3. Semiconductor heterostructures and nanostructures

Transcription

1 3. Semiconductor eterostructures and nanostructures We discussed before ow te periodicity of a crystal results in te formation of bands. or a 1D crystal, we obtained: a (x) x In 3D, te crystal lattices can be of different type. or instance, for cubic lattices simple body centered face centered 110

2 us, for 3D crystals, complex functions in te 3D reciprocal space are obtained. e rillouin zones, wic determine periodic volumes in te space, are complex polyedra. rillouin zones of te face centred & body centred cubic lattices Points of ig symmetry are denoted by Γ, L, X, etc. are ten plotted as curves along tese symmetry directions 111

3 e important distinction between a semiconductor and a metal is made on te basis of te occupancy of electronic states Wen te igest occupied state is identical wit te upper energetic edge of an electronic band wic is separated from iger bands by an absolute gap semiconductor no conduction at 0K, as te electron cannot be excited due to te presence of te gap In metals, te igest occupied energy level falls in te continuum of an allowed energy band (partially filled band) Metal Semiconductor 11

4 We ave seen before tat 1 m * ij for a cubic lattice 1 m * 1 1 d d i j inverse of te curvature of te energy band is related to te effective mass m * 1 µ m effective mass and mobility are related by * te carrier mobility can be directly calculated from te curvature of te energy bands 113

5 lose to te edge of te valence and conduction band, (parabolic dependence). We ave already calculated te density of states for a 3D crystal wit a parabolic dependence (slides 4 & 43), wic now referred to te edge of te conduction ( ) and valence ( ) bands can be written as ρ (m ) π * 3 n 3 ( ) > ρ (m ) * 3 p 3 π ( ) < 114

6 115 o calculate te number of electrons in te conduction band, and oles in te valence band: d n ) ( I ρ [ ]d p ) ( 1 I ρ [ ] + I e exp 1 1 ) ( ) ( m n n exp 3 * π wit m p p exp 3 * π N N effective density of states in te valence band effective density of states in te conduction band

7 116 At 0K, all electrons are in te valence band, te conduction band is empty zero conductivity (a completely filled band cannot conduct current) At finite temperatures, electrons ave a finite probability to be in te conduction band. Due to carge neutrality pn + + * * ln 4 3 n p m m m m p n exp exp 3 * 3 * π π

8 en, n i G pi N N exp intrinsic carrier concentration exponential dependence on te band gap e number of carriers at room temperature is extremely low for large band gaps 117

9 Impurities (donors and acceptors) are used to introduce carriers in te bands: Donors: provide extra electrons compared to te atoms of te semiconductor matrix (for instance Si doped wit As) Acceptors: ave a lower number of electrons to be used in te bonds, compared to te atoms of te semiconductor matrix (for instance Si doped wit ) e presence of impurities can cange te position of te ermi level 118

10 Semiconductor alloys e alloying of two or more semiconductor materials, for instance Si and Ge or GaAs and AlAs, is a powerful tool in order to control te band structure. onsider an alloy consisting of two components: A wit a fraction x, and, wit a fraction 1 x. If A and ave similar crystalline lattices, one can expect tat te alloy A x 1 x as a lattice constant a c given by a c a x + a ( 1 x) egard s law A irtual crystal approximation: certain parameters of te alloy, suc as band gap, can be caracterized as a function of te fraction x. alloy g g (x) 119

11 Semiconductor eterostructures Semiconductor structures wit two or more abrupt interfaces at te boundaries between te regions of different materials abrupt cange in te energy gap and te positions of conduction () and valence () band. Now it is important to consider te absolute energy position of conduction and valence bands. e reference level is te vacuum level, wic coincides wit te energy of te electron outside of a material. lectron affinity χ of a material: energy distance between te bottom of te conduction band and te vacuum level energy required to remove an electron from te bottom of te conduction band to outside te material If we now χ for different materials, we now te position of te conduction band bottoms wit respect to eac oter. 10

12 Semiconductor eterostructures 11

13 Semiconductor eterostructures us, we can calculate te discontinuity in te conduction band at an abrupt eterojunction of two materials A and,, A ( χ ) ( χ ) vac vac A χ χ A Similarly, te discontinuity of te valence band,, A χ χ A + g wit g g, A g, Different band alignment can be expected for two semiconductor materials, depending on te electron affinities and te bandgaps a) type I eterostructure. Δ and Δ ave opposite sign, and te lowest states occur in te same part of te structure as te igest states. (GaAs/Al x Ga 1 x As, for x<0.4) b) type II eterostructure. te lowest states ( min ) and te igest ( max ) states occur in different parts & min max < min { ga, g } (some Si/SiGe) c) broen gap line up. min < max (InAs/GaSb) 1

14 Lattice matced and pseudomorpic eterostructures e quality of te material near eterointerfaces depends strongly on te ratio of lattice constants for te two materials. i) lattice matced structures: te lattice constants of te materials are nearly matced (tis is te case of AlGaAs/GaAs) ii) lattice mismatced structures: tere exists a finite lattice mismatc of te lattice constants of te materials if te upper layer adopts te lattice of te substrate, by elastic deformation of its lattice, te resulting structure is strained (pseudomorpic). 13

15 Lattice matced and pseudomorpic eterostructures Due to te elastic deformation, a layer grown on a substrate wit mismatced lattice sould possess extra elastic energy, el, caused by te strain. el d S U elastic energy increases wit layer ticness d layer area elastic energy density If te layer accumulates a large el imperfections are formed (relaxation of te strain by formation of defects). e generation of defects, imperfections required certain energy im. us, for certain ticnesses, el (d) < im, wic means tat tere is no sufficient strain energy and imperfections are not generated. d cr is te critical ticness for wic te layers accommodate te strain witout defect generation pseudomorpic eterostructures 14