Semiconductor Manufacturing Plasma Science

Overview

This document covers the physics, chemistry, and engineering of plasma processes in semiconductor manufacturing—the foundation of modern chip fabrication.

1. Fundamentals of Plasma Physics

1.1 What is Plasma?

Plasma is the fourth state of matter—an ionized gas containing:

• Free electrons ($e^-$)
• Positive ions ($\text{Ar}^+$, $\text{Cl}^+$, $\text{F}^+$, etc.)
• Neutral species (atoms, molecules, radicals)

In semiconductor processing, we use non-equilibrium or cold plasmas where:

$$ T_e \gg T_i \approx T_n \approx T_{\text{room}} $$

Where:

• $T_e$ = electron temperature (~1–10 eV, equivalent to $10^4$–$10^5$ K)
• $T_i$ = ion temperature (~0.025–0.1 eV)
• $T_n$ = neutral temperature (~300 K)

This asymmetry allows chemically reactive species to be generated without thermally damaging the substrate.

1.2 Key Plasma Parameters

1.3 Debye Length

The Debye length characterizes the distance over which charge separation can occur:

$$ \lambda_D = \sqrt{\frac{\varepsilon_0 k_B T_e}{n_e e^2}} $$

Where:

• $\varepsilon_0$ = permittivity of free space ($8.85 \times 10^{-12}$ F/m)
• $k_B$ = Boltzmann constant ($1.38 \times 10^{-23}$ J/K)
• $T_e$ = electron temperature (K)
• $n_e$ = electron density (m$^{-3}$)
• $e$ = electron charge ($1.6 \times 10^{-19}$ C)

1.4 Plasma Frequency

The plasma frequency is the natural oscillation frequency of electrons:

$$ \omega_{pe} = \sqrt{\frac{n_e e^2}{\varepsilon_0 m_e}} $$

Or in practical units:

$$ f_{pe} \approx 9 \sqrt{n_e} \text{ Hz} \quad \text{(with } n_e \text{ in m}^{-3}\text{)} $$

2. The Plasma Sheath

2.1 Sheath Formation

The plasma sheath is the most critical region for semiconductor processing. At any surface in contact with plasma:

1. Electrons (lighter, faster) escape more readily than ions 2. A positive space charge region forms adjacent to the surface 3. This creates a potential drop that accelerates ions toward the substrate

2.2 Sheath Potential

The Bohm criterion requires ions entering the sheath to have a minimum velocity:

$$ v_{\text{Bohm}} = \sqrt{\frac{k_B T_e}{M_i}} $$

Where $M_i$ is the ion mass.

The floating potential (potential of an isolated surface) is approximately:

$$ V_f \approx -\frac{k_B T_e}{2e} \ln\left(\frac{M_i}{2\pi m_e}\right) $$

For argon plasma with $T_e = 3$ eV:

$$ V_f \approx -15 \text{ V} $$

2.3 Child-Langmuir Law

The ion current density through a collisionless sheath is given by:

$$ J_i = \frac{4\varepsilon_0}{9} \sqrt{\frac{2e}{M_i}} \frac{V^{3/2}}{d^2} $$

Where:

• $V$ = sheath voltage
• $d$ = sheath thickness

2.4 Sheath Thickness

The sheath thickness scales approximately as:

$$ s \approx \lambda_D \left(\frac{2eV_s}{k_B T_e}\right)^{3/4} $$

Where $V_s$ is the sheath voltage.

3. Plasma Etching

3.1 Etching Mechanisms

Three primary mechanisms contribute to plasma etching:

1. Chemical etching (isotropic): $$ \text{Rate}_{\text{chem}} \propto \Gamma_n \cdot S \cdot \exp\left(-\frac{E_a}{k_B T_s}\right) $$ Where $\Gamma_n$ is neutral flux, $S$ is sticking coefficient, $E_a$ is activation energy

2. Physical sputtering (anisotropic): $$ Y(E) = \frac{0.042 \cdot Q \cdot \alpha^* \cdot S_n(E)}{U_s} $$ Where $Y$ is sputter yield, $E$ is ion energy, $U_s$ is surface binding energy

3. Ion-enhanced etching (synergistic): $$ \text{Rate}_{\text{total}} > \text{Rate}_{\text{chem}} + \text{Rate}_{\text{phys}} $$

3.2 Etch Rate Equation

A general expression for ion-enhanced etch rate:

$$ \text{ER} = \frac{1}{n} \left[ k_s \Gamma_n \theta + Y_{\text{phys}} \Gamma_i + Y_{\text{ion}} \Gamma_i (1-\theta) + Y_{\text{chem}} \Gamma_i \theta \right] $$

Where:

• $n$ = atomic density of material
• $\Gamma_n$ = neutral flux
• $\Gamma_i$ = ion flux
• $\theta$ = surface coverage of reactive species
• $Y$ = yield coefficients

3.3 Ion Energy Distribution Function (IEDF)

For sinusoidal RF bias, the IEDF is bimodal with peaks at:

$$ E_{\pm} = eV_{dc} \pm eV_{rf} \cdot \frac{\omega_{pi}}{\omega_{rf}} $$

Where:

• $V_{dc}$ = DC self-bias voltage
• $V_{rf}$ = RF amplitude
• $\omega_{pi}$ = ion plasma frequency
• $\omega_{rf}$ = RF frequency

The peak separation:

$$ \Delta E = 2eV_{rf} \cdot \frac{\omega_{pi}}{\omega_{rf}} $$

3.4 Common Etch Chemistries

3.5 Selectivity

Selectivity is the ratio of etch rates between target and mask (or underlayer):

$$ S = \frac{\text{ER}_{\text{target}}}{\text{ER}_{\text{mask}}} $$

For oxide-to-nitride selectivity in fluorocarbon plasmas:

$$ S_{\text{ox/nit}} = \frac{\text{ER}_{\text{SiO}_2}}{\text{ER}_{\text{Si}_3\text{N}_4}} \propto \frac{[\text{F}]}{[\text{CF}_x]} $$

4. Plasma Sources

4.1 Capacitively Coupled Plasma (CCP)

Configuration: Parallel plate electrodes with RF power

Power absorption: Primarily through stochastic (collisionless) heating:

$$ P_{\text{stoch}} \propto \frac{m_e v_e^2 \omega_{rf}^2 s_0^2}{v_{th,e}} $$

Where $s_0$ is the sheath oscillation amplitude.

Dual-frequency operation:

• High frequency (27–100 MHz): Controls plasma density
• Low frequency (100 kHz–13 MHz): Controls ion energy

Ion energy scaling:

$$ \langle E_i \rangle \propto \frac{V_{rf}^2}{n_e^{0.5}} $$

4.2 Inductively Coupled Plasma (ICP)

Power transfer: Through induced electric field from RF current in coil:

$$ E_\theta = -\frac{\partial A_\theta}{\partial t} = j\omega A_\theta $$

Skin depth (characteristic penetration depth of fields):

$$ \delta = \sqrt{\frac{2}{\omega \mu_0 \sigma_p}} $$

Where $\sigma_p$ is plasma conductivity:

$$ \sigma_p = \frac{n_e e^2}{m_e u_m} $$

Power density:

$$ P = \frac{1}{2} \text{Re}(\sigma_p) |E|^2 $$

Advantages:

• Higher plasma density: $10^{11}$–$10^{12}$ cm$^{-3}$
• Lower operating pressure: 1–50 mTorr
• Independent control of ion flux and energy

4.3 Plasma Density Comparison

5. Plasma-Enhanced Deposition

5.1 PECVD Fundamentals

Reaction rate in PECVD:

$$ R = k_0 \exp\left(-\frac{E_a}{k_B T_{eff}}\right) [A]^a [B]^b $$

Where $T_{eff}$ is an effective temperature combining gas and electron contributions.

The plasma reduces the effective activation energy by providing:

• Electron-impact dissociation
• Ion bombardment energy
• Radical species

5.2 Common PECVD Reactions

Silicon dioxide from silane and nitrous oxide:

$$ \text{SiH}_4 + 2\text{N}_2\text{O} \xrightarrow{\text{plasma}} \text{SiO}_2 + 2\text{N}_2 + 2\text{H}_2 $$

Silicon nitride from silane and ammonia:

$$ 3\text{SiH}_4 + 4\text{NH}_3 \xrightarrow{\text{plasma}} \text{Si}_3\text{N}_4 + 12\text{H}_2 $$

Amorphous silicon:

$$ \text{SiH}_4 \xrightarrow{\text{plasma}} a\text{-Si:H} + 2\text{H}_2 $$

5.3 Film Quality Parameters

Film stress in PECVD films:

$$ \sigma = \frac{E_f}{1- u_f} \left( \alpha_s - \alpha_f \right) \Delta T + \sigma_{\text{intrinsic}} $$

Where:

• $E_f$ = film Young's modulus
• $

u_f$ = film Poisson's ratio

• $\alpha_s, \alpha_f$ = thermal expansion coefficients (substrate, film)
• $\sigma_{\text{intrinsic}}$ = intrinsic stress from deposition process

5.4 Plasma-Enhanced ALD (PEALD)

Growth per cycle (GPC):

$$ \text{GPC} = \frac{\theta_{\text{sat}} \cdot \Omega}{A_{\text{site}}} $$

Where:

• $\theta_{\text{sat}}$ = saturation coverage
• $\Omega$ = molecular volume
• $A_{\text{site}}$ = area per reactive site

Self-limiting behavior requires:

$$ \Gamma_{\text{precursor}} \cdot t_{\text{pulse}} > \frac{N_{\text{sites}}}{S_0} $$

Where $S_0$ is the initial sticking coefficient.

6. Advanced Topics

6.1 Aspect Ratio Dependent Etching (ARDE)

Etch rate decreases with increasing aspect ratio due to:

1. Ion shadowing: Reduced ion flux at feature bottom 2. Neutral transport: Knudsen diffusion limitation 3. Product redeposition: Reduced volatile product escape

Knudsen number for feature transport:

$$ Kn = \frac{\lambda}{w} $$

Where $\lambda$ is mean free path, $w$ is feature width.

For $Kn > 1$ (molecular flow regime):

$$ \Gamma_{\text{bottom}} = \Gamma_{\text{top}} \cdot K(\text{AR}) $$

Where $K(\text{AR})$ is the Clausing factor, approximately:

$$ K(\text{AR}) \approx \frac{1}{1 + \frac{3}{8}\text{AR}} $$

For high aspect ratio features.

6.2 Atomic Layer Etching (ALE)

Self-limiting surface modification:

$$ \theta(t) = \theta_{\text{sat}} \left[1 - \exp\left(-\frac{t}{\tau}\right)\right] $$

Etch per cycle (EPC):

$$ \text{EPC} = \frac{N_{\text{modified}} \cdot a}{n_{\text{film}}} $$

Where:

• $N_{\text{modified}}$ = surface density of modified atoms
• $a$ = atoms removed per modified site
• $n_{\text{film}}$ = atomic density of film

6.3 Plasma-Induced Damage

Charging damage occurs when:

$$ V_{\text{antenna}} = \frac{J_e - J_i}{C_{\text{gate}}/A_{\text{antenna}}} \cdot t > V_{\text{breakdown}} $$

Antenna ratio limit:

$$ \text{AR}_{\text{antenna}} = \frac{A_{\text{antenna}}}{A_{\text{gate}}} < \text{AR}_{\text{critical}} $$

UV damage from vacuum UV photons ($\lambda < 200$ nm):

$$ N_{\text{defects}} \propto \int I(\lambda) \cdot \sigma(\lambda) \cdot d\lambda $$

7. Plasma Diagnostics

7.1 Langmuir Probe Analysis

Electron density from ion saturation current:

$$ n_e = \frac{I_{i,sat}}{0.61 \cdot e \cdot A_p \cdot \sqrt{\frac{k_B T_e}{M_i}}} $$

Electron temperature from the exponential region:

$$ T_e = \frac{e}{k_B} \left( \frac{d(\ln I_e)}{dV} \right)^{-1} $$

EEDF from second derivative of I-V curve:

$$ f(\varepsilon) = \frac{2m_e}{e^2 A_p} \sqrt{\frac{2\varepsilon}{m_e}} \frac{d^2 I}{dV^2} $$

7.2 Optical Emission Spectroscopy (OES)

Actinometry for radical density measurement:

$$ \frac{n_X}{n_{\text{Ar}}} = \frac{I_X}{I_{\text{Ar}}} \cdot \frac{\sigma_{\text{Ar}} \cdot Q_{\text{Ar}}}{\sigma_X \cdot Q_X} $$

Where:

• $I$ = emission intensity
• $\sigma$ = electron-impact excitation cross-section
• $Q$ = quantum efficiency

8. Process Control Equations

8.1 Residence Time

$$ \tau_{\text{res}} = \frac{p \cdot V}{Q \cdot k_B T} $$

Where:

• $p$ = pressure
• $V$ = chamber volume
• $Q$ = gas flow rate (sccm converted to molecules/s)

8.2 Mean Free Path

$$ \lambda = \frac{k_B T}{\sqrt{2} \pi d^2 p} $$

For argon at 10 mTorr and 300 K:

$$ \lambda \approx 0.5 \text{ cm} $$

8.3 Power Density

Effective power density at wafer:

$$ P_{\text{eff}} = \frac{\eta \cdot P_{\text{source}}}{A_{\text{wafer}}} $$

Where $\eta$ is power transfer efficiency (typically 0.3–0.7).

9. Critical Equations