6.720J/3.43J Integrated Microelectronic Devices Fall 2002 Lecture 21 Lecture 2 Carrier Statistics in Equilibrium Contents: September 5, 2002 1. Conduction and valence bands, bandgap, holes 2. Intrinsic semiconductor 3. Extrinsic semiconductor 4. Conduction and valence band density of states Reading assignment: del Alamo, Ch. 2, 2.12.4 (2.4.1) Announcement: Go to http://weblab.mit.edu and register. Put 6.720 student in the Description field.
6.720J/3.43J Integrated Microelectronic Devices Fall 2002 Lecture 22 Key questions What are these energy band diagrams? What are these holes? In a perfectly pure semiconductor, how many electrons and holes are there? Can one engineer the electron and hole concentrations in a semiconductor?
6.720J/3.43J Integrated Microelectronic Devices Fall 2002 Lecture 23 1. Conduction and valence bands, bandgap, holes energy conduction band E c E c E v bandgap E v E g valence band space coordinate Conduction and valence bands: bonding electrons occupy states in valence band free electrons occupy states in conduction band holes: empty states in valence band CB electrons and VB holes can move around: carriers a) b) c)
6.720J/3.43J Integrated Microelectronic Devices Fall 2002 Lecture 24 electron energy hole energy E c E v Elements of energy band diagrams: at edges of bands, kinetic energy of carriers is zero electron energies increase upwards hole energies increase downwards electron energy hυ υ = Eg hυ υ > Eg hυ υ > Eg E c E v a) b)
6.720J/3.43J Integrated Microelectronic Devices Fall 2002 Lecture 25 2. Intrinsic semiconductor Define intrinsic semiconductor, or ideal semiconductor: perfectly crystalline (no perturbations to periodic lattice) perfectly pure (no foreign atoms) no surface effects Question: How many electrons and holes are there in an intrinsic semiconductor in thermal equilibrium? Answer requires fairly elaborate model [lecture 3], but key dependencies can be readily identified. Define: n o equilibrium (free) electron concentration in conduction band [cm 3 ] p o equilibrium hole concentration in valence band [cm 3 ] Certainly in intrinsic semiconductor: n o = p o = n i n i intrinsic carrier concentration [cm 3 ]
6.720J/3.43J Integrated Microelectronic Devices Fall 2002 Lecture 26 Key dependencies of n i : Temperature: Bandgap: T n i E g n i What is detailed form of dependencies? Use analogy of chemical reactions.
6.720J/3.43J Integrated Microelectronic Devices Fall 2002 Lecture 27 Electronhole formation can be thought of as chemical reaction: bond e h similar to water decomposition reaction: H 2 O H OH Lawofmass action relates concentration of reactants and reaction products. For water: K = [H ][OH ] [H 2 O] exp( E kt ) E is energy consumed or released in reaction. This is a thermally activated process rate of reaction limited by need to overcome energy barrier E (activation energy). In analogy, for electronhole formation: K = n op o [bonds] exp( E g kt ) [bonds] is concentration of unbroken bonds. Note: activation energy is E g.
6.720J/3.43J Integrated Microelectronic Devices Fall 2002 Lecture 28 In general, relatively few bonds are broken. Hence: [bonds] n o,p o and [bonds] constant Then: Two important results: First, n o p o exp( E g kt ) n i exp( E g 2kT ) As expected: T n i E g n i To get prefactor, need detailed model [lecture 3].
6.720J/3.43J Integrated Microelectronic Devices Fall 2002 Lecture 29 Arrhenius plot for Si [experiments of Misiakos and Tsamakis, 1993]: 1E15 300 K 1E10 1E05 ni (cm 3 ) 1E00 1E05 1E10 0.612 ev 1E15 1E20 30 50 70 90 110 130 150 1/kT (ev 1 ) In Si at 300 K, n i 1.1 10 10 cm 3. Second important result: n o p o = n 2 i Equilibrium np product in a semiconductor at a certain temperature is a constant specific to the semiconductor.
6.720J/3.43J Integrated Microelectronic Devices Fall 2002 Lecture 210 3. Extrinsic semiconductor Can electron and hole concentrations be engineered? Insert dopants in substitutional positions in the lattice: Donors: introduce electrons to conduction band without introducing holes to valence band Acceptors: introduce holes to valence band without introducing electrons to conduction band If any one carrier type overwhelms n i extrinsic semiconductor
6.720J/3.43J Integrated Microelectronic Devices Fall 2002 Lecture 211 Donor in Si, atom from column V (As, P): a) neutral donor b) ionized donor Acceptor in Si, atom from column III (B): a) neutral acceptor b) ionized acceptor
6.720J/3.43J Integrated Microelectronic Devices Fall 2002 Lecture 212 Representation of donor and acceptor states in energy band diagram: Ec Ed ED Ea Ev EA E d,e a 40 60 mev, for common dopants Near room temperature, all dopants are ionized: N D N D N A N A Typical doping levels: N A,N D 10 15 10 20 cm 3
6.720J/3.43J Integrated Microelectronic Devices Fall 2002 Lecture 213 ntype semiconductor n o N D p o n2 i N D These equations are valid at intermediate temperatures. log no intrinsic regime full ionization N D Eg/2 carrier freezeout extrinsic regime Ed/2 1/kT ptype semiconductor p o N A n o n2 i N A
6.720J/3.43J Integrated Microelectronic Devices Fall 2002 Lecture 214 4. Conduction and valence band density of states 6E22 Si 5E22 4E22 DOS (cm 3 ev 1 ) 3E22 2E22 1E22 0E00 5 4 3 2 1 0 1 2 3 4 5 6 EE v (ev) [Calculations by M. Fischetti, IBM] Close to edges: g c (E) E E c E E c g v (E) E v E E E v g(e) gv(e) gc(e) 0 Ev Ec > E
6.720J/3.43J Integrated Microelectronic Devices Fall 2002 Lecture 215 Common expressions for DOS: g c (E) =4π 2m de h 2 3/2 E Ec E E c g v (E) =4π 2m dh h 2 3/2 Ev E E E v m de density of states electron effective mass m dh density of states hole effective mass Three comments: m de, m dh values given in terms of m o, electron rest mass i.e. for Si at 300 K, m de =1.09m o, m dh =1.15m o m o =5.69 10 16 ev s 2 /cm 2 other effective masses defined: conductivity, longitudinal, transversal, lighthole, heavyhole, etc; only DOS m should be used to compute DOS values of DOS m are definition sensitive
6.720J/3.43J Integrated Microelectronic Devices Fall 2002 Lecture 216 Key conclusions Concept of (free) electron: electron in conduction band. Concept of hole: empty state in valence band. Intrinsic semiconductor: ideally pure semiconductor. n o = p o = n i exp( E g 2kT ) To first order, for a given semiconductor n o p o is a constant that only depends on T : n o p o = n 2 i Equilibrium carrier concentrations can be engineered through shallow dopants extrinsic semiconductor. ntype semiconductor: ptype semiconductor: n o N D, p o n2 i N D p o N A, n o n2 i N A Around edges, conduction and valence bands in semiconductors feature DOS E. Order of magnitude of key parameters for Si at 300 K: intrinsic carrier concentration: n i 10 10 cm 3 typical doping level range: N D,N A 10 15 10 20 cm 3
6.720J/3.43J Integrated Microelectronic Devices Fall 2002 Lecture 217 Self study Charge neutrality