Crystal Properties Crystal Lattices: Periodic arrangement of atoms Repeated unit cells (solid-state) Stuffing atoms into unit cells Determine mechanical & electrical properties High performance, high current N O ZnO GaN 1
Crystal Structure The simplest structural unit is the unit cell. The periodic lattice: The basic There may be several different shapes of the combination of the circles, however, there is only one which is the basic. Unit Cell The lattice parameters a, b, and c are unit cell lengths. The lattice parameters α, β, and γ are angles between adjacent unit-cell axes
Seven Crystal Systems & (Reference) Fundamental crystal geometry has the seven crystal systems and 14 crystal lattices. First, there are only seven unique unit-cell shape that can be stacked together to fill three-dimensional space. These are called Seven Crystal Systems.
Fourteen Bravais Lattices (Reference) Second, consider the points where the atoms are stacked in a given unit-cell. These points are called lattice points. And there is 14 possible ways of arrangement of the lattice points, called 14 Bravais Lattices. (14 possible way of arrangement of the lattice points in unit-cells.)
The Silicon lattice: Si atom: 14 electrons occupying the lowest 3 energy levels: 1s, s, p orbitals filled by 10 electrons 3s, 3p orbitals filled by 4 electrons Each Si atom has four neighbors Diamond lattice : Two fcc structures How many atoms per unit cell? Ge is also diamond structure Zinc blende lattice GaAs, AlAs, InP Two intercalated fcc lattices Wurtzite lattice GaN, CdS, ZnS 5
Crystal Planes Crystallographic planes and Si wafers Standard Projection of Si (100) wafer 0 Si wafers usually cut along {100} plane with a notch or flat side to orient the wafer during fabrication Standard Projection of Si (110) wafer? 6 Crystal planes and directions of Semiconductor are very important since many properties (e.g. etching rate, anisotropy etching, mobility etc)
Crystal Planes vs Mobility (Reference) 7
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Si anisotropy etching theory (KOH or TMAH) Etch rate = (110): (100): (111)= 300:600:1 Anisotropy of etching attributed to the density of free bonds, flatness and oxidation of Si during wet etching 1. Density of free bonds=100:110:111=1:0.71:0.58. Flatness of the surface (111)>(100)>(110) 3. Oxidation of Si during wet etching Crystal plane & Wet etching (111) surface covered with instant silicon oxide (hydrated silicon oxide, (111) has most fast oxidation property) during the wet etching and this blocked etching speed. ( Reference: Ann. Rev. Mater. Sci. 1979. 9:373-403) Si etching chemistry in base (KOH or TMAH) Oxidation Si + OH - + 4h + -> Si(OH) + Reduction 4H O 4OH - + H + 4h + Overall Si + OH - + 4H O Si(OH) + + H (bubble)+ 4OH - Si(110) wafer 9
Atoms and Electrons In 190, many observation revealed that atom and electrons did not obey the classical laws of mechanics. New approach, called quantum mechanics, was proposed. Heisenberg uncertainty principle: ( x )( p x ) ( E )( t ) Simultaneous measurement of Position and momentum or of Energy and time are inherently inaccurate. We must look for not the accurate position but the probability of finding an electron at a certain position. Schrödinger Wave Equation (196): Quantum mechanics as a wave mechanics ( :Laplacian) Time-independent equation : Wave function :Probability 10
Atoms and Electrons E n n ml 11
Case : Hydrogen Atom and extending periodic table B.C: nlm V ( r,, ) (4 1 0 ) q r ( r,, ) R ( r) ( ) ( ) n Atoms and Electrons l m Restrictions: n=1,,3. l=0,1,,, (n-1) m=-l,,-,-1,0,1,,...,l s= 1
Case 3: Crystal Lattice Energy Bands Energy levels when atoms are far apart: Energy levels when atoms are close together (their potential wells interact): Energy levels from discrete atoms to crystal lattice: cf) To solve energy band gap of crystal lattice in QM, Kronig-Penny model is used along with wave equation 13
E-k Diagram For free electrons in metals, E k m Now, superimpose a simple periodic potential. 14 a You learned X-ray Bragg s reflection and it is equally applicable to electron wave. n a & k Electrons with k=nπ/a cannot propagate through the crystal, since they are reflected back due to Bragg s diffraction.
E-k Diagram The energy band becomes discontinuous when internal diffraction occurs at k=nπ/a. Thus, superimposing periodic potential introduces energy band gap into E-k diagram. A displacement of k by nπ/a does not change the graph because of the periodic nature of crystalline structure. The periodicity of a crystal is different for each direction. Therefore the energy diagram would look different depending on the direction of k. Thus we need to calculate E-k diagram for k along all directions to obtain a complete picture of energy band diagram 15
Energy Bands E-k diagram for Ge, Si, and GaAs Indirect Indirect Direct Indirect: phonon + photon Direct: photon (LED or Laser) (phonon: lattice vibration heat) Constant energy surface of Si conduction band minima along X directions Ellipsoidal for 3 dimensional k vectors m l, m t different. k z m l m t k y m t k x 16 Effective mass: Due to interaction with periodic potential of the lattice, electron motion in crystal lattice is not same as that of free space. Effective mass compensates the influence of the lattice, so that the electrons and holes can be treated as almost free carriers in most computations. E 1 mv p m k m d E dk m
E-k Diagram Small band gap: semiconductor No band gap: metal 17
E-k Diagram After summing E-k curves for all crystallographic directions, three dimensional real E-k diagram can be obtained. 6 equivalent conduction minima at X Compare with free electrons. 18
Energy Bands Band structure at 0K Lower bands are filled with electrons, higher bands are empty in a semiconductor & insulator The difference lies in Eg. Small bandgap of semiconductor allows excitation. In metal, Eg is very small, or conduction band and valance band is overlapped. Typical examples of bandgap 19 10 5 Ohm-cm 10 16 Ohm-cm 10-5 Ohm-cm
Intrinsic vs Extrinsic Intrinsic Semiconductor For semiconductor, where are all electrons at T=0 K? No charge carriers at T=0K What about at T=300K? In thermal equilibrium, generation = recombination At room temperature (T=300K), intrinsic concentrations: ni ~ x 10 6 electrons and holes per cm 3 in GaAs 1.5 x 10 10 cm -3 in Si x 10 13 cm -3 in Ge ri g i n p n i (T n & p increase equally) (for intrinsic) By kt=0.06ev(rt), valence electrons become conduction electrons. What fraction of Si atoms contributes to intrinsic electron concentration at RT? (a=5.43 x10-8 cm -3 ) 0 8 a 3 1.5 x10 = 5 x10 #/cm 3 5 x10 10 = 3 x10-13
Intrinsic vs Extrinsic Extrinsic Semiconductor Doping = purposely introducing impurities into the crystal. 1 Does n-type Si have only electron carriers and no holes? No.
n i vs Temp Doping vs Resistivity ex) Intrinsic Si x10 5 Ohm-cm, 10 15 As atoms/cm 3 doping in Si 5 Ohm-cm
Ionization energy (ev) Donor in Si P As Sb Ionization energy (ev) 0.045 0.054 0.039 Acceptor in Si B Al Ga In Binding energy (ev) 0.045 0.067 0.07 0.16 3