Key Questions. ECE 340 Lecture 4 : Bonding Forces and Energy Bands 1/28/14. Class Outline: v Crystal Diffraction Crystal Bonding

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1 ECE 340 Lectue 4 : onding Foces and Enegy ands v Cystal Diffaction Class Outline: Things you should know when you leave Key Questions Why is the oh model useful? What is the Schödinge equation? What is a wavefunction? What ae the most common chemical bonds? What ae enegy bands? What is an enegy gap? Enegy and Fomation Cystal Diffaction Last time, we discussed cystal lattices and stuctue. How can we detemine this? Cystal stuctue is detemined in expeiment by studying the diffaction of suitable wave souces fom paticula sets of cystal planes. X-ay diffaction is most widely exploited in pactice and aises fom the inteaction of the incoming wave with the electon cloud of each atom in the cystal. SCHEMTIC ILLUSTRTION OF X-RY DIFFRCTION Y N TOM IN CRYSTL. THE ELECTRIC FIELD SSOCITED WITH THE INCOMING X-RY CCELERTES THE CLOUD OF ELECTRONS THT SURROUND THE TOM. THE CCELERTED ELECTRONS THEN EMIT SO-CLLED SECONDRY X-RYS WITH THE SME PHSE ND WVELENGTH S THE INCIDENT RDITION. WE MY THEREFORE CONSIDER ECH TOM IN THE CRYSTL TO E SOURCE OF SECONDRY WVES. THE STRENGTH OF THE SECONDRY X-RYS IS DETERMINED Y THE SCTTERING POWER OF THE TOM WHICH QUNTITY IS KNOWN S THE TOMIC FORM FCTOR. Cystal Diffaction The incident ays will scatte off the cystal. The CRUCIL featue of the atomic fom facto is that it is a STRONG function of scatteing angle because the atom does NOT scatte X-ays in an isotopic manne. The eason fo the anisotopic scatteing is the FINITE size of the atom itself which is COMPRLE to the wavelength of the X-ays. Consequently seconday X-ays that leave the atom fom diffeent pats of the electon cloud ae NOT in exact phase with each othe. Since the phase diffeence is moe significant fo the moe backscatteed waves the fom facto deceases with inceasing scatteing angle TOMIC FORM FCTOR SCTTERING NGLE THE FORM FCTOR ESSENTILLY MESURES THE SCTTERING POWER OF N TOM THE FORM FCTOR DECRESES WITH INCRESING SCTTERING NGLE INDICTING THT X-RYS RE MOST LIKELY TO E SCTTERED IN THE FORWRD DIRECTION THIS VRITION IS ILLUSTRTED SCHEMTICLLY HERE 1

2 Cystal Diffaction Now conside an incident beam of x-ays Each plane causes a eflection of a small potion of the incident ays. This numbe is usually a few pecent of the total incident beam. Waves in successive planes intefee with one anothe This pocess is citical fo the obsevation of cystal diffaction. SCHEMTIC IMGE ILLUSTRTING THE REFLECTION OF N INCOMING X-RY EM FROM PRTICULR SET OF CRYSTL PLNES ECUSE OF THE STRONG NGULR DEPENDENCE OF THE TOMIC FORM FCTOR WE MY CONSIDER THT ONLY SMLL FRCTION OF THE INCOMING WVE IS REFLECTED Y NY GIVEN PLNE Cystal Diffaction Typically, ays incident fom an abitay angle intefee destuctively Fo eflection fom a set of cystal planes with spacing d and with incident illumination of wavelength. Thee is a special set of angles fo which the intefeence is constuctive and so fo which stong eflection of the X-ay beam occus. This set of angles is given by the agg condition which is obtained by consideing the intefeence of waves eflected fom two successive planes. d dsin dsin I II y the time the two waves leave the cystal wave-ii has taveled a distance dsinθ LONGER than wave-i * PHSE-DIFFERENCE theefoe exists between the two waves which may be witten as ( d sin ) / λ * When this phase-diffeence is equal to an INTEGER the waves intefee constuctively and this leads to the RGG CONDITION d sin = nλ, n = 1,,3, Cystal Diffaction How is it done expeimentally? In the figue below, we show the measued X-RY DETECTOR vaiation of the eflected X-ay intensity at an SOURCE incident angle fo a cystalline sample of silicon cabide (SiC) majo peak is obseved at Θ = 36 at an PTH OF incident wavelength of 1.9 Ǻ coesponding to DETECTOR an inteplane spacing of 3. Ǻ. This agees with the known lattice spacing of CRYSTL silicon (111). The oh model of Hydogen +q To explain the spectum of photon emissions in hydogen oh poposed the following: 1. Electons exist in cetain stable obits. This assumption implies that the obiting electon does not give off adiation as classical electomagnetics would equie of a chage expeiencing angula acceleation. -q (111) PLNE IN SILICON Y. Sun et al. J. ppl. Phys. 8, 334 (1997) 4 m0q 13. 6eV E H = = ( 4πε0! n) n n = 1,, 3,. The electon may shift to an obit of highe o lowe enegy, theeby gaining o losing enegy equal to the enegy diffeence between the two layes. 3. The angula momentum of the electon in an obit is always an integal multiple of Planck s constant divided by Π

3 The Schödinge Equation: The oh model explains some things but is inadequate to explain many othe obsevable phenomena. Using the Schödinge Equation: Fee electon gas 0 L Use wave mechanics Schödinge equation ased on thee essential postulates: 1. Each paticle in the system is defined by a wavefunction. The wavefunction and its space deivative ae continuous, finite and single valued.. We must expess the nomal classical quantities with the new quantum mechanical fomulations. 3. The pobability of finding a paticle with a given wavefunction within a volume should be unity. V = 0 =ψ ( x + L, y, =ψ ( x, y + L, =ψ ( x, y, z + L) The solution wavefunctions: Momentum space becomes discetized ψ ( x) = exp ( i) x π 4π = 0; ± ; ± ;... L L 0 L n x,y,z = k y k z Ionic onding: NaCl Metallic onding : Sodium (Na) e - e - e - e- Sodium (Na) + Chloine (Cl) - z y x a Oute shell is only patially filled Sceening by othe chages makes the valence electon vey loosely bound. Oute electon contibuted to cystal as a whole. onding can be vey complex depending on the compound involved. 3

4 The most impotant type of bonding : Covalent Look to the simplest mateials Hydogen y assuming that only one electon is pesent, we can neglect the inteaction of electons in the molecule. We can still undestand the citical featues. Conside the case of two atoms and with gound state wavefunctions shown below: The esulting molecule is a supeposition of the two atomic wavefunctions ψ ± = ψ ± ψ ut anothe combination of wavefunctions exists. The antisymmetic wavefunction gives a educed pobability of finding the electon between the two atoms. The solution to the Schödinge equation shows that the symmetic state lies lowe in enegy. The symmetic state is efeed to as the bonding state and the antisymmetic state is efeed to as the anti-bonding state. ψ ψ ψ ψ + ψ * ψ + ψ + Symmetic State ψ * ψ ψ Visualizing the bonding vs. anti-bonding states E The vaiation of the total enegy of the hydogen molecule as a function of inteatomic sepaation is shown fo the bonding and anti-bonding states. Thee is no minimum pesent in the anti-bonding state indicating the absence of a stable molecula state. In the bonding state thee is a minimum which defines a stable molecula state at a cetain inte-atomic sepaation. When this occus thee is an enhanced pobability of finding the electon between the two atoms. WITH THE TOMS FR PRT FROM ECH OTHER THE ONDING ND NTI-ONDING STTES RE EQUL IN ENERGY. Now we can examine elemental and compound semiconductos. Each atom shaes bonds with 4 othe atoms. Silicon bonds ae covalent, but compound semiconductos have a mix of ionic and covalent bonds. Depends on sepaation on peiodic table. 1s s p x p y p z 3s 3p x 3p y 3p z DIMOND CRYSTL STRUCTURE WITH TETRHEDRL ONDING EXMPLES INCLUDE SILICON & GERMNIUM E - INTER-TOMIC SEPRTION T SUCH LRGE SEPRTIONS THE TOTL ENERGY OF THE TWO TOMS IS EQUL TO THT OF PIR OF INDEPENDENT HYDROGEN TOMS. S THE INTER-TOMIC SEPRTION IS REDUCED HOWEVER THE ENERGY OF THE ONDING STTE DECRESES FSTER THN THT OF THE NTI-ONDING STTE. When silicon atoms COMINE to fom a cystal the s- and p- obitals HYRIDIZE to fom so-called sp 3 ORITLS that ae mixtues of the s- and p-obitals. THE EXISTENCE OF MINIMUM ENERGY IN THE ONDING STTE DETERMINES THE EQUILIRIUM E + SEPRTION OF THE HYDROGEN TOMS IN THE MOLECULE. s-oritl p-oritl sp 3 -ORITL 4

5 Enegy and Fomation ing atoms togethe, the wavefunctions begin to ovelap. Enegy and Fomation POTENTIL Lage Sepaation IN SINGLE TOM ELECTRONS RE TRPPED IN POTENTIL WELL WHEN MNY TOMS COMINE ND FORM CRYSTL THE TOMICPOTENTILS OVERLP GIVING RISE TO PERIODIC VRITION Small Sepaation Enegy ands We conclude by looking at the fee electon model llowed enegy values: n x,y,z = ( kx + k y kz ) E k =! + m So, let s look at a 1D cystal with peiodic bounday conditions: =ψ ( x + L, y, Wavefunction solutions ae in the fom of tavelling waves: ψ ( x) = exp ( i) Solutions to the Schödinge equation discetize the allowed values: a x π 4π = 0; ± ; ± ;... L L Enegy ands If the wavevectos at = ± Π/a, satisfy the agg condition d dsin ρ( + ) = ψ ( + ) cos dsin Examine the density: ρ( + ) = ψ ( ) sin a I a II agg Condition: d sin = nλ, n = 1,,3, Then the esultant wave is a combination of left and ight going waves (Standing wave): i i ψ ( + ) = exp + exp cos a a a i i ψ ( ) = exp exp i sin a a a 5

6 Enegy ands To find the enegy gap, we look at the enegies: Let s assume that the potential in ou cystal can be witten as: U ( x) = U cos The enegy gap is the enegy diffeence between these two standing waves: E g 1 [ ] = dxu( x) ψ ( + ) ψ ( ) 0 a 1 = dxu cos a cos sin a a 0 = U Enegy ands y putting the atoms togethe, we get an enegy gap Distance (x) The top band is efeed to as the conduction band. t low tempeatues it is mostly empty of electons. The bottom band is efeed to as the valence band. t low tempeatues it is almost entiely filled with electons. E c E v E g Enegy 6