Fall 2007 Fundamentals of Semiconductor Physics 万 歆 Zhejiang Institute of Modern Physics xinwan@zimp.zju.edu.cn http://zimp.zju.edu.cn/~xinwan/
Transistor technology evokes new physics The objective of producing useful devices has strongly influenced the choice of the research projects with which I have been associated. It is frequently said that having a more-or-less specific practical goal in mind will degrade the quality of research. I do not believe that this is necessarily the case and to make my point in this lecture I have chosen my examples of the new physics of semiconductors from research projects which were very definitely motivated by practical considerations. -- William B Shockley Nobel Lecture, December 11, 1956
Chapter 3. Junctions & Contacts 3.1 Metal-semiconductor contacts 3.2 p-n junctions 3.3 Heterojunctions* Total 6 hours.
The birth of transistor William Bradford Shockley John Bardeen Nobel Prize in Physics 1956 Walter Houser Brattain
Why Contacts? Bringing materials with vastly different properties together can produce remarkable effects. Examples: Cu/Fe: temperature controlled switch Superconductor/metal: Andreev reflection Ferromagnet/semiconductor: spin injection Ferromagnetic/normal metal: GMR
The birth of nonotechnology/spintronics Nobel Prize 2007: Albert Fert & Peter Gruenberg For the discovery of Giant Magnetoresistance
Chapter 3. Junctions & Contacts 3.1 Metal-semiconductor contacts A review of the principles Idealized metal-semiconductor junctions Current-Voltage characteristics Ohmic contacts 3.2 p-n junctions 3.3 Heterojunctions
Separated M-S Systems
Bring M-S into Contact f D1, 2 = 1 1 exp [ E E f1, 2 / kt ] E f1=e f2 At thermal equilibrium the Fermi levels is constant throughout a system.
Inhomogeneously Doped SC E0 E0 Ec Ef Ec Ei Ef Ei Ev Ev
Inhomogeneously Doped SC
Ideal Density-of-States Typical Metal Typical Semiconductor
Energy levels
Charge Transfer Equilibrium Electrons depleted Assume band structures not changed near the surface. Real situation: Surface states
No Hope to Solve Analytically Poisson s equation 2 d q q = = p n N N, n=n exp d a i 2 kt dx s s the electron density, the hole density and the donor and acceptor densities. To solve the equation, we have to express the electron and hole density, n and p, as a function of the potential, φ 2 d q q = 2 n sinh N N i a d 2 kt dx s { } { } E f E i E f Ei E i E f n=ni exp, p=ni exp, = kbt kbt q
Depletion Approximation For the idealized n-type semiconductor, hole density neglected Electron density n << Nd, thus depleted, from the interface to x = xd Beyond xd, n = Nd Negative charge right at the metal surface
Field, Potential and Charge (Gauss law)
Applied Bias Up to this point, we have been considering thermal equilibrium conditions at the metal-semiconductor junction. Now, we study the case of an applied voltage; i.e., a nonequilibrium condition. Electrons transferring from metal to semiconductor see a barrier.
Applied Bias To the first order, the barrier height is independent of bias because no voltage can be sustained across the metal. Bias changes the curvature of the semiconductor bands, modifying the potential drop from φi.
Junction Charge & Capacitor
Small-Signal Capacitance
Variable Doping
Semiconductor Profiler
Schottky Barrier Lowering We now explore the statement that the barrier to electron flow from metal to semiconductor is to first order unchanged by bias. Approximations: Free electron theory Metal as plane conducting sheet Semiconductor: effective mass, relative permittivity Root: Metal plane = image charge of opposite sign
Schottky Barrier Lowering
I-V Characteristics w/out Math At equilibrium, rate at which electrons cross the barrier into the semiconductor is balanced by rate at which electrons cross the barrier into the metal. (flow = counter flow). When a bias is applied, the potential drop within the semiconductor is changed and we can expect the flux of electrons from the semiconductor toward the metal to be modified. The flux of electrons from the metal to the semiconductor is not affected. The difference is the net current.
I-V Characteristics The current of electrons from the semiconductor to the metal is proportional to the density of electrons at the boundary. q B q i n s = N C exp = N D exp kt kt q B =q i E C E f In equilibrium, q i J MS = J SM =K N D exp KT
I-V Characteristics When a bias is applied to the junction, q i V a n s = N D exp kt Now, q i V a q i J =J MS J SM = K N D exp K N D exp KT KT Therefore, the ideal diode equation reads [ ] qv a J =J 0 exp 1 KT
Comments The ideal diode equation arises when a barrier to electron flow affects the thermal flux of carriers asymmetrically. The essence of the ideal diode equation predicts a saturation current J0 for negative Va and a very large steeply rising current when Va is positive. J0 in the above ideal case is independent of the applied bias. More careful analysis will modify this slightly.
More Detailed Analysis Schottky: Integrating the equations for carrier diffusion and drift across the depletion region near the contact. Assumes that the dimensions of the space-charge region are sufficiently large (a few electron mean-free path) so that the use of a diffusion constant and a mobility value are meaningful (small field, no drift velocity saturation). Bethe: Based on carrier emission from the metal. Valid even when these abovementioned constraints are not met.
Diffusion and Drift Currents [ J x =q n n E x D n ] [ dn qn d dn =q D n dx kt dx dx ]
Trick for Integration [ J x =q n n E x D n xd J x 0 exp ] [ dn qn d dn =q Dn dx kt dx dx [ ] q q dx=q D n n exp kt kt ] xd 0 0 =0 ; x d = i V a = B n V a q B q n n 0 = N C exp ; n x d =N D =N C exp kt kt J x= q B q D n N C exp kt xd 0 ] [ ] [ q x exp dx kt qv a qv a exp 1 = J 0 exp 1 kt kt
Schottky Barrier
Mott Barrier
Comments One important implicit assumption is that the system is in quasiequilibrium, i.e., almost at thermal equilibrium even though currents are flowing. Electron density at the interface when the bias is applied. Using Einstein relation to relate drift and diffusion current. The ultimate test: Agreements between measurements and predictions. Derivation not valid when Va ~ φi (no barrier). A fraction of the voltage is dropped across the resistance of the semiconductor. The forward voltage across the junction is reduced.
Schottky Diodes
Surface Effects
Pinning of Fermi Energy To account for the surface effects, the metal-semiconductor contact is treated as if it contained an intermediate region sandwiched between the two crystals. For a large density of surface states, the Fermi energy is said to be pinned by the high density of states.
Nonrectifying (Ohmic) Contacts Definition: The contact itself offers negligible resistance to current flow when compared to the bulk. The voltage dropped across the ohmic contacts is negligible compared to voltage drops elsewhere in the device. No power is dissipated in the contacts. Ohmic contacts can be described as being in equilibrium even in when currents are flowing. All free-carrier densities at an Ohmic contact are unchanged by current flow. The densities remain at their thermal-equilibrium values.
Tunneling Contacts By heavily doping the semiconductor, so that the barrier width is very small and tunneling through the barrier can take place.
Schottky Ohmic Contacts
Chapter 3. Junctions & Contacts 3.1 Metal-semiconductor contacts 3.2 p-n junctions 3.3 Heterojunctions
Metal-Semiconductor Contacts A system of electrons is characterized by a constant Fermi level at thermal equilibrium. Systems that are not in thermal equilibrium approach equilibrium as electrons are transferred from regions with a higher Fermi level to regions with a lower Fermi level. The transferred charge causes the buildup of barriers against further flow, and the potential drop across these barriers increases to a value that just equalizes the Fermi levels. Similar phenomena in a single crystal with non-uniform doping.
p-n Junction E0 E0 electrons Ec Ec Ef Ei Ei Ef Ev Ev Systems that are not in thermal equilibrium approach equilibrium as electrons are transferred from regions with a higher Fermi level to regions with a lower Fermi level.
p-n Junction E0 Ec Ei Ef Ev E0 E Ec Ef Ei Ev A system of electrons is characterized by a constant Fermi level at thermal equilibrium.
p-n Junction E0 Ec Ei Ef Ev φi E0 Ec Ef Ei Ev The transferred charge causes the buildup of barriers against further flow, and the potential drop across these barriers increases to a value that just equalizes the Fermi levels
Graded Impurity Distributions We assume initially the majority carrier density equals the dopant density everywhere. We ask how equilibrium is approached. A gradient in the mobile carrier density diffusion of carriers. Carrier diffusion leaves dopant ions behind. Separation of charge field opposing the diffusion flow. Equilibrium is reached when diffusion is balanced by the field.
Potential The separation of the Fermi level from the conduction-band edge (or intrinsic Fermi level) represents the potential energy of an electron.
Field dn J n=q n n x q D n =0 in equilibrium dx Using mass-action law,
Density vs Potential Barrier
Poisson s Equation 2 d q = = p n N d N a 2 s s dx q q n=ni exp ; p=ni exp kt kt 2 d q q = 2 ni sinh N a N d 2 s kt dx Cannot be solved in the general case!
(i) Small Gradient Case Quasi-neutrality approx.: constant field: kt qλ
(ii) p-n Junction Depletion approximation quasi-neutral approximation
Depletion of Mobile Charge
Potential Barrier p = Na n= 2 i n Na p= 0 n= 0 d 2φ q = ( Nd Na ) 2 dx εs n = Nd p= ni2 Nd
Two Idealized Cases Linearly graded junction: a continuous gradient in dopant between n-type and p-type regions. Step junction (abrupt junction): a constant n-type dopant density changes abruptly to a constant p-type dopant density for example, formed by epitaxial deposition
Step Junctions
Step Junction Analysis In n-type region, Integration leads to Similarly, Continuation at x = 0 requires This is just the charge neutrality in the depletion region.
Step Junction Analysis In n-type region, In p-type region, Totally,
Comments At high dopant concentrations, use F-D distribution in stead. However, the practical result is that the Fermi level is very near the band edge. For example, for heavily doped p-type silicon and lightly doped silicon: Total depletion-region width is: There is only partial depletion near the edge of the depletion region.
Partial Depletion
Debye Length 16 3 Typically, N d ~10 cm, L D ~40 nm
Linearly Graded Junction
Do It Yourself!
Applied Bias Everything works as before. Just replace φ i by φ i Va For positive Va, the barrier to the majority carrier is reduced; depletion region is narrowed; the junction is forward biased; appreciable currents can flow under small forward bias. For negative Va, the barrier to the majority carrier increases; depletion region widens; the junction is reverse biased; there is very little current flow under reverse bias.
Depletion Width & Maximum Field Abrupt p-n junction Linearly graded junction
Capacitance
Varactor Under reverse bias VR C i V R n Question: Which one is more sensitive? Linearly graded junction? Or abrupt junction? LGJ: n = 1/3 AJ: n = 1/2 (more sensitive) Can you design an even more sensitive varactor?
Junction Breakdown
Avalanche Breakdown Q: Conservation of energy and momentum requires the original electron possess kinetic energy of at least? 3 Eg 2 Avalanche breakdown is confines to the central portion of the space-charge region, where the field is sizeable.
Zener Breakdown The WKB approximation
Currents in p-n junctions Generation and recombination Continuity equation Current-voltage characteristics Charge storage and diode transients
Recombination through Traps Additional reading: Shockley-Hall-Read recombination
Generation and Recombination Recombination rate: n ' n n0 U= = τn τn Excess carrier concentration Carrier lifetime
Minority Carriers Matter Quasi-neutral zone Quasi-neutral zone ==> current flow (I-V characteristics) & charge storage (transient behavior)
Ideal-Diode Analysis Consider excess holes injected into the n-regions, where bulk recombination through generation-recombination centers is dominant. 2 pn pn pn E = pn p p E D p G p R p 2 t x x x In the quasi-neutral region, we have roughly E = 0. 2 pn p n pn p n0 =Dp 2 p t x Stationary:
Minority-Carrier Boundary Values Change of majority-carrier populations negligible. Detailed balance nearly applied for small enough applied bias. φi
Minority-Carrier Boundary Values Change of majority-carrier populations negligible. Detailed balance nearly applied for small enough applied bias.
Case 1: Long-Base Diode Length scale: diffusion length Lp
Hole Current
Total Current Minority-carrier electron current injected into p-region: Total current:
Case 2: Short-Base Diode Length scale: lengths of n- and p-type regions, WB and WE Ohmic contact
Current Assumption: No recombination occurs in n-type region. Hole current through n-type region: Total current:
Higher Order Corrections The ideal-diode equation is based on events in the quasi-neutral regions. The space-charge region is purely a barrier to diffusion of majority carriers and it plays a role only in the establishment of minority-carrier density at its boundaries. This is a reasonable first-order description of all the events. It is inaccurate, especially for silicon p-n junctions. Corrections due to events in the space-charge region are required.
Qualitative Picture
Diode Transients On: buildup time = stored charge / source current Off: turn-off time limited by how fast charge can be removed from the quasineutral region.
Charge Storage Mechanism 1: Majority carriers near the edges of the depletion region move as the depletion region expands or contracts in response to a changing bias. The charge storage in the depletion region is modeled by a small-signal capacitance, e.g., the abruptjunction depletion capacitance. Mechanism 2: Minority-carrier charge changes in the quasi-neutral region when applied bias is switched on or off. This can be modeled by another small-signal capacitance - diffusion capacitance. Equivalent circuit
Chapter 3. Junctions & Contacts 3.1 Metal-semiconductor contacts 3.2 p-n junctions 3.3 Heterojunctions
Will be discussed in the preparation of twodimensional electron gas.