qNv_d\qquad A/cm^2\\
+or\quad J=q\sum^N_{i=1}v_i
+$$
+if a force is applied to a particle and the particle moves.
+$$
+dE=Fdx=Fv\cdot dt
+$$
+
+
+### 3.2.3 Electron Effective Mass
+
+$$
+F_{\text{total}}=F_{\text{ext}}+F_{\text{int}}=ma
+$$
+
+and we add $m^*$ as effective mass
+$$
+F_{ext}=m^*a
+$$
+then back to the Equation, we obtain,
+$$
+\frac{dE}{dk}=\frac{\hbar^2k}{m}=\frac{\hbar p}{m}\\
+\frac1{\hbar}\frac{dE}{dk}=\frac{p}m=v\\
+$$
+and, if we take the second derivative of $E$ with respect to $k$
+
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@@ -70,22 +70,28 @@ 数字逻辑设计 notes
-SOP：sum of minterms
-对函数求complement
-Implicant: squares in the map
- product term obtained by combining adjacent squares in the map into a rectangle with the number of squares a power of 2.
-A Prime Implicant is a product term obtained by combining the maximum possible number of adjacent squares in the map into a rectangle with the number of squares a power of 2.
-A prime implicant is called an Essential Prime Implicant if it is the only prime implicant that covers (includes) one or more minterms.
-简单来说就是如果没有这个圈有的格子就包裹不进去，就是 essential prime implicant
-1.3 implicant -蕴含项其实就是我们文章最开始提到的“圈”，每一种可能的圈就是一个蕴含项
-
上图中就有7个一次蕴含项，6个二次蕴含项和1个四次蕴含项，共计14个蕴含项
-1.4 prime implicant -质蕴含项就是不能与其它蕴含项合并的蕴含项，在1.3中，我们发现四次蕴含项中有4个二次蕴含项，那么它们就不是质蕴含项
-
可以看到上图中的每一个圈都不能和其它圈合并，所以上图有共计4个质蕴含项
-1.5 essential prime implicant -实质本源蕴含项中必须含有至少一个没被包含在其它蕴含项中的项，而且不能被更大的圈包裹
-
上图中有三个实质本源蕴含项，而我们找到最简SOP的方式就是找到所有的实质本源蕴含项
-cost criteria
-Literal cost L
-gate input cost G
-gate input cost with NOTs GN
-propagation delay \(t_{pd} = max{(t_{PLH},t_{PHL})}\)
-\(t_{PHL}\)是
-propagation : 传输
-分为transport delay 和 inertia delay（有rejection time属性）
-
三个串联？四个可以吗
-74LS00与非门传输延迟测量：3个74LS00门串联，4个可以吗？
-\((\overline{A B})\)
-
2-12怎么化简
-4位全加器的延时较长
-因此采用carry look ahead adder
-16位加法器使用4个CLA串联
-64位 组超前进位
-将G0~3, P0~3传给下一层？
-C4 = G3 +P3G2 + P3P2G1+P3P2P1G0+P3P2P1P0C0
- = G0~3 + P0~3*C0
-C8 = G4~7 + P4~7*C4
-同理，C12 = G8~11+ P8~11*C8
-C16 = G12~15+ P12~15*C12
-依赖关系和4位的CLA相似
-
小测1 复习
-1、逻辑表达式化简：
-\(A+\overline{A}B=(A+B)(A+\overline{A})=A+B\)
-\(逻辑表达式可重复使用\)
-2、门代价计算：
-3、卡诺图化简
-4变量的卡诺图，16个格子，圈的大小为8->1个变量
-
-$$
-T(N)=O(|V|^2)
-$$
-Improvement :
