24&25 秋冬期中历年卷

Impurity ion scattering is the deflection(偏转) of mobile charge carriers (electrons or holes) by the fixed, ionized impurity atoms in a semiconductor.

• Dopant Concentration: A higher concentration of dopant ions creates more scattering sites, which significantly increase Impurity Scattering and decreases carrier mobility.
• Temperature: This scattering effect is strong at low temperatures and becomes weaker as temperature increases, because carriers with higher thermal energy are less easily deflected.

The probability that a carrier will be scattered by mechanism i within a time period dt is $\frac{dt}{\tau_i}$

where $\tau_i$ is the mean time between scattering events due to mechanism $i$

The probability that a carrier will be scattered within a time period dt is $\sum_i\frac{dt}{\tau_i}$(assume that each mechanism i is independent to each other)

So the total possibility of carrier scattering is $dt / \tau_{\text{total}} = \sum_i\frac{dt}{\tau_i}$

Then we get $1/\tau_{\text{total}} = 1/\tau_{\text{phonon}} + 1/\tau_{\text{impurity}}$

As $\mu=\frac{qt}{m^*}$, we could get $1/\mu_{\text{total}} = 1/\mu_{\text{phonon}} + 1/\mu_{\text{impurity}}$

The effective mass ($m^*$) is the mass parameter that makes an electron in a periodic crystal respond to forces(在周期性晶格势场中对于外界力量的质量参数) (electric, magnetic) as if it were a free particle.

It is influenced by:

1. The Material: Its value is determined by the material's unique electronic band structure.
2. Crystal Direction: Different crystal directions have different band curvature (能带弯曲)→ different $m^*$.
3. Strain: Changes lattice spacing/symmetry(改变晶格间距/对称性), reshaping band curvature ($\frac{d^2E}{dk^2}$) → different $m^*$.

Three Ways to Increase Mobility:

1. Decrease Dopant Concentration: Reduces impurity scattering.
2. Lower the lattice temperature to suppress(抑制) phonon scattering (phonon density↓, mobility↑).
3. Use Strain Engineering: Straining the crystal lattice (e.g. strained silicon) can alter the band structure to reduce effective mass, thereby improving mobility

Impact on Device Performance:

Higher carrier mobility directly results in higher current drive ($I \propto \mu$) and lower resistance. This allows transistors to switch faster, leading to higher operating frequencies ($f_T$) and improved overall device speed.

When the kinetic energy(动能) of a carrier exceeds a critical value(达到某个临界值), it generates an optical phonon(光学声子) and loses the kinetic energy. Therefore, the kinetic energy is capped at large E, and the velocity does not rise above a saturation velocity, $v_{sat}$

In short-channel transistors, this effect is what causes the drain current to saturate, limiting the maximum current and switching speed.

• Drift

charged particle motion under the influence of an electric field.

• Diffusion

particle motion due to concentration gradient or temperature gradient.

• Thermal velocity is the random motion(随机) of carriers in all directions due to thermal energy, while drift velocity is the small average velocity (很小的平均漂移速度) superimposed(叠加) on this random motion when an electric field is applied.
• Thermal velocity depends on temperature and effective mass $V_{th} = \sqrt{\frac{3kT}{m^*_n}}$. But drift velocity depends on the electric field's value , meaning free time and effective mass $|V_{d}| = \frac{qE\tau_{min}}{m^*_n}$.

Carrier generation is the process that creates electron–hole pairs in a semiconductor by exciting electrons from the valence band to the conduction band (possibly via defect levels可能经过缺陷能级).

Three possible mechanisms

• Band-to-Band Generation（带-带产生）Thermal energy or photons with energy $\ge E_g$ directly excite an electron from the valence band $E_v$ to the conduction band $E_c$, creating an electron–hole pair.
  • Thermal: energy comes from lattice vibrations (heat/phonons).
  • Optical (photogeneration): energy comes from absorbed photons(注意这是photon不是phonon).

然后可以画个图：

• R-G Center Generation（通过 R-G 中心产生）Recombination–generation (R-G) centers are defect(缺陷)/impurity(杂质)/surface states(表面态) that introduce an energy level $E_T$ inside the bandgap. With thermal energy, carriers are generated in two steps via this level (e.g. $E_v \rightarrow E_T \rightarrow E_c$), producing an electron–hole pair. R-G centers are a major source of thermal generation and of reverse saturation current in devices.

• Impact Ionization（碰撞电离） In a strong electric field, carriers are accelerated and gain high kinetic energy between collisions. A “hot” carrier can collide with the lattice and knock an electron from the valence band to the conduction band, creating a new electron–hole pair. This is the basic mechanism behind avalanche multiplication and avalanche breakdown.

• On the n-side (n-type region):

The injected excess hole concentration(注入过剩空穴浓度) is much smaller than the equilibrium majority electron concentration:

so the total electron concentration is essentially unchanged, $n_n \approx n_{n0}$.

• On the p-side (p-type region):

The injected excess electron concentration is much smaller than the equilibrium majority hole concentration:

so the total hole concentration is essentially unchanged, $p_p \approx p_{p0}$

Excess carriers are the extra electrons and holes above their thermal-equilibrium values in non-equilibrium concentration, defined as

where in a charge-neutral region $\Delta n = \Delta p$

Electric field, magnetic field, mechanical stress and light.

On the n-side, n is the majority carrier, p is the minority carrier.

On the p-side, p is the majority carrier, n is the minority carrier.

Quasi-Fermi levels are separate effective Fermi levels for electrons and holes used when a semiconductor is out of thermal equilibrium. They are defined so that the carrier concentrations can still be written as

In equilibrium, $E_{Fn}=E_{Fp}=E_F$; under bias they split and describe non-equilibrium electron and hole populations separately.

首先画出平衡状态下的能带图，因为thermal equilibrium, 费米能级是相同的，然后越接近$E_c$说明电子越多，画出P区与N区，中间弯折，注意一下$E_i=\frac{E_c+E_v}{2}$也会一起弯折

正向偏置的时候，P与N之间的势垒减小，能带曲线更平缓，$E_{Fn}$也跟随着往上走，$E_{FP}$跟随着往下走

反向偏置的时候，P与N之间的势垒增大，能带曲线更陡峭，$E_{Fn}$也跟随着往下走，$E_{FP}$跟随着往上走