**1. **Diffusion 1 Dopant Diffusion (Jaeger Chapter 4 and Campbell Chapter 3) As indicated previously the main front-end processing in building a device or integrated circuit is to selectively introduction of dopant atoms into silicon wafer. Dopant introduction by high temperature diffusion is one of the two major processes for achieving this. Diffusion is carried out at high temperature ( 800 to 1000 ?C) in a dopant-rich gaseous ambient.
Diffusion also covers the redistribution in the wafer of dopant atoms introduced into the wafer by other methods such as ion implantation.
For simplicity the diffusion of dopant in silicon is modelled by the simple diffusion theory described by Fick?s law in which the flux of dopant atoms at any point is assumed to be proportional to the gradient of the dopant concentration at that point. The proportionality constant is the diffusion coefficient, D . This is assumed to be a constant for a given dopant species at a given temperature. Actually D is not constant by is dependent on dopant concentration.
This section covers the diffusion techniques used and the simple 1-D analysis describing the dopant distribution into the wafer.

**2. **Diffusion 2 Diffusion System
The most common practice of introducing dopant into wafers by diffusion is the horizontal quartz tube furnaces similar to that used for thermal oxidation of silicon wafers. A separate furnace, wafer holder etc are reserved for a particular dopant to avoid contamination by other dopants.
In general the dopant atoms are introduced in two separate diffusion steps, each in its own tube furnace. The first step is referred to as pre-deposition diffusion, it is aimed to introduce a controlled amount of dopant atoms into the silicon surface (dopant atoms per unit surface area). The second step known as the drive-in diffusion which is aimed (at least theoretically) to redistribute the dopant atoms introduced during predeposition further into the silicon and reducing the dopant concentration near the surface.
Predeposition Diffusion
In pre-deposition diffusion the wafer is exposed to excess amount of dopant atoms in a gaseous ambient to ensure that the wafer surface has the maximum concentration of dopant atoms determined by the diffusion temperature (referred to as the solid solubility limit). Not all introduced dopant atoms are electrically active.

**3. **Diffusion 3 The dopant atoms can be introduced in solid, liquid or gaseous form. The following table gives the sources used for Boron and Phosphorus: Dopant Boron PhosphorusFormSolid B2O3,BN P2O5 Liquid BBr3 POCl3Gaseous BCl3 PH3The BN solid source (in solid disc form) is actually converted to B2O3 in an oxidising ambient in a separate furnace and then loaded in the predeposition furnace with each BN disc facing the silicon wafers. The B2O3 is then transferred to the surface of the wafers.

**4. **Diffusion 4 These diagrams show how these are done.

**5. **Diffusion 5 Drive-in DiffusionThe drive-in diffusion is typically carried out in an oxidising ambient but with no dopant atoms. This is carried out after in a furnace very similar to an oxidation furnace usually at a temperature higher than the predeposition temperature. Prior to the drive-in diffusion, a very thin layer of dopant rich silicon and/or silicon dioxide formed during predeposition is etched away (hydrofluoric acid ?dip?).

**6. **Diffusion 6 Prior to the drive-in diffusion, a very thin layer of dopant rich silicon and/or silicon dioxide formed during predeposition is etched away (hydrofluoric acid ?dip?). The drive-in oxide layer formed on the wafer is to ?seal in? the dopant atoms. In simple analysis. It is assumed that no dopant atoms are lost either to the ambient or to the oxide. Due to the conversion of silicon into silicon dioxide and the redistribution of dopant atoms at Si-SiO2 interface during oxidation, there is always loss of dopant atoms, more so for born due to the ?suck-out? effect.
As mentioned before, during drive-in the dopant atoms diffuse further into the silicon and it also reduces the surface dopant concentration. The resistance (more correctly the sheet resistance) of the diffused layer increases and the junction depth increases.

**7. **Diffusion 7 Mathematical Modelling of Diffusion ? 1-dimensional

**8. **Diffusion 8 Predeposition Diffusion For this process, the dopant concentration at the surface N(x=0) is limited to the solid solubility of the dopant at the predeposition temperature No for all t >0. Assuming that there are no dopant atoms of this element in the silicon prior to diffusion (N =0 for all x at t <0) and that the silicon is infinitely thick such that N(x=infinity) =0 for t >0.Then it can be shown that the N(x) for t>0 is given by:

**9. **Diffusion 9 This profile is often referred to as the complementary error function profile or the predeposition profile. The term 2v(Dt) is known as the diffusion length (not to be confused with the carrier diffusion length v(Dt) when we consider carrier diffusion in the operation of semiconductor devices). In some literature v(Dt) is referred to as the characteristic length of the diffusion process. Increasing v(Dt) extends the dopant atoms further into the bulk of the semiconductor wafer ? this simply means that by increasing the predeposition diffusion time we have more dopant atoms moving further into the semiconductor while keeping the surface concentration constant at No.

**10. **Diffusion 10 The diffusion coefficient, D, is a function of diffusion temperature following the Arrhenius relationship:
Typical Do and activation energy EA as well as D-temperature plots are given here. For first order approximation, D is taken to be independent of the actual dopant level.

**11. **Diffusion 11 For a given predeposition diffusion conditions (i.e. fixed No, D and t) the total amount of dopant atoms introduced into the silicon wafer, Q, measured in atoms per unit surface area of the diffusion window, is determined by predeposition diffusion. Q is given by:
Concept of Junction Depth, xj
When dopant of one type (p or n) is diffused into a wafer originally doped with dopant of the opposite type, a junction will be formed. The distance of the junction from the surface is known as the junction depth, xj. At xj the concentration of the diffused dopant = the concentration of the substrate dopant (assumed to have a uniform concentration). Consider a p-type substrate with uniform dopant concentration NA subjected to the n-type predeposition diffusion with the profile:

**12. **Diffusion 12 For x <xj the material is n-type and for x>xj it remains p-type. The net dopant concentration on n-side is ND(x) ?NA and on p-side it is NA-ND(x). See Left plot below. Because the dopant concentration level varies several order of magnitude, the dopant concentration axis is usually plotted on log scale. See npn bjt dopant concentration plot below right.

**13. **Diffusion 13 Drive-in Diffusion Dopant diffusion during drive-in is governed by the same diffusion equation i.e. Fick?s 2nd Law for the diffusion of dopant atoms inside the silicon, i.e.
The only difference is the boundary conditions for this equation. It is usually assumed that a layer of SiO2 is formed immediately at the surface and this serves as a mask preventing the inward (from ambient to silicon) or outward (silicon to ambient) diffusion of dopant atoms. Mathematically, the boundary condition is that the dopant atom flux at x=0 is zero for all t >0 i.e.
where D1 and t1 are the diffusion coefficient and diffusion time for predeposition diffusion. In order to obtain analytical solution for the final profile, we have to make simplifying assumptions based on our knowledge about the diffusion sequence.

**14. **Diffusion 14 In general, predeposition diffusion is carried out at a lower temperature than drive-in diffusion and also for a shorter time. If D2 and t2 and the diffusion coefficient and diffusion time for drive-in, then
D2t2 >> D1t1.
Under this condition, it is usual to treat the predeposition diffusion profile as a delta function at x=0 with the ?area? of the delta function = Q, the dopant atoms per unit surface area introduced by predeposition i.e.
N(x,t=0)= Q.d(x),
where,
In this case, the solution of the drive-in diffusion is:
The right hand side is the well known Gaussian function and the profile is referred to as the Gaussian diffusion profile. The peak of the profile occurs at x =0, i.e. at the surface as we would expect it. However notice that the magnitude of this peak decreases as the diffusion time t increases. This is the consequence that as diffusion time increases, the fixed amount of dopant atoms, Q, is spread out further into the silicon wafer. The plot of Gaussian dopant profile is shown below:

**15. **Diffusion 15 Erfc and Gaussian profiles are only approximate description of the actual profiles but they are used extensively for estimation of junction depth, resistance (known as sheet resistance) etc of diffused layers.
Some reasons for deviations from erfc and Gaussian profiles:
Dopant concentration dependent diffusion coefficient
Lateral diffusion of dopant atoms
Boron suck-out and phosphorus pile-up during drive-in
Shallow junctions where delta function approximation of pre-deposition profile is not accurate.

**16. **Diffusion 16 Sheet Resistance and Irvin Curves With varying dopant concentration N(x), the conductivity and resistivity of a diffused layer also change with distance: s(x)= q(mn.n+mp.p) =1/?(x). However the layer can carry current parallel to the wafer surface and will obey Ohm Law. For the purpose of characterising this layer and for the use of the layer to form resistors, the layer is characterised by a quantity known as sheet resistance, Rs (or ?s). This is the resistance between the two opposite sides of a unit square (side length L) of the diffused layer as shown below. It is assumed that current flows parallel to the surface.
Breaking this layer into elemental layers each of thickness dx. They are conductors in parallel.

**17. **Diffusion 17 The elemental layer between x and x+dx has net dopant concentration [N(x) ? NB]. It has a conductance:
We have assumed here that only the majority carriers contribute to the current flow. We have also ignored the effect of the space charge region of the junction.
Since all elemental layers are in parallel, the total conductance of the diffused layer, GS, the sheet conductance, is:
Or, the sheet resistance, RS, is:
RS has the unit of O and it is independent of the actual value of L ? i.e. so long it is a square piece of the diffused layer. RS is more commonly referred to in O per square.
Also notice that if we have the diffused layer of width L (direction perpendicular to current flow) and length (direction parallel to current flow) 2L, its resistance is that of two square units connected in series i.e. R = 2RS.

**18. **Diffusion 18 Diffused layers are used to produce resistors in bipolar technology, typically the base diffusion is used for this purpose because it has higher sheet resistance. The value of a diffused resistor is given by:
R = Rs. (L/W)
Where Rs is the sheet resistance of the diffused layer and L and W are the length and width of the diffused area as seen from the top.

**19. **Diffusion 19 Sheet Resistance Measurement
Sheet resistance of a diffused layer ( or an implanted layer) can be measured fairly easily using what is referred to as the (linear four-point probe measurement on a test wafer (of dopant type opposite to that of the diffusion) on which the diffusion is carried out on the whole wafer. The measurement is setup is shown in diagram. The four probes are in line equal distance a apart (typically 1mm). Constant current I is flowing into wafer at probe 1 and out of wafer at probe 4. Potential difference between probes 2 and 3, V is measured by a voltmeter.
Consider first the situation that there is only current I into probe 1 (no current in or out of probe 4). We expect this current to flow uniformly radially outward in the diffused layer. At a radius r from probe 1, the current density (sheet current density) is:

**20. **Diffusion 20 The sheet current density is related to the radial electric field E through Ohm?s Law:
E is related to the radial change in potential dV/dr by:
By integrating the above expression from probe 2 to probe 3, we gat the voltage drop from probe 2 to probe 3 due to current I flowing into probe 1:
By similar consideration of a current I flowing out at probe 4 (with no current in or out of probe 1), we get the voltage across probe 2 and 3 as:
The combination of the two above situations is equivalent to the case of the four point probe measurement. By superposition theorem, the measured voltage between probe 2 and 3 for the four point probe measurement is:
If the current is set at 4.532 mA, then the voltage measured is the sheet resistance in k? per square.