Difference between revisions of "Planarization of Substrate Topography by Spin Coating"

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The authors analyze the four major forces acting on a film during spin coating.  Centrifugal, capillary, and gravitational forces all drive flow outward; viscosity, on the other hand, resists flow.  After spinning is complete, only capillarity and gravity drive flow--and as we soon see, gravity can be neglected.  The authors assume a Newtonian fluid (viscosity independent of shear rate) and assume no evaporation of the liquid occurs during spinning.  They state that whether a spin coated layer is conformal (equal thickness everywhere) or planar (flat surface) is to first order determined by the balance of centrifugal force versus capillary and gravitational forces.  With two sets of competing forces (ignoring viscosity for the moment), there must exist some transitional condition when they balance.  This condition is shown in Fig. 1.   
 
The authors analyze the four major forces acting on a film during spin coating.  Centrifugal, capillary, and gravitational forces all drive flow outward; viscosity, on the other hand, resists flow.  After spinning is complete, only capillarity and gravity drive flow--and as we soon see, gravity can be neglected.  The authors assume a Newtonian fluid (viscosity independent of shear rate) and assume no evaporation of the liquid occurs during spinning.  They state that whether a spin coated layer is conformal (equal thickness everywhere) or planar (flat surface) is to first order determined by the balance of centrifugal force versus capillary and gravitational forces.  With two sets of competing forces (ignoring viscosity for the moment), there must exist some transitional condition when they balance.  This condition is shown in Fig. 1.   
  
The capillary force is proportional to the surface tension <math>\sigma</math> of the liquid film and to the curvature <math>R</math> of the air-film interface.  Applying the Pythagorean Theorem in Fig. 1, we get
+
The capillary force is proportional to the surface tension <math>\sigma</math> of the liquid film and to the curvature <math>1/R</math> of the air-film interface.  Applying the Pythagorean Theorem in Fig. 1, we get
  
 
  <math>R^2 = \frac{w^2}{4}+(R-h)^2</math>
 
  <math>R^2 = \frac{w^2}{4}+(R-h)^2</math>
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where <math>\gamma</math> is surface tension.  Gravity or <math>\rho gh</math> hydrostatic pressure tends to level the film as well.  But we are in the capillary regime when  
 
where <math>\gamma</math> is surface tension.  Gravity or <math>\rho gh</math> hydrostatic pressure tends to level the film as well.  But we are in the capillary regime when  
  
  <math>w < \left(\frac{8\gamma}{\rho g}^\frac{1}{2}\right)</math>
+
  <math>w < \left(\frac{8\gamma}{\rho g}\right)^\frac{1}{2}</math>
  
 
Assuming typical values, the capillary force wins when the width of the trench is less than 5 mm, a very large distance!  Hence we neglect gravity and balance the centrifugal and capillary forces to determine what the critical trench width is.  This is done by comparing the pressure gradients due to each force.   
 
Assuming typical values, the capillary force wins when the width of the trench is less than 5 mm, a very large distance!  Hence we neglect gravity and balance the centrifugal and capillary forces to determine what the critical trench width is.  This is done by comparing the pressure gradients due to each force.   
  
 
  <math>\nabla p_{centrifugal} = \rho \omega ^2 r = \nabla p_{capilllary} = \frac{p_{cap}}{1/2 w} = \frac{16\gamma h}{w^2}</math>
 
  <math>\nabla p_{centrifugal} = \rho \omega ^2 r = \nabla p_{capilllary} = \frac{p_{cap}}{1/2 w} = \frac{16\gamma h}{w^2}</math>
 +
 +
The critical width <math>w_c</math> when the pressure gradients are equal and the transitional state exists, is given by
 +
 +
<math>w_c = \left(\frac{16\gamma h}{\rho \omega ^2 r}\right)^{1/2}</math>
 +
 +
Plugging in representative values of <math>\gamma</math> = 30 dyn/cm h = 1 um, <math>\rho</math> = 1 g/cm^3,  = <math>\omega</math> = 4000 rpm, and r = 2.5 cm, this recovers the empirical value of 50 um as the (approximate) critical planarizing width.  Film profiles midway between the center and edge of a spinning wafer will conformally cover a 1 um deep gap wider than 50 um but planarize a 1 um deep gap narrower than 50 um.  A gap near the center of the spinning substrate will be planarized to a greater degree than a gap at the edge of the substrate since the centrifugal force is lower near the center, while the capillary force is invariant. 
 +
 +
The authors also estimate the time required for leveling of topographic features by capillary action, after spinning ceases, from the Stokes equation for viscous flow:
 +
 +
<math>\nabla p = -\eta \nabla ^2 v</math>
 +
 +
where <math>\eta</math> is the viscosity of the fluid and v is the radial velocity along the surface of the substrate.  With <math>\nabla p</math> constant, we have a laminar flow velocity profile assuming no slip and no drag at the interfaces.  The volumetric flow rate per unit length (Q) into a gap is
 +
 +
<math>Q = \int v dz = \frac{\nabla p}{3\eta} h^3</math> \Rightarrow Fill time = t = \frac{1/2 wh}{Q} = \frac{3}{32} \frac{w^4 \eta}{\gamma h^3}
  
 
[[Image:planarization2.png|thumb|right|300px|'''Fig. 2''' Helical ]]
 
[[Image:planarization2.png|thumb|right|300px|'''Fig. 2''' Helical ]]

Revision as of 02:16, 16 March 2009

Planarization of Substrate Topography by Spin Coating

Authors: L. E. Stillwagon, R. G. Larson, and G. N. Taylor

J. Electrochem. Soc. 134, 8, 2030-2037

Soft matter keywords

Spin coating, thin films, capillarity, viscosity


By Alex Epstein


Abstract from the original paper

The results of a study of topographic substrate planarization with films applied by a spin-coating process are reported. It is shown that spin coating produces conformal film profiles over topographic gaps on the substrate that are wider than about 50 um and that leveling of these gaps can only occur after spinning ceases if the film is able to flow over large distances. A comparison of the major forces acting on the film leads to the conclusion that the flow is driven primarily by capillarity when the width of the gap is less than 5000 um. A theory is developed that relates the time required to level the gaps to their width and to the thickness and viscosity of the film. The results of experiments performed to test the theory are presented and discussed.

Soft matters

Brief background

Spin coating is a widely used technique for coating a flat substrate with a thin film of a liquid material. It is particularly used in the electronics industry, where integrated circuits are manufactured by a sequence of photolithographic steps. However, these processing steps often result in not-so-flat topography. Modern IC's feature interlevel interconnects that can extend microns upward. The empirical "Moore's Law" tells us that maximum transistor density has been doubling in new computing hardware every 18 months since the early 1970s. Now consider basic physics as it applies to photolithography: resolution or minimum linewidth (l) and depth of focus (d) are related to the exposure wavelength (<math>\lambda</math>) and the numerical aperture (NA) of the lens as follows.

<math>l \propto \lambda/NA</math>
<math>d \propto \lambda/(NA)^2</math>

To achieve higher and higher resolution, we can decrease exposure wavelength or increase the numerical aperture; but either approach also reduces the depth of field. Reduced depth of field means that the process is more sensitive to non-flat spin coat topography resulting from a complex multi-level substrate. This has been worked around by using multilayer resist schemes, in which a thick spin-coated bottom layer planarizes the substrate topography, and only a thin top layer is used for imaging. Early studies of planarization of substrate topography revealed definite limitations. One important limitation, which this paper explains, is that the planarizing range of spin coated films is limited to a distance of about 50 um.

Fig. 1 Drawing of the film profile over a gap when the centrifugal and capillary forces are exactly balanced.

Force analysis

The authors analyze the four major forces acting on a film during spin coating. Centrifugal, capillary, and gravitational forces all drive flow outward; viscosity, on the other hand, resists flow. After spinning is complete, only capillarity and gravity drive flow--and as we soon see, gravity can be neglected. The authors assume a Newtonian fluid (viscosity independent of shear rate) and assume no evaporation of the liquid occurs during spinning. They state that whether a spin coated layer is conformal (equal thickness everywhere) or planar (flat surface) is to first order determined by the balance of centrifugal force versus capillary and gravitational forces. With two sets of competing forces (ignoring viscosity for the moment), there must exist some transitional condition when they balance. This condition is shown in Fig. 1.

The capillary force is proportional to the surface tension <math>\sigma</math> of the liquid film and to the curvature <math>1/R</math> of the air-film interface. Applying the Pythagorean Theorem in Fig. 1, we get

<math>R^2 = \frac{w^2}{4}+(R-h)^2</math>

The authors assume a wide trench, <math>w>>h</math>, which [I scratch my head at the math trick here] reduces <math>R</math> to

<math>R = \frac{w^2}{8h}</math>

And our Laplace pressure, driving liquid into the gap to reduce curvature and flatten the film, becomes

<math>p_{capillary} = \frac{\gamma}{R} = \frac{8\gamma h}{w^2}</math>

where <math>\gamma</math> is surface tension. Gravity or <math>\rho gh</math> hydrostatic pressure tends to level the film as well. But we are in the capillary regime when

<math>w < \left(\frac{8\gamma}{\rho g}\right)^\frac{1}{2}</math>

Assuming typical values, the capillary force wins when the width of the trench is less than 5 mm, a very large distance! Hence we neglect gravity and balance the centrifugal and capillary forces to determine what the critical trench width is. This is done by comparing the pressure gradients due to each force.

<math>\nabla p_{centrifugal} = \rho \omega ^2 r = \nabla p_{capilllary} = \frac{p_{cap}}{1/2 w} = \frac{16\gamma h}{w^2}</math>

The critical width <math>w_c</math> when the pressure gradients are equal and the transitional state exists, is given by

<math>w_c = \left(\frac{16\gamma h}{\rho \omega ^2 r}\right)^{1/2}</math>

Plugging in representative values of <math>\gamma</math> = 30 dyn/cm h = 1 um, <math>\rho</math> = 1 g/cm^3, = <math>\omega</math> = 4000 rpm, and r = 2.5 cm, this recovers the empirical value of 50 um as the (approximate) critical planarizing width. Film profiles midway between the center and edge of a spinning wafer will conformally cover a 1 um deep gap wider than 50 um but planarize a 1 um deep gap narrower than 50 um. A gap near the center of the spinning substrate will be planarized to a greater degree than a gap at the edge of the substrate since the centrifugal force is lower near the center, while the capillary force is invariant.

The authors also estimate the time required for leveling of topographic features by capillary action, after spinning ceases, from the Stokes equation for viscous flow:

<math>\nabla p = -\eta \nabla ^2 v</math>

where <math>\eta</math> is the viscosity of the fluid and v is the radial velocity along the surface of the substrate. With <math>\nabla p</math> constant, we have a laminar flow velocity profile assuming no slip and no drag at the interfaces. The volumetric flow rate per unit length (Q) into a gap is

<math>Q = \int v dz = \frac{\nabla p}{3\eta} h^3</math> \Rightarrow Fill time = t = \frac{1/2 wh}{Q} = \frac{3}{32} \frac{w^4 \eta}{\gamma h^3}
Fig. 2 Helical
Fig. 3 Helical
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Fig. 5 Helical
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Fig. 8 Helical