Q2IC Technology

Diffusion of dopant impurities in silicon is described mathematically by Fick's laws of diffusion. Fick's first law states that the flux J of diffusing particles is proportional to the negative gradient of concentration, J = -D(dC/dx), where D is the diffusion coefficient (diffusivity) of the impurity species in silicon at the given temperature, C is the impurity concentration, and x is position; the negative sign reflects that diffusion always acts to move particles from regions of high concentration to regions of low concentration. Fick's second law, obtained by combining the first law with the continuity (mass conservation) equation, describes how concentration changes with time at any point: dC/dt = D(d^2C/dx^2), assuming D is independent of concentration and position (a simplifying assumption valid at low to moderate doping concentrations).

This partial differential equation has two standard analytic solutions depending on the boundary condition assumed at the wafer surface. For a constant surface concentration (infinite source, as in a predeposition step where the dopant source continuously replenishes the surface concentration Cs), the solution is the complementary error function profile, C(x,t) = Cserfc(x/(2sqrt(Dt))). For a limited, fixed total dose Q introduced at time zero with no further replenishment (as in a subsequent drive-in step following an ion implant or a short predeposition), the solution is the Gaussian profile, C(x,t) = (Q/sqrt(piDt))exp(-x^2/(4Dt)), where the surface concentration Cs = Q/sqrt(piDt) now decreases over time as the fixed dose spreads deeper into the wafer.

Factors Affecting the Diffusion Profile

• Temperature: diffusivity D depends exponentially on temperature through the Arrhenius relation D = D0*exp(-Ea/(kT)), where Ea is the activation energy for diffusion of that particular dopant species; even modest increases in process temperature produce large increases in diffusivity, making temperature the single most sensitive process parameter controlling diffusion depth.
• Time: for a given temperature (and hence fixed D), the diffusion depth scales with the square root of time, so achieving deeper junctions requires proportionally much longer diffusion times.
• Dopant species: different dopant atoms (boron, phosphorus, arsenic, antimony) have different diffusivities in silicon at a given temperature due to differences in atomic size and the specific point-defect-mediated mechanism (vacancy versus interstitial) by which each species preferentially diffuses.
• Concentration-dependent effects: at very high doping concentrations (approaching or exceeding the intrinsic carrier concentration at the diffusion temperature), the simple constant-D assumption breaks down because the local density of charged point defects (vacancies or interstitials) that mediate diffusion becomes concentration-dependent itself, causing the diffusivity to become a function of local concentration and producing profiles that deviate from the ideal erfc or Gaussian shape, often exhibiting a more box-like or kinked profile at high concentrations.
• Presence of other dopants or defects: co-diffusion of multiple dopant species can produce interactions (such as the emitter-push effect in bipolar transistor fabrication, where phosphorus diffusion enhances the diffusion of an underlying boron base region) that alter the profile from what would be predicted for a single isolated dopant.
• Ambient atmosphere during diffusion: an oxidizing ambient during a diffusion step generates excess point defects (injecting excess self-interstitials into the silicon, known as oxidation-enhanced diffusion), which can measurably increase the effective diffusivity of certain dopants compared to diffusion carried out in an inert ambient.

Diffusion Profile Measurement: Spreading Resistance Profiling

Spreading resistance profiling (SRP) is a widely used technique for measuring the actual dopant concentration profile as a function of depth in a processed wafer. The technique involves beveling the wafer at a shallow, precisely known angle (typically a fraction of a degree) to expose the doping profile along an extended, magnified length on the sample surface, then stepping two closely spaced probes along this bevel and measuring the spreading resistance (the resistance associated with current spreading out from a small point contact into the surrounding semiconductor) at each successive point. Since spreading resistance at a given point is inversely related to the local carrier concentration (and hence to the local net dopant concentration) at that depth, a calibration curve relating measured spreading resistance to known resistivity (and hence to known dopant concentration) allows the raw resistance-versus-position data to be converted into a dopant concentration-versus-depth profile, providing a direct, empirical measurement of the actual profile shape and depth achieved by a diffusion or implantation process, which can then be compared against the profile predicted by the erfc or Gaussian analytic solutions discussed above.

It is also useful to note that the spreading resistance profiling technique described here is complemented in practice by other profile measurement techniques such as secondary ion mass spectrometry (SIMS), which directly measures the total (not just electrically active) dopant atom concentration as a function of depth by sputtering away the sample surface layer by layer and mass-analyzing the ejected ions, offering higher depth resolution and the ability to detect dopant species regardless of their electrical activation state, in contrast to spreading resistance profiling, which measures only the electrically active carrier concentration and therefore cannot distinguish inactive (un-activated or precipitated) dopant atoms from those actually contributing free carriers.

It is further worth noting that these diffusion profile measurement and characterization techniques are not merely academic exercises but form an essential feedback loop for process development and control in a production fab, since the actual achieved junction depth and dopant profile of a diffusion or implant-and-anneal step directly determines critical device parameters such as transistor threshold voltage, source-drain series resistance, and short-channel effect immunity, meaning any systematic deviation between the measured profile and the intended target profile revealed by spreading resistance profiling or a similar technique must be traced back to its process root cause (incorrect temperature, time, or dose) and corrected before the deviation propagates into yield-impacting electrical parameter shifts across the production lot.

In summary, Fick's diffusion equation and the profile measurement techniques used to verify its predictions together form the essential theoretical and experimental toolkit for controlling dopant introduction in silicon device fabrication, a toolkit whose correct application requires careful attention to which boundary condition (constant surface source or fixed dose) actually applies to a given process step.

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