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ALD step coverage and exposure calculator

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To coat the bottom of a deep hole uniformly, ALD needs (1 + 19a/4 + 3a²/2) times the exposure of a flat surface, where a is the hole's aspect ratio (Gordon et al.). A hole with an aspect ratio of 50 needs about 4,000 times the flat-surface exposure, and pulse time grows roughly with the square of aspect ratio. This is the main reason ALD throughput drops for 3D NAND and DRAM capacitors.

Inputs

Precursor and film (typical values)Precursor molecular weight (g/mol)Growth per cycle GPC (Å/cycle)Film density (g/cm³)Film formula weight (g/mol)Metal (or Si) atoms per formula unitDeposition temperature (°C)StructureDepth (µm)Hole diameter (nm)Precursor partial pressure (Pa)Safety factor (multiplier on the model value) (×)

Aspect ratio (depth ÷ width)40.0 : 1

Effective aspect ratio a used in the model40.0

Precursor surface density adsorbed per cycle2.78×10¹⁴molecules/cm²

Exposure to saturate a flat surface5.00×10⁻⁴Pa·s

Exposure to saturate the bottom of the structure1.3Pa·s

Same, in langmuirs9,710L (10⁻⁶ Torr·s)

Multiple of the flat-surface exposure2591×

Required precursor pulse time (with safety factor)259 ms

For a flat surface only (with safety factor)0.1 ms

Gordon's model gives a minimum that assumes the precursor adsorbs until the surface saturates and that molecules travel down the hole by bouncing off the walls (molecular flow). Real sticking probabilities are below 1, and byproduct readsorption and precursor decomposition also occur, so production conditions usually use several times the model exposure. The surface density is an approximation back-calculated from film density and GPC.

Aspect ratio and required pulse time

TEMAHf → HfO₂, 275 °C, partial pressure 10 Pa, safety factor 2×.
Aspect ratio	Vs. flat surface	Required exposure	Langmuirs	Pulse time
1:1	7×	3.6×10⁻³ Pa·s	27 L	0.001 s
5:1	62×	0.031 Pa·s	230 L	0.006 s
10:1	199×	0.099 Pa·s	740 L	0.020 s
20:1	696×	0.35 Pa·s	2,600 L	0.070 s
30:1	1494×	0.75 Pa·s	5,600 L	0.149 s
50:1	3989×	2 Pa·s	15,000 L	0.399 s
75:1	8795×	4.4 Pa·s	33,000 L	0.879 s
100:1	15476×	7.7 Pa·s	58,000 L	1.5 s
150:1	34464×	17 Pa·s	1.3×10⁵ L	3.4 s
200:1	60951×	30 Pa·s	2.3×10⁵ L	6.1 s

Why exposure grows with the square of aspect ratio

ALD stops reacting once the surface is saturated, so given enough time it coats the bottom of deep holes to the same thickness. Inside a hole, however, precursor molecules are adsorbed and consumed by the walls on the way down, so the flow reaching the bottom is thinner than at the entrance. Wall area grows with depth and the fraction of molecules that make it through falls with depth, so the required exposure grows roughly with the square of the aspect ratio. Ten times the aspect ratio needs roughly a hundred times the pulse time.

This is the main reason ALD slows down in 3D NAND memory holes (aspect ratios of 50 to 100 or more), DRAM capacitors and deep trench liners. Equipment makers secure throughput by shrinking chamber volume to shorten purges, engineering the precursor flow and spreading the time across many wafers in batch tools. Throughput by tool architecture can be estimated with the ALD throughput comparison calculator.

FAQ

Where does the Gordon model come from?

R. G. Gordon, D. Hausmann, E. Kim and J. Shepard, "A Kinetic Model for Step Coverage by Atomic Layer Deposition in Narrow Holes or Trenches," Chemical Vapor Deposition 9 (2003) 73.

Why use several times the model exposure in production?

The model is a minimum that assumes ideal sticking. Real sticking probabilities are below 1, and byproduct readsorption and precursor decomposition also occur.

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Last updated: October 23, 2026