Thermal Evaporation Physics
Thermal evaporation heats materials above their sublimation/evaporation temperature in high vacuum. Atoms/molecules escape the melt surface, travel ballistically, and condense on substrates.
Vapor Pressure - Clausius-Clapeyron
log₁₀(P_vapor) = A - B/T
P_vapor: equilibrium vapor pressure (Torr)
T: absolute temperature (K)
A, B: material constants
Examples at 1400°C (1673 K):
Al: A=12.95, B=15,900 → P≈10⁻³ Torr
Au: A=12.23, B=19,710 → P≈10⁻⁵ Torr
Cu: A=12.24, B=17,520 → P≈10⁻⁴ Torr
Ag: A=12.01, B=14,850 → P≈10⁻³ Torr
Rule of thumb: Need T where P > 10⁻² Torr
for practical evaporation rates
Evaporation Rate - Langmuir Equation
Γ_evap = α × P_vapor × √(M / (2πRT))
Γ_evap: molecular flux (molecules/cm²/s)
α: evaporation coefficient (0.01-1.0)
M: molecular mass (g/mol)
R: gas constant = 8.314 J/(mol·K)
Mass evaporation rate:
m_dot = Γ_evap × M × A_source
Thickness rate at source:
R_evap = (Γ_evap × M) / (N_A × ρ)
N_A: Avogadro's number
ρ: film density
Typical R_evap: 1-100 Å/s
(depends on T, material)
Mean Free Path - Knudsen Number
λ_mfp = kT / (√2 × π × d² × P)
d: molecular diameter (≈ 3 Å)
k: Boltzmann constant
For T=300K, d=3Å:
P = 10⁻³ Torr → λ_mfp ≈ 50 m
P = 10⁻⁵ Torr → λ_mfp ≈ 5000 m
P = 10⁻⁷ Torr → λ_mfp ≈ 500 km
Knudsen number:
Kn = λ_mfp / L_chamber
Kn >> 1: Molecular flow (ballistic)
Kn << 1: Viscous flow (collisional)
Evaporation requires Kn > 10
→ P < 10⁻⁴ Torr typical
Angular Distribution - Cosine Law
Flux density at angle θ:
Φ(θ) = Φ₀ × cos^n(θ)
θ: angle from source normal
For ideal point source:
n = 1 (Lambert's cosine law)
For extended source:
n > 1 (over-cosine)
n ≈ 1-3 (typical thermal sources)
Directional e-beam:
n ≈ 2-5 (focused beam)
Physical origin:
Surface atoms evaporate preferentially
along surface normal (momentum conservation)
Thickness Distribution on Substrate
Point source approximation:
d(r) = d₀ × cos^n(α) × cos(β) / h²
α: angle at source to point r
β: angle at substrate to point r
h: source-substrate distance
r: lateral distance from axis
On-axis (r=0):
d(0) = d₀
Off-axis:
cos(α) = h / √(h² + r²)
cos(β) = h / √(h² + r²) (parallel)
d(r) / d(0) = (1 + r²/h²)^(-(n+2)/2)
Uniformity degrades as r increases
Substrate Rotation for Uniformity
Planetary rotation:
• Substrate holder rotates
• Individual substrates rotate
Time-averaged thickness:
d_avg = (1/T) ∫₀^T d(r(t)) dt
For circular rotation at radius R:
r(t) = R + r₀ cos(ωt)
Uniformity improvement:
σ/μ < 2% (with rotation)
vs σ/μ > 20% (no rotation)
Optimal conditions:
R/h ≈ 0.5-0.7
ω × t_dep >> 10 revolutions
Electron Beam Evaporation
E-beam heating:
P_beam = V_accel × I_beam × η
V_accel: 5-10 kV
I_beam: 0.1-1 A
η: coupling efficiency (0.6-0.9)
Advantages vs thermal:
• Higher temperatures (>3000°C)
• Refractory materials (W, Ta, Mo)
• Localized heating (less contamination)
• Water-cooled crucible (less impurities)
Beam sweeping:
Magnetic deflection for uniform melt
Sweep frequency: 100-1000 Hz
Typical power density:
10⁴-10⁶ W/cm² at melt surface
Multi-Component Evaporation
For alloy or compound AB:
Evaporation rates differ:
Γ_A ≠ Γ_B
Film composition:
x_A(film) = Γ_A / (Γ_A + Γ_B)
≠ x_A(source) due to:
1. Different vapor pressures
2. Different sticking coefficients
3. Melt composition changes
Solutions:
• Co-evaporation (separate sources)
• Flash evaporation (fast heating)
• Composition monitoring (EQCM)
• Feedback control
Sticking Coefficient
s = (atoms stuck) / (atoms arriving)
Temperature dependent:
s(T) = s₀ × exp(-E_des / kT)
E_des: desorption energy
For metals on cold substrate:
s ≈ 1 (every atom sticks)
For low T_melt materials:
Need T_substrate << T_melt
e.g., Ag at RT: s ≈ 0.5
For molecules (organics):
s < 1 (re-evaporation)
Need substrate cooling
Condensation coefficient:
Often assumed s=1 for PVD
Film Growth Modes
Volmer-Weber (island growth):
γ_film > γ_substrate + γ_interface
Common: metals on insulators
Frank-van der Merwe (layer-by-layer):
γ_film + γ_interface < γ_substrate
Common: epitaxial growth
Stranski-Krastanov (mixed):
Few layers, then islands
Strain-driven
Adatom mobility:
D_surface ∝ exp(-E_diff / kT_substrate)
High T_substrate:
→ Smooth, large grains
Low T_substrate:
→ Rough, columnar, small grains
Deposition Rate Control
Quartz Crystal Microbalance (QCM):
Δf / f₀ = -(ρ_film × d_film) / (ρ_quartz × d_quartz)
Frequency shift:
Δf ∝ mass per unit area
Sensitivity:
6 MHz crystal: ~1 Å resolution
Feedback control:
PID loop adjusts source power
to maintain setpoint rate
Accuracy: ±1-2% thickness
Reproducibility: ±0.5%
Tooling factor:
Corrects QCM vs substrate geometry
Tf = d_substrate / d_QCM
Residual Gas Effects
Impurity incorporation:
At P = 10⁻⁶ Torr, Γ_gas ≈ 10¹⁶ cm⁻²s⁻¹
For R_dep = 10 Å/s ≈ 10¹⁵ atoms/cm²/s:
Gas/deposit ratio ≈ 10:1
If sticking s_gas = 0.01:
Impurity level ≈ 0.1 atomic%
Main contaminants:
H₂O → OH, O in film
CO, CO₂ → C, O in film
N₂ → N in film
Require:
Base pressure < 10⁻⁷ Torr
for high purity films (<0.01% impurity)
Energy of Arriving Atoms
Thermal evaporation:
E_atom ≈ 3kT_source / 2
For T_source = 1400°C = 1673K:
E_atom ≈ 0.22 eV
Very low energy compared to:
• Sputtering: 1-100 eV
• IBAD: 100-1000 eV
Consequences:
• Limited adatom mobility
• Porous films (low T_substrate)
• Poor step coverage
• High defect density
Need T_substrate > 0.3 T_melt
for dense, smooth films
Crucible Materials and Compatibility
Crucible selection depends on the material being evaporated, temperature requirements, and contamination constraints. Different evaporants require specific crucible materials to prevent reactions.
Crucible material selection:
Tungsten (W):
• T_max = 3000°C
• Compatible: Al, Cu, Au, Ag
• Incompatible: Si, reactive metals
• Pros: High temp, low cost
• Cons: Brittle, oxidizes
Molybdenum (Mo):
• T_max = 2600°C
• Compatible: Most metals
• Incompatible: Si, carbon
• Pros: Machinable, reusable
• Cons: Expensive
Graphite (C):
• T_max = 3000°C (vacuum)
• Compatible: Al, Cr, Fe
• Incompatible: Ti, Zr, carbide formers
• Pros: High temp, inexpensive
• Cons: Carbon contamination
Alumina (Al₂O₃):
• T_max = 2000°C
• Compatible: Most materials
• Incompatible: Si at high T
• Pros: Inert, insulating
• Cons: Thermal shock sensitive
BN (Boron Nitride):
• T_max = 2000°C
• Compatible: Most materials
• Pros: Inert, machinable
• Cons: Expensive
Electron Beam Gun Design
E-beam evaporation uses magnetically deflected electron beams to heat materials in water-cooled crucibles, enabling deposition of refractory materials with minimal contamination.
E-beam gun parameters:
Accelerating voltage: V_accel = 5-10 kV
Beam current: I_beam = 0.1-1 A
Power: P = V × I = 0.5-10 kW
Electron energy:
E_e = e × V_accel (eV)
Penetration depth (Kanaya-Okayama):
R = (0.0276 × A × E_e^1.67) / (ρ × Z^0.89)
A: atomic mass
ρ: density (g/cm³)
Z: atomic number
For 10 keV e⁻ on Cu:
R ≈ 1-2 μm
Magnetic deflection:
r_beam = (m_e × v) / (e × B)
B: magnetic field (Tesla)
Typical deflection:
B ≈ 0.01-0.1 T
θ_deflect = 90-270° (bent beam)
Beam sweeping:
f_sweep = 100-1000 Hz
Pattern: raster or circular
Purpose: uniform melt
Power density:
P_density = P / A_spot
A_spot ≈ 0.1-1 cm²
P_density ≈ 10⁴-10⁶ W/cm²
Multi-Source Co-Evaporation
Simultaneous evaporation from multiple sources enables alloy deposition, multilayers, and compositional gradients with precise flux control.
Co-evaporation from n sources:
Γ_total = Σ Γ_i × cos(θ_i) / r_i²
Film composition:
x_A = Γ_A / (Γ_A + Γ_B + ...)
Thickness rate:
R_total = Σ (Γ_i × M_i) / (ρ_film × N_A)
Source positioning:
• Angular separation: 30-60°
• Height staggering
• Rotation for uniformity
Flux calibration:
• Individual QCM sensors
• Tooling factors per source
• Cross-contamination minimized
Feedback control:
PID loops per source
Adjust T or power
Real-time composition
Applications:
• Ternary alloys (CuInGa)
• Multilayers (Al/Cr stack)
• Graded buffers (SiGe)
• Doped films (Al:ZnO)
Challenges:
• Cross-talk between sources
• Stability matching
• Thickness uniformity
• Calibration complexity
Reactive Evaporation
Introducing reactive gases during evaporation enables formation of compounds (oxides, nitrides) from metallic sources, but requires careful pressure and composition control.
Reactive evaporation process:
Metal evaporation + O₂ → M_xO_y
Metal evaporation + N₂ → M_xN_y
Reaction locations:
1. Gas phase (molecular)
2. Transit (atomic)
3. Substrate surface (adsorbed)
Oxygen incorporation:
P_O₂ / P_total ≈ 10⁻⁵ - 10⁻⁴
Higher P_O₂:
• Better stoichiometry
• Lower deposition rate
• Risk of target oxidation
Sticking probability:
s(O) >> s(M) typically
Excess O required
Partial pressure control:
P_O₂ = (Q_O₂ / Q_total) × P_total
Q_O₂: oxygen flow rate
Hysteresis behavior:
Similar to reactive sputtering
• Metallic mode: low P_O₂
• Oxide mode: high P_O₂
• Transition mode: optimal
Process window:
Narrow for stoichiometric films
Requires active feedback
Monitor optical transmission
or emission spectroscopy
Directional Deposition and Liftoff
The ballistic nature of evaporation enables directional deposition for liftoff lithography, where shadowing prevents sidewall coverage.
Liftoff geometry:
Undercut profile required:
w_top < w_bottom
Evaporation angle:
θ_evap = arctan(h_resist / w_undercut)
Critical angle:
θ_crit ≈ 70-80°
Above → sidewall coverage
Below → successful liftoff
Resist profile optimization:
• Bilayer resist (LOR/photoresist)
• Image reversal
• Angled development
Film thickness limit:
t_film < 0.3 × w_undercut
For w_undercut = 1 μm:
t_max ≈ 300 nm
Liftoff process:
1. Pattern resist
2. Evaporate metal
3. Dissolve resist (acetone, NMP)
4. Ultrasonic agitation
5. Rinse, dry
Yield issues:
• Fence formation (sidewall)
• Incomplete liftoff
• Pattern lifting
Mitigation:
• Optimize resist profile
• Control film stress
• Gentle liftoff chemistry
• Post-rinse inspection
Vacuum System Design and Pumping
Evaporation requires high vacuum (< 10⁻⁵ Torr) to ensure ballistic transport and minimize contamination. Proper pump selection and chamber design are critical.
Pump types for evaporation:
Roughing pump (mechanical):
• Rotary vane or scroll
• P_ultimate ≈ 10⁻³ - 10⁻² Torr
• Speed: 50-500 L/s
• Initial pumpdown
High vacuum pump:
• Turbomolecular pump (typical)
• Diffusion pump (legacy)
• Cryopump (UHV)
Turbo pump performance:
S_pump = 500-5000 L/s (N₂)
P_ultimate < 10⁻⁹ Torr
Compression ratio: 10⁹ (N₂)
Pumpdown time:
t = (V / S_eff) × ln(P_initial / P_final)
For V = 100 L, S = 1000 L/s:
t ≈ 0.1 × ln(760/10⁻⁵) ≈ 1.6 s
(theoretical, ignoring outgassing)
Effective pumping speed:
1/S_eff = 1/S_pump + 1/S_line
S_line = C (conductance)
Conductance (aperture):
C = 11.6 × A × √(T/M) L/s
A: area (cm²)
T: temperature (K)
M: molecular mass
Outgassing rate:
q = q₀ × t^(-α)
q₀: initial rate (Torr·L/s/cm²)
α ≈ 1 (typical)
For stainless steel:
q₀ ≈ 10⁻⁶ Torr·L/s/cm²
After 1 hour: q ≈ 10⁻⁷
Ultimate pressure:
P_ultimate = q_total / S_pump
Bakeout benefits:
T_bake = 150-200°C
t_bake = 12-48 hours
Reduces outgassing by 10-100×
Achieves P < 10⁻⁸ Torr
Load lock:
Separate chamber for loading
Prevents venting main chamber
Improves throughput
Reduces contamination
Rate Monitoring and Process Control
Precise thickness control requires real-time monitoring of deposition rate using quartz crystal microbalances (QCM) or optical methods.
Quartz Crystal Microbalance (QCM):
Sauerbrey equation:
Δf / f₀ = -2 × Δm / (A × ρ_q × t_q)
Δf: frequency shift (Hz)
f₀: resonant frequency (5-6 MHz)
Δm: mass deposited
ρ_q = 2.648 g/cm³ (quartz density)
t_q: quartz thickness
Mass sensitivity:
C_f = -2f₀² / (ρ_q × v_q)
v_q = 3.34×10⁵ cm/s (shear wave)
For 6 MHz crystal:
C_f ≈ -56.6 Hz·cm²/μg
Thickness from frequency:
d = -Δf × ρ_q × Z / (ρ_film × f₀² × N)
Z: acoustic impedance
N: frequency constant
Accuracy limitations:
• Assumes uniform film
• ρ_film must be known
• Temperature sensitivity (Δf/ΔT ≈ -20 ppm/°C)
• Valid up to ~10% Δf/f₀
Tooling factor:
TF = d_substrate / d_QCM
Accounts for geometric difference
TF = 0.5-2.0 (typical)
Calibration needed
Optical monitoring:
• Transmission/reflection
• Interference fringes
• Ellipsometry (ex-situ)
Feedback control:
PID controller adjusts power
Setpoint: target rate (Å/s)
Maintains ±1-2% stability
Process Integration and Future Trends
Advanced process integration combines multiple deposition techniques for optimal device performance.
Integration strategies:
• Multi-chamber cluster tools
• Sequential processing without vacuum break
• In-situ surface preparation
• Real-time endpoint detection
Future directions:
• Atomic layer control (ALD-like precision)
• Machine learning process optimization
• Digital twin modeling
• Autonomous process control
Vapor Pressure vs Temperature
Mean Free Path vs Pressure
Evaporation Rate vs Power
Film Microstructure Zones