3. Two-dimensional systems

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Jessie Bryan
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, graphite (with C nanostructures) Type II: artificial structures, heterojunctions

1 3. Two-dimensional systems Image from IBM-Almaden 1 Introduction Type I: natural layered structures, e.g., graphite (with C nanostructures) Type II: artificial structures, heterojunctions Great technological importance: microelectronics industry based on electrons at interfaces semiconductor-semiconductor, insulator or metal. 2

2 Contents Surfaces and interfaces Junctions (metal-metal, metal-semiconductor, semiconductorsemiconductor, metal-oxide-semiconductor (MOS)) Quantum wells and supperlattices. Quantum Hall effect. Applications Semiconductor transistors (bipolar, field-effect, modulationdoped devices) Opto-electronic devices (solar cells, photodetectors, lightemitting diodes, semiconductor lasers) Summary 3 Surfaces and interfaces Material interfaces: Boundary between material and vacuum : - Break of lattice symmetry - Bloch wave picture break down, but it is recovered far from interface Boundary between one type of material and another : - Much more complicated. - Many possibilities 4

3 Surfaces and interfaces Important concepts: Work function! The amount of energy required to take an electron from the Fermi level within a material and remove it to infinity. Fermi level, EF The energy of the highest occupied single particle state in the material at T=0, must be negative. Ws,Work required to remove an electron through the surface layer:!= difference between energy at infinity and energy of bound electron = -E F +Ws. Vacuum level,! vac Vacuum potential level, relative to the band bottom W vacuum! EF Band width, W 5 Surfaces and interfaces Why is there a surface layer effect? Charge spills out into empty space, resulting in a dipole layer pointed toward the metal. Surface layer effect can depend on particular crystal face and polarity of bonding. Single atomic layers of junk can strongly affect! by altering the surface charge layer. In practice, work function is measured empirically -photoemission, thermionic emission. (~ 5eV for Au) positive background + - dipolar barrier electronic charge 6

4 Surfaces and interfaces 7 Surfaces and interfaces Surface states Occupied free surface states of Cu (111), confined by a ring of iron atoms. Image from IBM-Almaden Spillage of electrons across interface " bound states tied to the interface. Tamm states -general consequence of breaking periodic potential. Surface states are bound in z, but may be free in x and y. Disorder (impurity, unsatisfied chemical bond, vacancy) can lead to surface states that are localized in all 3 directions. Surface states may be empty or full, depending on Fermi level. 8

5 Surfaces and interfaces Tamm states Kronig-Penney model V(x) Proposed in previous lecture for infinite crystal b x Tamm model: -V 0 a K-P model in a crystal with periodic boundary " bands of allowed states, eachw ith real values of k. Allowed single-particle states are Bloch waves, and are delocalized ; the wavefunctions extend throughout the sample. Tamm model: to meet b.c., k has to become complex, " wavefunction to exponentially decay away from surface. mm model: V(x) V 1 Results: b For large crystal, complex part of k is small for states in the x middle of a band; decay length is long compared to crystal -V size. These states are relatively unaffected. 0 a A new state appears, one for each band, in the gap. State has large complex component of k, and is spatially localized at sample edge. 9 Metal-metal junction!! Join two metals with different work functions: EF EF Electrons flow from system of higher chemical potential to that of lower chemical potential. Total electrochemical potential (including voltage) must end up being uniform across junction. Conventional way of drawing: shift bands to allow EF to be uniform across sample Bands effectively bend because of double charged layer interface Double charge layer because departing electrons leave behind ion cores. Thickness of charge layer called depletion width ; atomic scale in metals. 10

6 Metal-metal 11 vacuum Before contact: EF EC EV After contact: Depletion width much larger than in metal case. Schottky barrier Schottky barrier makes it difficult to inject electrons from metal into semiconductor" nonlinear IV behavior diode (used for devices). Very small barriers can result in almost Ohmic contact. 12

7 13 Φ S > Φ M Ohmic contact Φ S < Φ M Blocking contact Electron affinity Energy barrier Electron accumulation layer Shottky barrier Depletion layer Ohmic contact Facile charge flow between metal and semic. (Chemical potential! in conduction band) Barrier height E B,n = Φ M χ S =(Φ M Φ S )+(E C E F ) 14

Shottky barrier involving n-type and p-type semiconductors Shottky barrier depletion region accumulation region But barrier height not so dependent on metal workfunction, strong dependence for

8 Shottky barrier involving n-type and p-type semiconductors Shottky barrier depletion region accumulation region But barrier height not so dependent on metal workfunction, strong dependence for wide-gap semiconductor (ZnS) but very weak for small-gap semiconductor (Si, GaAs) The Fermi level is pinned at a particular energy level in a band of interface states which lie in the gap Barrier height strong for wide-gap semiconductor (ZnS) 15 Metal induced gap states (MIGS) The filling of some interface states created dipoles " vacuum levels of metal and semic. displaced δ M 16

9 Effect of external potential V on Shottky barrier (n-type semic.) Now electrochemical potential η = µ ev is constant Forward bias (semic. +) Reduce barrier Reverse bias (semic. -) Increase barrier 17 "A Schottky barrier is a rectifying contact (non-ohmic): Forward V "allow I large (e- form semic. to metal) j = j o [e ev/k BT 1] Reverse V " only small current density due to thermionic emission over barrier) j 0 = AT 2 e E B,n/k B T Richardson-Dushman eq. A = 4πem ek 2 B/h 3 Doping heavily the surface " n+ (or p+) layer " allow electron tunnelling through barrier "similar I for both V polarities " quasi-ohmic contacts 18

direct gap (c-gaas) indirect gap (c-si, Ge 19 Semiconductor-semiconductor

band-gaps) 2) Homojunctions (the same semiconductors with different dopings)

" band-bending Commonly used structure: GaAs/AlGaAs interface (intrinsic)

10 direct gap (c-gaas) indirect gap (c-si, Ge 19 Semiconductor-semiconductor Two possibilities : 1) Heterojunctions (semiconductors with different band-gaps) 2) Homojunctions (the same semiconductors with different dopings) For example: the p-n junction p-type n-type Depletion region If different " " band-bending Commonly used structure: GaAs/AlGaAs interface (intrinsic) (n-type) Electrons confined in GasAs condunction band potential well (2D electron gas) 20

Semiconductor-semiconductor Homojunctions: p-n junction Host: c-si; n: e- carriers in cond.

equal Mirror images; V for q>0, band for e-, <0) d = dn + dp Φc Contact potential associated to charged double layer Φc = kb T ln e!

1016 m 3 ) Intrinsic carrier concentration 21 Semiconductor-semiconductor Effect of external potential V on p-n junction electron flow Reduce

11 Semiconductor-semiconductor Homojunctions: p-n junction Host: c-si; n: e- carriers in cond. band ; p: holes in valence band Contact " diffusion of e- into p; holes into n "! equal Mirror images; V for q>0, band for e-, <0) d = dn + dp Φc Contact potential associated to charged double layer Φc = kb T ln e! Nd Na n2i " Responsible for band bending donor, acceptor concentrations T! 300K Φc! 0.7 V, for asi doped Nd = Na = 1022 m 3 (ni! 1016 m 3 ) Intrinsic carrier concentration 21 Semiconductor-semiconductor Effect of external potential V on p-n junction electron flow Reduce "c barrier"increase I (Now electrochemical potential is constant) Increase "c barrier"reduce I e- and h diffusion coeficients Rectifying behaviour as in Shottky barrier! " De Dh 2 e- and holes. Total current : sum of Io = eni A + ev /kb T I = Io [e 1] Le Na 22 Lh Nd diffusion lengths

Semiconductor-semiconductor In V bias high & high p-n doping "tunneling of e- and h through " c barrier Depletion layer d decrease " enhance junction current Large

In very heavily doped p-n junction semiconductor deg. "cond. band min. of n layer below valence band max of p-layer (!

12 Semiconductor-semiconductor In V bias high & high p-n doping "tunneling of e- and h through " c barrier Depletion layer d decrease " enhance junction current Large reverse V " conduction band in n lower the band in p " for narrow d, tunnelling of e- form p valence into conduction band : Zener breakdown 23 Semiconductor-semiconductor In very heavily doped p-n junction semiconductor deg. "cond. band min. of n layer below valence band max of p-layer (! in one or other bands out of d) Leo Esaki The Nobel Prize in Physics 1973 The tunel diode (first quantum electron device) d very small "tunneling of e- from cond. band in n layer to valence band of p layer " for small V large I 24

13 Metal-oxide-semiconductor (MOS) 25

Figure 3.1 (p. 141) Figure 3.2 (p. 142)

Figure 3.1 (p. 141) Figure 3.2 (p. 142) Figure 3.1 (p. 141) Allowed electronic-energy-state systems for two isolated materials. States marked with an X are filled; those unmarked are empty. System 1 is a qualitative representation of a metal;