Lecture 4 PN Junction and MOS Electrostatics(I) Semiconductor Electrostatics in Thermal Equilibrium Outline • Non-uniformly doped semiconductor in thermal equilibrium • Relationships between potential, φ(x) and equilibrium carrier concentrations, po(x), no(x) –Boltzmann relations & “60 mV Rule” • Quasi-neutral situation Reading Assignment: Howe and Sodini; Chapter 3, Sections 3.1-3.2 6.012 Spring 2007 Lecture 4 1 1. Non-uniformly doped semiconductor in thermal equilibrium Consider a piece of n-type Si in thermal equilibrium with non-uniform dopant distribution: Nd Nd(x) x What is the resulting electron concentration in thermal equilibrium? n-type ⇒ lots of electrons, few holes ⇒ focus on electrons 6.012 Spring 2007 Lecture 4 2 OPTION 1: electron concentration follows doping concentration EXACTLY ⇒ no(x) = N d(x) no, Nd Nd(x) no(x)=Nd(x)? x Gradient of electron concentration ⇒ net electron diffusion ⇒ not in thermal equilibrium! 6.012 Spring 2007 Lecture 4 3 OPTION 2: electron concentration uniform in space no(x) = nave ≠ f(x) no, Nd Nd(x) no = f(x)? x Think about space charge density: ρ(x) ≈ qN[ d(x) − no(x)] If Nd(x) ≠ no(x) ⇒ρ(x) ≠ 0 ⇒ electric field ⇒ net electron drift ⇒ not in thermal equilibrium! 6.012 Spring 2007 Lecture 4 4 OPTION 3: Demand that Jn = 0 in thermal equilibrium at every x (Jp = 0 too) Diffusion precisely balances Drift drift diff J n(x) = J n (x) + Jn (x) = 0 What is no(x) that satisfies this condition? partially uncompensated donor charge no, Nd Nd(x) + no(x) - net electron charge x Let us examine the electrostatics implications of no(x) ≠ Nd(x) 6.012 Spring 2007 Lecture 4 5 Space charge density ρ(x) = qN[ d(x) − no(x)] partially uncompensated donor charge no, Nd Nd(x) + no(x) - net electron charge x ρ + − x 6.012 Spring 2007 Lecture 4 6 Electric Field Poisson’s equation: dE ρ(x) = dx εs Integrate from x = 0: x 1 E(x)− E(0) = ρ(x ′ )dx ′ εs ∫ 0 no, Nd Nd(x) + no(x) - x ρ + − x E x 6.012 Spring 2007 Lecture 4 7 Electrostatic Potential dφ =−E(x) dx Integrate from x=0: x φ(x)− φ(0) =−∫ E(x ′ )dx′ 0 Need to select reference (physics is in the potential difference, not in absolute value!); Select φ(x = 0) = φref no, Nd Nd(x) + no(x) - x ρ + − x E x φ φ ref x 6.012 Spring 2007 Lecture 4 8 2. Relationships between potential, φ(x) and equilibrium carrier concentrations, po(x), no(x) (Boltzmann relations) dn J = 0 = qn µ E + qD o n o n n dx µ dφ 1 dn n • = • o Dn dx no dx Using Einstein relation: q dφ d(ln n ) • = o kT dx dx Integrate: q no ()φ − φref = ln no − ln no,ref = ln kT no,ref Then: ⎡ ⎤ q(φ − φref ) n = n exp⎢ ⎥ o o,ref ⎢ kT ⎥ ⎣ ⎦ 6.012 Spring 2007 Lecture 4 9 Any reference is good In 6.012, φref = 0 at no,ref = ni Then: qφ kT no = nie If we do same with holes (starting with Jh = 0 in thermal 2 equilibrium, or simply using nopo = ni ); −qφ kT po = nie We can re-write as: kT n φ = • ln o q ni and kT p φ =− • ln o q ni 6.012 Spring 2007 Lecture 4 10 “60 mV” Rule At room temperature for Si: n n φ = ()25mV • ln o = ()25mV • ln() 10 •log o ni ni Or n φ ≈ ()60mV • log o ni EXAMPLE 1: 18 −3 no = 10 cm ⇒ φ = (60mV)× 8 = 480 mV 6.012 Spring 2007 Lecture 4 11 “60 mV” Rule: contd. With holes: p p φ =−()25mV • ln o =−()25mV • ln() 10 • log o ni ni Or p φ ≈−()60 mV •log o ni EXAMPLE 2: 18 −3 2 −3 no = 10 cm ⇒ po =10 cm ⇒ φ =−()60mV ×−8 = 480mV 6.012 Spring 2007 Lecture 4 12 Relationship between φ, no and po: −3 po, equilibrium hole concentration (cm ) p-type intrinsic n-type 10191018 1016 1014 1012 1010 108 106 104 102 101 φ φ φ p+ (mV) n+ −550 −480 −360 −240 −120 0 120 240 360 480 550 −550 −480 −360 −240 −120 0 120 240 360 480 550 φ φ p+ φ (mV) n+ 101 102 104 106 108 1010 1012 1014 1016 1018 1019 p-typeintrinsic n-type −3 no, equilibrium electron concentration (cm ) Note: φ cannot exceed 550 mV or be smaller than -550 mV. (Beyond this point different physics come into play.) 6.012 Spring 2007 Lecture 4 13 Example 3: Compute potential difference in thermal 17 -3 equilibrium between region where no = 10 cm and no = 1015 cm-3. 17 −3 φ(no = 10 cm ) = 60 × 7 = 420 mV 15 −3 φ(no = 10 cm ) = 60 × 5 = 300 mV 17 −3 15 −3 φ(no = 10 cm ) − φ(no = 10 cm ) = 120 mV Example 4: Compute potential difference in thermal 20 -3 equilibrium between region where po = 10 cm and po = 1016 cm-3. 20 −3 φ( po = 10 cm ) = φmax =−550mV 16 −3 φ( po = 10 cm ) =−60 × 6 =−360mV 20 −3 16 −3 φ( po = 10 cm ) − φ(po = 10 cm ) =−190 mV 6.012 Spring 2007 Lecture 4 14 3. Quasi-neutral situation If Nd(x) changes slowly with x ⇒ no(x) also changes slowly with x. WHY? Small dno/dx implies a small diffusion current. We do not need a large drift current to balance it. Small drift current implies a small electric field and therefore a small space charge Then: no(x) ≈ Nd (x) no(x) tracks Nd(x) well ⇒ minimum space charge ⇒ semiconductor is quasi-neutral no, Nd Nd(x) no(x) = Nd(x) x 6.012 Spring 2007 Lecture 4 15 What did we learn today? Summary of Key Concepts • It is possible to have an electric field inside a semiconductor in thermal equilibrium – ⇒ Non-uniform doping distribution. • In thermal equilibrium, there is a fundamental relationship between the φ(x) and the equilibrium carrier concentrations no(x) & po(x) – Boltzmann relations (or “60 mV Rule”). • In a slowly varying doping profile, majority carrier concentration tracks well the doping concentration. 6.012 Spring 2007 Lecture 4 16.