3.225 Electrical,Optical, and Magnetic Properties of Materials

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1 3.5 Electrical,Optical, and Magnetic Properties of Materials Professor Eugene Fitzgerald Purpose: connect atoms and structure to properties Semi-historical context What was understood first from the micro to the macro? What was missing to explain other materials?

2 Origin of Conduction Range of Resistivity Electrical Resistivity (ohm-m) Log Electrical Conductivity (S/m) Why? intercalated graphite graphite (in-plane) graphite (out of plane) polyacetylene (doped) TTF-TCNQ polyacetylene (undoped) Bakelite polypyrrole Lucite (PMMA) polyvinyl chloride polyethylene, teflon copper, silver iron -6 6 stainless steel, metallic glass -5 5 YBa Cu 3 O 7 (ab plane) -4 4 YBa Cu 3 O 7 (c-axis) -3 3 silicon (doped) ZnO (doped) - seawater - Fe 3 O germanium - - silicon 3-3 lnsb water AgCl ZnO (undoped) NaCl Al O mica silica 6-6 diamond Figure by MIT OpenCourseWare.

3 Response of material to applied potential I V=f(I) I V R e- Rectification, Non-linear, Non-Ohmic Linear, Ohmic V V=IR Metals show Ohmic behavior Microscopic origin?

4 Remove geometry of material V In general, L V=IR=IR/L R=L/(σA) r J= ~ σe r All material info In cubic material, W Isotropic material Anisotropic material J x σxx Jy = σxy J z σxz σ σ σ xy yy yz J=σE σ xz Ex σ yz Ey σ zz Ez E E I J J

5 Microscopic Origin: Can we predict Conductivity of Metals? Drude model: Sea of electrons all electrons are bound to ion atom cores except valence electrons ignore cores electron gas J=σE=-nev, by definition of flux through a cross-section n=number of electrons per volume v=velocity of the carriers due to electric field--> drift velocity Therefore, σ=-nev/e and we define v=-μe μ is mobility, since the electric field creates a force on the electron F=-eE σ = neμ

6 Does this microscopic picture of metals give us Ohm s Law? E F=-eE F=ma m(dv/dt)=-ee v=-(ee/m)t J=σE=-nev=ne Et/m σ=ne t/m v,j,σ,i E t Constant E gives ever-increasing J t No, Ohm s law can not be only from electric force on electron!

7 Hydrodynamic representation of e- motion p=momentum=mv dp () t pt () = + F () t + F ( t ) +... dt τ Response (ma) Drag Driving Force Restoring Force... dp () t pt () dt τ ee Add a drag term, i.e. the electrons have many collisions during drift /τ represents a viscosity in mechanical terms

8 In steady state, p -eeτ dp () t = 0 dt t pt () = p ( e τ ) p = ee τ If the environment has a lot of collisions, mv avg =-eeτ v avg =-eeτ/m τ t Now we have Ohm s law σ = ne τ m μ = τ e m

9 Mobility μ in Free-Electron Theory Between collisions with the lattice atoms, each electron experiences a force from the electric field given by F = -ee and therefore an acceleration given by a = F/m = -ee/m Theory assumes that the energy picked up from the electric field by the electron between collisions is delivered to the lattice in each collision, so acceleration must start again. v D If the average time between collisions is τ, the average velocity is t v = (ee/m)τ D μ = v = eτ D E m

10 Predicting conductivity using Drude n theory from the periodic table (# valence e- and the crystal structure) n theory =A V Zρ m /A, where AV is 6.03x0 3 atoms/mole ρ m is the density Z is the number of electrons per atom A is the atomic weight For metals, n theory ~0 cm -3 If we assume that this is correct, we can extract τ Element Z n (0 /cm 3 o ) r s (A) Li (78 K) Na (5 K) K (5 K) Rb (5 K) Cs (5 K) Cu Ag Au Be Mg Ca Sr Ba Nb Fe Mn (α) Zn Cd Hg (78 K) Al Ga In Tl Sn Pb Bi Sb r s /a Table by MIT OpenCourseWare.

11 Extracting Typical τ for Metals τ~0-4 sec for metals in Drude model Element 77 K 73 K 373 K Li Na K Rb Cs Cu Ag Au Be Mg Ca Sr Ba Nb Fe Zn Cd Hg Al Ga In Tl Sn Pb Bi Sb Drude relaxation times in units of 0-4 second Table by MIT OpenCourseWare.

12 Thermal Velocity So far we have discussed drift velocity v D and scattering time τ related to the applied electric field Thermal velocity v th is much greater than v D x x L=v D t x mvth = 3 kt v th = 3kT m Thermal velocity is much greater than drift velocity

13 Additivity of Resistivity In metals & alloys, the various contributions to the scattering of free electrons, and therefore to resistivity, are approximately additive, i.e., = ρ ρ i ρ σ i (ohm-m) Cu-%Sn, deformed dislocations x0-8 0 Cu Cu-%Sn solute atoms lattice vibrations Temperature (K)

14 Example: Conductivity Engineering Objective: increase strength of Cu but keep conductivity high σ = ne τ μ = eτ Scattering length m m connects scattering time l = vτ to microstructure Dislocation (edge) e- l decreases, τ decreases, σ decreases

15 Example: Conductivity Engineering Can increase strength with second phase particles As long as distance between second phase< l, conductivity marginally effected L L+S S α α+l L α+β β+l β Sn Cu X Cu microstructure S L β α L>l dislocation Material not strengthened, conductivity decreases Dislocation motion inhibited by second phase; material strengthened; conductivity about the same

16 Example: Conductivity Engineering Scaling of Si CMOS includes conductivity engineering One example: as devices shrink vertical field increases τ decreases due to increased scattering at SiO /Si interface increased doping in channel need for electrostatic integrity: ionized impurity scattering τ SiO <τ impurity if scaling continues properly G SiO S D E vert Ionized impurities (dopants)

Electrical conduction in solids

Equations of motion Electrical conduction in solids Electrical conduction is the movement of electrically charged particles through a conductor or semiconductor, which constitutes an electric current.