Lecture 7 - Carrier Drift and Diffusion (cont.) February 20, Non-uniformly doped semiconductor in thermal equilibrium

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1 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 7-1 Lecture 7 - Carrier Drift and Diffusion (cont.) February 20, 2007 Contents: 1. Non-uniformly doped semiconductor in thermal equilibrium Reading assignment: del Alamo, Ch. 4, 4.5

2 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 7-2 Key questions Is it possible to have an electric field in a semiconductor in thermal equilibrium? What would that imply for the electron and hole currents? Is there a relationship between mobility and diffusion coefficient? Given a certain non-uniform doping distribution, how does one compute the equilibrium carrier concentrations? Under what conditions does the equilibrium majority carrier concentration follow the doping level in a non-uniformly doped semiconductor?

3 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture Non-uniformly doped semiconductor in thermal equilibrium It is possible to have an electric field in a semiconductor in thermal equilibrium non-uniform doping distribution Gauss Law: electrical charge produces an electric field: de = ρ ɛ ρ volume charge density [C/cm 3 ] if ρ = 0 de =0 in a certain region, it is possible to have E =0 with there are charges outside the region of interest ρ =0if In semiconductors: + ρ = q(p n + N D N A ) + If N D N D and N A N A, de q = (p n + N D N A ) ɛ

4 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 7-4 de q = (p n + N D N A ) ɛ Uniformly-doped semiconductor in TE: far away from any surface charge neutrality: ρ o =0 de o =0 since no field applied from the outside E o =0

5 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 7-5 Non-uniformly doped semiconductor in TE (n-type): n o, N D N D (x) n o (x)? n o (x)? x de o q = (N D n o ) ɛ Three possibilities: n o (x) = N D (x) net diffusion current n o (x) uniform net drift current n o = f (x) but n o (x) N D (x) in a way that there is no net current

6 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 7-6 n o, N D partially uncompensated donor charge a) - N D (x) + n o (x) net electron charge ρ o x b) + x c) ε ο x φ o d) φ ref x E c q[φ ο (x)-φ ref ] e) E F E v

7 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 7-7 Goal: understanding physics of non-uniformly doped semiconductors in TE principle of detailed balance for currents in TE Einstein relation Boltzmann relations general solution quasi-neutral solution

8 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 7-8 In thermal equilibrium: J = J e + J h =0 Detailed balance further demands that: J e = J h =0 [study example of 4.5.2]

9 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 7-9 Einstein Relation μ relates to ease of carrier drift in an electric field. D relates to ease of carrier diffusion as a result of a concentration gradient. Is there a relationship between μ and D? Yes, Einstein relation: D e D h kt = = μ e μ h q Relationship between D and μ only depends on T. [study derivation and restrictions in 4.5.2]

10 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 7-10 Boltzmann Relations In TE, diffusion = drift there must be a conexion between electrostatics and carrier concentrations. In TE, J e =0: dn o J e = qμ e n o E o + qd e =0 Then, relationship between E o and equilibrium carrier concentration: Express in terms of φ: Integrate: kt 1 dn o kt d(ln n o ) E o = = q n o q dφ o kt d(ln n o ) = q φ o (x) φ(ref ) = kt n o (x) ln q n o (ref) Relationship between the ratio of equilibrium carrier concentration and difference of electrostatic potential at two different points. At 300 K: factor of 10 in n o kt ln 10 = 60 mv of Δφ q 60 mv/decade rule

11 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 7-11 If φ(ref) = 0 where n o = n i : qφ o n o = n i exp kt Similarly: qφ o p o = n i exp kt 2 Note: n o p o = n i E c q[φ ο (x)-φ ref ] E F

12 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 7-12 Equilibrium carrier concentration: general solution in the presence of doping gradient (n-type) de o q = (N D n o ) ɛ Relationship between E o and n o in thermal equilibrium: Then: kt d(ln n o ) E o = q ɛkt d 2 (ln n o ) = no N D q 2 2 One equation, one unknown. Given N D (x), can solve for n o (x) (but in general, require numerical solution).

13 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 7-13 Quasi-neutral situation ɛkt d 2 (ln n o ) = no N D q 2 2 If N D (x) changes slowly with x n o (x) will also change slowly, then: n o (x) N D (x) The majority carrier concentration closely tracks the doping level. When does this simple result apply? or ɛkt d 2 (ln n o ) ND q 2 2 or ɛkt d 2 (ln N D ) ND q 2 2 ɛkt d 2 (ln N D ) 1 q 2 N D 2

14 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 7-14 Define Debye length: ɛkt L D = q 2 N D Rewrite condition: L 2 d E o kt D q Note: L D de o is change in electric field over a Debye length L 2 de o is change in electrostatic potential over a Debye length D For a non-uniformly doped profile to be quasi-neutral in TE, the change in the electrostatic potential over a Debye length must be smaller than the thermal voltage.

15 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 7-15 ɛkt L D = q2 N D L D characteristic length for electrostatic problems in semiconductors. Similar arguments for p-type material. 1E-04 Si at 300K 1E-05 LD (cm) 1E-06 1E-07 1E-08 1E+15 1E+16 1E+17 1E+18 1E+19 1E+20 1E+21 N (cm -3 ) Since L D 1/ N: N L D QN condition easier to fulfill.

16 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 7-16 Key conclusions Detailed balance in TE demands that J e = J h = 0 everywhere. Einstein relation: D μ = kt q Boltzmann relations: in TE, difference of electrostatic potential between two points is proportional to ratio of carrier concentration at same points: kt n o (x) kt p o (x) φ o (x) φ(ref) = ln = ln q n o (ref) q p o (ref) In quasi-neutral non-uniformly doped semiconductor in TE: n o (x) N D (x) Debye length: key characteristic length for electrostatics in quasineutral semiconductor. Order of magnitude of key parameters for Si at 300K: Debye length: L D cm (depends on doping)

17 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 7-17 Self study transit time Derive Einstein relation. Study example supporting detailed balance for J e and J h in TE. Work out exercises