Models of diffusion in solids

 

 

At high temperatures point defects such as vacancies and self-interstitial atoms are generated in a single-crystal solid. Diffusion in a solid can be visualized as atomic movement of the diffusant in the crystal lattice by vacancies or self-interstitials.

 

Diffusion by a vacancy

 

At elevated temperature the lattice atoms vibrate around the equilibrium lattice sites. Occasionally a host atom acquires sufficient energy to leave the lattice site, becoming a self-interstitial atom and creating a vacancy. When a neighboring atom migrates to the vacancy site, it is called diffusion by a vacancy. [2]

If the migrating atom is a host atom, the diffusion is self-diffusion.

If the migrating atom is an impurity atom, the diffusion is impurity diffusion.

 

Divalency

 

If the impurity atom moves to a second vacancy that is at the nearest neighbor of the original vacancy site, it is called diffusion assisted by a double vacancy or divacancy.

 

Fig. 1 Models of atomic mechanism for a two-dimensional lattice (a) Vacancy and interstitial mechanism (b) Interstitialcy mechanism [2]

 

Interstitialcy mechanism

 

Interstitialcy or the extended interstitial mechanism is shown in (b) above.

A self-interstitial atom displaces an impurity (step 1) which in turn becomes an interstitial atom (step 2). Subsequently the impurity atom displaces another host atom (step 3) and the second host atom becomes a self-interstitial (step 4).

 

Smaller ionic radii impurities diffuse faster through interstitial mechanism.

Atomic diffusion mechanism in silicon

 

 

         Dopant impurity atoms occupying a substitutional site in silicon cannot move without the presence of point defects.

 

         The generation, annihilation, and movement of point defects and their interactions with impurity atoms affect the diffusion results and the measured diffusivities.

 

         Diffusion in silicon can be described by mechanisms involving impurity.

 

         And point-defect interactions, such as vacancies and interstitials, at different charge states.

 

Ionized Point Defects

 

Point defects can become electrically active when they accept or lose electrons. A vacancy can act as an acceptor by acquiring a negative charge V-.

 

V + e V-.

 

Similarly a self-interstitial atom can act as an acceptor by acquiring a negative charge I-

 

I + e I-

 

Where V represents a vacancy, I represents a Si self-interstitial.

A vacancy represents a lattice site where the silicon atom is missing.

A self-interstitial is a silicon atom that is not on a lattice site.[2]

 

Kickout Mechanism

 

If an impurity atom occupying a substitutional site is "kicked out" by a silicon self-interstitial atom, the impurity becomes an interstitial atom. It could move to another vacancy site, kick out another lattice silicon atom some distance away from the original lattice site or diffuse interstitially for some distance.

 

As + ISi AI

Where As is an impurity atom occupying a lattice site.

ISi is the silicon self-interstitial atom

AI is the impurity atom, which is not occupying a lattice site.[6]

 

Dissociative Mechanism

 

If an impurity atom occupying a substitutional site has left that site, and if it becomes an interstitial atom and a vacancy is left behind, the reaction can be described by the following equation, which is called the dissociative mechanism.

 

AS AI + V

 

If the impurity in AS AI + V is replaced by a silicon atom, then it would describe the creation of a Frenkel pair, which is a vacancy and a self-interstitial pair. For a perfect crystal at thermal equilibrium, the Frenkel pair implies that the number of vacancies and interstitials are equal.

 

Intrinsic Diffusivity

 

When the impurity concentration C(x), is less than ni, the intrinsic carrier concentration, the diffusion results can be described by a concentration independent diffusivity called intrinsic diffusivity Di

 

Extrinsic Diffusivity

 

When the impurity concentration, including both substrate doping and the dopant impurity, is greater than ni (T), intrinsic carrier concentration at the diffusion temperature T, the silicon is considered extrinsic silicon and diffusivity is considered as the extrinsic diffusivity De.

 

Diffusivity D can be expressed as follows:

 

D=D + D-(n/ni) + D=(n/ni)2 + D+(n/ni) +

 

Where D represents the intrinsic diffusivity of impurity interaction with a neutral point defect, D- represents the intrinsic diffusivity of impurity interaction with a singly charged acceptor point defect, D+ represents the intrinsic diffusivity of impurity interaction with a singly charged donor point defect and D= represents the intrinsic diffusivity of impurity interaction with a doubly charged acceptor defects.

 

In case of intrinsic silicon, C(x) << ni, so n=ni

 

Therefore DD + D- + D= + D+ + and D depends on temperature alone from the relation D=D0exp (-EA/kT)

 

In case of extrinsic silicon, C(x) >> ni, so n>>ni and so D becomes a function of temperature and concentration of the dopant impurity C(x).