MM 765: Semiconductor Devices

1D Nonrelativistic Schrödinger's Equation

Time-Dependent Form (1926)

- \frac{ℏ^{2}}{2 m} \frac{\partial^{2} Ψ (x, t)}{\partial x^{2}} + V (x) Ψ (x, t) = j ℏ \frac{\partial Ψ (x, t)}{\partial t}

By separating the time and position dependent variables, assume that

Ψ (x, t) = ψ (x) ϕ (t)

Simplifying Schrödinger's equation,

\frac{- ℏ^{2}}{2 m} ϕ (t) \frac{d^{2} ψ (x)}{d x^{2}} + V (x) ψ (x) ϕ (t) = j ℏ ψ (x) \frac{d ϕ (t)}{d t}

Dividing both sides by total wave function $Ψ (x, t) = ψ (x) ϕ (t)$ ,

\frac{- ℏ^{2}}{2 m} \frac{1}{ψ (x)} \frac{d^{2} ψ (x)}{d x^{2}} + V (x) = j ℏ \frac{1}{ϕ (t)} \frac{d ϕ (t)}{d t}

LHS function of $x$ only, RHS function of $t$ only

Each side must be equal to constant

Solving time-dependent portion:

η = j ℏ \frac{1}{ϕ (t)} \frac{d ϕ (t)}{d t}

, where

η

is a separation constant

Now,

E = h ν = h \frac{ω}{2 π} = η

Separation constant is equal to E (total energy of the particle)

\frac{d^{2} ψ (x)}{d x^{2}} + \frac{2 m}{ℏ^{2}} (E - V (x)) ψ (x) = 0

Physical Meaning of Total Wave Function

Total wave function:

Ψ (x, t) = ψ (x) ϕ (t) = ψ (x) e^{- j ω t}

Max Born in 1926: The function ${| Ψ (x, t) |}^{2}$ is the probability of finding the particle between $x$ and $x + dx$ at a given time.

Particle in an Infinite Potential Well

"I could be bounded in a nutshell, and count myself a king of infinite space..."

Consider a particle trapped in a rigid box (potential well). In classical mechanics, it bounces back and forth with any energy. Quantum mechanics reveals a different reality dictated by confinement.

The Mathematical Solution

The Time-independent Schrödinger Equation (S.E.) is:

\frac{d^2\psi(x)}{dx^2} + \frac{2m}{\hbar^2}(E - V(x))\psi(x) = 0

Applying Boundary Conditions

At $ x = 0 $, $ \psi(0) = 0 \implies A = -B $.
At $ x = a $, $ \psi(a) = 0 \implies \sin(ka) = 0 $. This requires $ ka = n\pi $ (where $ n = 1, 2, 3... $).
Applying the normalization conditionThe total probability of finding the particle within the well (0 to a) must be 1. $ \int_{0}^{a} \psi\psi^* dx = 1 $.

Final Wave Function

\[ \psi_n(x) = \sqrt{\frac{2}{a}} \sin\left(\frac{n\pi x}{a}\right) \]

Interactive States Explorer

Select Energy Level

Wave Function $\psi_n(x)$

Probability Density $|\psi_n(x)|^2$

Potential Barrier & Tunnelling

"Alice laughed. 'There's no use trying,' she said. 'One can't believe impossible things.'
'I daresay you haven't had much practice,' said the Queen."

The Setup

A potential barrier (Region II) of width '$ a $' and height $ V_0 = 20 \text{ eV} $. Particles approach from Region I with energy $ E < V_0 $.

The Transmission Coefficient (T) is the ratio of transmitted to incident flux. For $ E \ll V_0 $:

\[ T \approx 16\left(\frac{E}{V_0}\right)\left(1 - \frac{E}{V_0}\right)e^{-2k_2a} \]

Tunnelling Simulator

Barrier Width $ a $ (Å) 3.0

Incident Energy $ E $ (eV) 2.0

Probability (T): 3.17e-6

The One-Electron Atom

We now extend our quantum mechanical toolkit from simple 1D boxes to the three-dimensional, spherically symmetric world of a one-electron atom (like Hydrogen).

The general Schrödinger equation incorporates the Laplacian operator $\nabla^2$, taking into account the effective mass $m_0$ (strictly the reduced mass $\mu \approx m_0$ since $m_p \gg m_0$):

\nabla^2\psi(r,\theta,\phi) + \frac{2m_0}{\hbar^2}(E - V(r))\psi(r,\theta,\phi) = 0

Deriving the Quantum Numbers

Step 1: Separation of Variables

General 3D Schrödinger Equation

\[ \nabla^2\psi + \frac{2m_0}{\hbar^2}(E - V(r))\psi = 0 \]

Assume Separable Solution

Because the Coulomb potential $V(r)$ is spherically symmetric.

\[ \psi(r, \theta, \phi) = R(r) \cdot \Theta(\theta) \cdot \Phi(\phi) \]

Radial Eq. $R(r)$

Colatitude Eq. $\Theta(\theta)$

Azimuthal Eq. $\Phi(\phi)$

Step 2: Magnetic Quantum Number ($m_l$)

Azimuthal Differential Equation

\[ \frac{d^2\Phi}{d\phi^2} + m_l^2 \Phi = 0 \implies \Phi(\phi) = A e^{j m_l \phi} \]

Physical Boundary Condition

Wavefunction must match itself exactly after a full $360^\circ$ ($2\pi$) rotation around the axis:

\[ \Phi(\phi) = \Phi(\phi + 2\pi) \]

Quantization Result

$ m_l = 0, \pm1, \pm2, ... $

($m_l$ must be an integer for the complex exponential equality to hold)

Step 3: Azimuthal Quantum Number ($l$)

Colatitude Equation ($\theta$)

Forms the mathematically rigorous Associated Legendre Differential Equation.

Physical Boundary Condition

Solution must remain finite (not blow up to infinity) at the mathematical poles ($\theta = 0, \pi$).

Legendre Polynomial Constraint: $ l \ge |m_l| $

Quantization Result

$ l = 0, 1, 2, ... $

($l$ is a positive integer strictly bounding the allowed magnetic states)

Step 4: Principal Quantum Number ($n$)

Radial Equation ($r$)

Incorporates the Coulomb potential $V(r)$. Solved via power series expansion, forming Associated Laguerre Polynomials.

Physical Boundary Condition

To keep the electron bounded to the nucleus, the wavefunction must decay to zero as $r \to \infty$. The infinite power series must therefore terminate.

Quantization Result

\[ E_n \propto \frac{1}{n^2} \]

$ n = 1, 2, 3, ... $

($n$ must be a positive integer where $n > l$)

$ l $ (Azimuthal / Angular)

Must be a non-negative integer ($0, 1, 2...$). It dictates the magnitude of orbital angular momentum. Constrained by: $ m_l \le l $.

$ n $ (Principal)

Takes positive integral values ($1, 2, 3...$). Primarily determines the total energy of the electron orbit.

Radial Probability Density Function $ P(r) $

The probability of finding an electron at a specific distance $r$ is proportional to $\psi \psi^*$ multiplied by the differential volume of the spherical shell ($4\pi r^2 dr$).

Highest probability occurs exactly at $ r = a_0 $

Spin & Pauli's Exclusion Principle

While Schrödinger’s Equation provides $n, l,$ and $m_l$, empirical observations of emission spectra splitting required a fourth quantum number.

Spin Quantum Number (s)

Indicates intrinsic angular momentum. It can only take values of $+\frac{1}{2}$ or $-\frac{1}{2}$.

"In any given system... no two electrons may occupy the same quantum state."

Reference Texts

1

Principles of Semiconductor Devices (2nd Edition)

Sima Dimitrijev, Oxford University Press, 2012.
2

Solid State Electronic Devices (7th Edition)

Ben G. Streetman and S. K. Banerjee, Pearson, 2016.

Practice Assessment

Lecture 2 Knowledge Check

Put your understanding of the energy-band model and quantum mechanics to the test.

Start Assessment

The Energy-Band Model
& Quantum Mechanics

1D Nonrelativistic Schrödinger's Equation

Physical Meaning of Total Wave Function

Particle in an Infinite Potential Well

The Mathematical Solution

Interactive States Explorer

Select Energy Level

Wave Function \(\psi_n(x)\)

Probability Density \(|\psi_n(x)|^2\)

Potential Barrier & Tunnelling

The Setup

Tunnelling Simulator

The One-Electron Atom

Deriving the Quantum Numbers

Step 1: Separation of Variables

Step 2: Magnetic Quantum Number (\(m_l\))

Step 3: Azimuthal Quantum Number (\(l\))

Step 4: Principal Quantum Number (\(n\))

\( l \) (Azimuthal / Angular)

\( n \) (Principal)

Radial Probability Density Function \( P(r) \)

Spin & Pauli's Exclusion Principle

Spin Quantum Number (s)

Reference Texts

Principles of Semiconductor Devices (2nd Edition)

Solid State Electronic Devices (7th Edition)

Lecture 2 Knowledge Check