非均匀掺杂半导体的扩散、电势和场

半导体中的杂质对电阻率的影响非常大。半导体中掺入微量杂质时,杂质原子附近的周期势场受到干扰并形成附加的束缚状态,在禁带中产加的杂质能级。同样的掺杂浓度,载流子的迁移率越大, 半导体材料的导电率越高。
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1. Lecture #5 ANNOUNCEMENT • Discussion Section 102 (Th 10-11AM) moved to 105 Latimer OUTLINE – Mobility dependence on temperature – Diffusion current – Relationship between band diagrams & V, – Non-uniformly doped semiconductor – Einstein relationship – Quasi-neutrality approximation Read: Chapter 3.2 Spring 2003 EE130 Lecture 5, Slide 1 Mechanisms of Carrier Scattering Dominant scattering mechanisms: 1. Phonon scattering (lattice scattering) 2. Impurity (dopant) ion scattering Phonon scattering mobility decreases when T increases: 1 1 µ phonon ∝ τ phonon ∝ ∝ ∝ T −3 / 2 phonon density × carrier thermal velocity T × T 1/ 2 µ = qτ / m v th ∝ T Spring 2003 EE130 Lecture 5, Slide 2 1

2. Impurity Ion Scattering Boron Ion Electron _ - - Electron + Arsenic Ion There is less change in the electron’s direction of travel if the electron zips by the ion at a higher speed. v th3 T 3/2 µ impurity ∝ ∝ NA + ND NA + ND Spring 2003 EE130 Lecture 5, Slide 3 Temperature Effect on Mobility 1 1 1 = + τ τ phonon τ impurity 1 1 1 = + µ µ phonon µ impurity Spring 2003 EE130 Lecture 5, Slide 4 2

3.Example: Temperature Dependence of ρ Consider a Si sample doped with 1017cm-3 As. How will its resistivity change when the temperature is increased from T=300K to T=400K? Solution: The temperature dependent factor in σ (and therefore ρ) is µn. From the mobility vs. temperature curve for 1017cm-3, we find that µn decreases from 770 at 300K to 400 at 400K. As a result, ρ increases by 770 = 1.93 400 Spring 2003 EE130 Lecture 5, Slide 5 Diffusion Particles diffuse from regions of higher concentration to regions of lower concentration region, due to random thermal motion. Spring 2003 EE130 Lecture 5, Slide 6 3

4. Diffusion Current dn dp J N,diff = qDN J P,diff = − qDP dx dx x x D is the diffusion constant, or diffusivity. Spring 2003 EE130 Lecture 5, Slide 7 Total Current J = JN + JP dn JN = JN,drift + JN,diff = qnµn + qDN dx dp JP = JP,drift + JP,diff = qpµp – qDP dx Spring 2003 EE130 Lecture 5, Slide 8 4

5.Band Diagram: Potential vs. Kinetic Energy increasing electron energy electron kinetic energy increasing hole energy Ec Ev hole kinetic energy Ec represents the electron potential energy: P.E. = Ec − Ereference Spring 2003 EE130 Lecture 5, Slide 9 Electrostatic Potential V 0.7V V(x) E 0.7V + – x N-Si 0 • Potential energy of a –q charged particle is related to the electrostatic potential V(x): P.E. = − qV 1 V= ( Ereference − Ec ) q Spring 2003 EE130 Lecture 5, Slide 10 5

6. Electric Field 0.7V V(x) E 0.7V + – x N-Si 0 dV 1 dEc =− = dx q dx • Variation of Ec with position is called “band bending.” Spring 2003 EE130 Lecture 5, Slide 11 Non-Uniformly-Doped Semiconductor • The position of EF relative to the band edges is determined by the carrier concentrations, which is determined by the dopant concentrations. • In equilibrium, EF is constant; therefore, the band energies vary with position: Ec(x) EF Ev(x) Spring 2003 EE130 Lecture 5, Slide 12 6

7.• In equilibrium, there is no net flow of electrons or holes JN = 0 and JP = 0 Î The drift and diffusion current components must balance each other exactly. (A built-in electric field exists, such that the drift current exactly cancels out the diffusion current due to the concentration gradient.) dn J N = qnµ n + qDN =0 dx Spring 2003 EE130 Lecture 5, Slide 13 Consider a piece of a non-uniformly doped semiconductor: n = N c e − ( Ec − EF ) / kT n-type semiconductor dn N dE = − c e −( Ec − EF ) / kT c Decreasing donor concentration dx kT dx Ec(x) n dEc =− EF kT dx n =− q Ev(x) kT Spring 2003 EE130 Lecture 5, Slide 14 7

8. Einstein Relationship between D and µ Under equilibrium conditions, JN = 0 and JP = 0 dn J N = qnµ n + qDN =0 dx qDN kT 0 = qnµ n − qn DN = µn kT q kT Similarly, DP = µp q Note: The Einstein relationship is valid for a non-degenerate semiconductor, even under non-equilibrium conditions Spring 2003 EE130 Lecture 5, Slide 15 Example: Diffusion Constant What is the hole diffusion constant in a sample of silicon with µp = 410 cm2 / V s ? Solution:  kT  DP =   µ p = (26 mV) ⋅ 410 cm 2 V −1s −1 = 11 cm 2 /s  q  Remember: kT/q = 26 mV at room temperature. Spring 2003 EE130 Lecture 5, Slide 16 8

9. Potential Difference due to n(x), p(x) • The ratio of carrier densities (n, p) at two points depends exponentially on the potential difference between these points: n  n  EF − Ei1 = kT ln 1  => Ei1 = EF − kT ln 1   ni   ni  n  Similarly, Ei2 = EF − kT ln 2   ni   n   n  n  Therefore Ei1 − Ei2 = kT ln 2  − ln 1  = kT ln 2    ni   ni   n1  1   V2 − V1 = (Ei1 − Ei2 ) = kT ln n2  q q  n1  Spring 2003 EE130 Lecture 5, Slide 17 Quasi-Neutrality Approximation • If the dopant concentration profile varies gradually with position, then the majority-carrier concentration distribution does not differ much from the dopant concentration distribution. Spring 2003 EE130 Lecture 5, Slide 18 9

10. Summary • Carrier mobility varies with temperature – decreases w/ increasing T if lattice scattering dominant – decreases w/ decreasing T if impurity scattering dominant • Electron/hole concentration gradient Æ diffusion dn dp J N,diff = qDN J P,diff = −qDP dx dx • Current flowing in a semiconductor is comprised of drift & diffusion components for electrons & holes J = JN,drift + JN,diff + JP,drift + JP,diff – In equilibrium, JN = JN,drift + JN,diff = 0 Spring 2003 EE130 Lecture 5, Slide 19 • The characteristic constants of drift and diffusion are related: D kT = µ q • In thermal equilibrium, EF is constant • Ec represents the electron potential energy Variation in Ec(x) Æ variation in electric potential V dEc dEv Electric field = = dx dx • E - Ec = electron kinetic energy Spring 2003 EE130 Lecture 5, Slide 20 10