Other aspects of conduction

Other aspects of conduction The transport of electrons/ions/vibrations in a metal have a number of other important effects. Thermal conductivity (metals electrons, non-metals phonons) Ionic conduction Skin effect high speed signals, RF resistivity. Thin film effects Interconnect design RC delays, technology changes. Electro-migration reliability of interconnects Fig 2.19

Thermal conduction In a metal two types of heat transport Electron gas (which is dominant). Lattice vibrations (phonons) Heat flow can be described by a diffusion equation (hot flows to cold). The temperature of lattice is what we feel as heat, the electrons are usually in thermal equilibrium with the lattice and Te = Tl,, but not always the case. In non-metals with few conduction electrons, phonons (lattice vibrations) dominant. The balls on spring models is used to model the process of vibrational waves traveling through the material. Very useful to view heat transfer like current flow and define a thermal resistance.

H O T C O LD H E A T E le c tro n G a s V ib ra tin g C u + io n s Thermal conduction in a metal involves transferring energy from the hot region to the cold region by conduction electrons. More energetic electrons (shown with longer velocity vectors) from the hotter regions arrive at cooler regions and collide there with lattice vibrations and transfer their energy. Lengths of arrowed lines on atoms represent the magnitudes of atomic vibrations. Fig 2.19

δt H O T C O L D dq dt H E A T A δx Heat flow in a metal rod heated at one end. Consider the rate of heat flow, dq/dt, across a thin section δ x of the rod. The rate of heat flow is proportional to the temperature gradient δ T/δ x and the cross sectional area A. Fig 2.20

Fourier s Law of Thermal Conduction dq δt Q = = κa dt δx Q = rate of heat flow, Q = heat, t = time, κ = thermal conductivity, A = area through which heat flows, dt/dx = temperature gradient Ohm s Law of Electrical Conduction δv I = Aσ δx I = electric current, A = cross-sectional area, σ = electrical conductivity, dv/dx = potential gradient (represents an electric field), δv = change in voltage across δx, δx = thickness of a thin layer at x

Fourier s Law T T Q = A κ = L (L / κ A) Q = rate of heat flow or the heat current, A = cross-sectional area, κ = thermal conductivity (material-dependent constant), T = temperature difference between ends of component, L = length of component Ohm s Law V V I= = R ( L / σa) I = electric current, V = voltage difference across the conductor, R = resistance, L = length, σ = conductivity, A = cross-sectional area

Definition of Thermal Resistance T Q = θ Q = rate of heat flow, T = temperature difference, θ = thermal resistance Thermal Resistance L θ= Aκ θ = thermal resistance, L = length, A = cross-sectional area, κ = thermal conductivity

Thermal Diffusion Eq Go the diff.pdf!

A g 400 A g -3 C u C u T h e r m a l c o n d u c tiv ity, κ (W K -1 m -1 ) 450 A g -2 0 C u 300 κ σ = T C A u W FL A l 200 W N i 100 B e M g M o B ra ss (C u -3 0 Z n ) B ro n z e (9 5 C u -5 S n ) S te e l (1 0 8 0 ) P d -4 0 A g H g 0 0 10 20 30 40 E le c tr ic a l c o n d u c tiv ity, σ, 1 0 50 6 Ω -1 6 0 m 70-1 Thermal conductivity, κ vs. electrical conductivity σ for various metals (elements and alloys) at 20 C. The solid line represents the WFL law with CWFL 2.44 108 W Ω K-2. Fig 2.21

Wiedemann-Franz-Lorenz Law κ 8 2 = CWFL = 2.45 10 W Ω K σt κ = thermal conductivity σ = electrical conductivity T = temperature in Kelvins CWFL = Lorenz number 1/T dependence is due to the difference relationship of the two conductivities to the velocity distribution of the particles

T h e r m a l c o n d u c tiv ity, κ (W K -1 m -1 ) 50000 C opper 10000 A lu m in u m 1000 B ra s s (7 0 C u -3 0 Z n ) 100 A l-1 4 % M g 10 1 10 100 T e m p e ra tu re (K ) 1000 Thermal conductivity vs. temperature for two pure metals (Cu and Al) and two alloys (brass and Al-14%Mg). Data extracted from Thermophysical Properties of Matter, Vol. 1: Thermal Conductivity, Metallic Elements and Alloys, Y.S. Touloukian et. al (Plenum, New York, 1970). Fig 2.22

Thermal conduction in non-metal is due to lattice vibrations E q u ilib riu m H ot C o ld E n e rg e tic a to m ic v ib ra tio n s Conduction of heat in insulators involves the generation and propogation of atomic vibrations through the bonds that couple the atoms. (An intuitive figure.) We shall find that the waves travel as packets that carry heat. (Phonons) Fig 2.23

Q = T /θ T H ot T C o ld Q Q Q A θ L (b ) (a ) Conduction of heat through a component in (a) can be modeled as a thermal resistance θ shown in (b) where Q = T/θ. Fig 2.24

Semiconductor As you know from 398 (and we shall look at in more depth) semi-conductors have two carriers Electrons negative charge, positive mass Holes positive charge, positive mass (virtual particle for accounting) Both are modeled as free gas with Boltzmann distribution. We define n and p as the concentrations of electrons and holes And the current flow is due to both particles Hall effect is modified due to the two different carriers Fig 2.25

E h o le a e - b c (a) Thermal vibrations of the atoms rupture a bond and release a free electron into the crystal. A hole is left in the broken bond which has an effective positive charge. (b) An electron in a neighbouring bond can jump and repair this bond and thereby create a hole in its original site; the hole has been displaced. (c) When a field is applied both holes and electrons contribute to electrical conduction. Fig 2.26

Conductivity of a Semiconductor σ = enµe + epµh σ = conductivity, e = electronic charge, n = electron concentration, µ e = electron drift mobility, p = hole concentration, µh = hole drift mobility Drift Velocity and Net Force µe ve = Fnet e ve = drift velocity of the electrons, µe = drift mobility of the electrons, e = electronic charge, Fnet = net force

B Jy = 0 y + + ee z E y x y Jx E vhx e v hxb x Jx z v ex ee z y e v exb z + + + + B z V Hall effect for ambipolar conduction as in a semiconductor where there are both electrons and holes. The magnetic field Bz is out from the plane of the paper. Both electrons and holes are deflected toward the bottom surface of the conductor and consequently the Hall voltage depends on the relative mobilities and concentrations of electrons and holes. Fig 2.27

Hall Effect for Ambipolar Conduction p µ h 2 nµ e 2 RH = e( pµh + nµe )2 RH = Hall coefficient, p = concentration of the holes, µh = hole drift mobility, n = concentration of the electrons, µe = electron drift mobility, e = electronic charge OR 2 p nb RH = e( p + nb)2 b = µe,/µh

Other types of conduction Non-metal also exhibit charge flow from other types of carriers Ionic crystals have charge atoms that can move through vacancies Impurities can be ionized Defects can bring about hole and electron transfer These processes are typically inhibited by potential barrier, thermal activated and characterized by and activation energy The total conduction is the sum of all the different processes.

E E V a c a n c y a id s th e d iff u s io n o f p o s itiv e io n O S i Na A n io n v a c a n c y + In te r s titia l c a tio n d iffu s e s a c ts a s a d o n o r (a ) (b ) Possible contributions to the conductivity of ceramic and glass insulators (a) Possible mobile charges in a ceramic (b) A Na+ ion in the glass structure diffuses and therefore drifts in the direction of the field. (E is the electric field.) Fig 2.28 24+

General Conductivity σ = Σqi ni µi σ = conductivity qi = charge carried by the charge carrier species i (for electrons and holes qi = e) ni = concentration of the charge carrier µi = drift mobility of the charge carrier of species i

Temperature Dependence of Conductivity Eσ σ = σ o exp kt σ = conductivity σο = constant Εσ = activation energy for conductivity k = Boltzmann constant, T = temperature

1 1 0-1 2 4 % N a 2 O -7 6 % S io 2 A s 3.0 T e 3.0 S i 1.2 G e 1.0 g l a s s Conductivity 1/( m) 1 1 0-3 P y rex 1 1 0-5 1 1 0-7 1 1 0 1 2 % N a 2 O -8 8 % S io 2-9 PV A c S io 2 1 1 0-1 1 P V C 1 1 0-1 3 1 1 0-1 5 1.2 1.6 2 2.4 2.8 1 0 3 /T (1 /K ) 3.2 3.6 4 Conductivity vs reciprocal temperature for various low conductivity solids. (PVC = Polyvinyl chloride; PVAc = Polyvinyl acetate.) Data selectively combined from numerous sources. Fig 2.29

Insulators Semiconductors Conductors Many ceramics Superconductors Alumina Diamond Inorganic Glasses Metals Mica Polypropylene PVDF Pure SnO2 Borosilicate PET SiO2 10-18 Soda silica glass 10-15 Amorphous As2Se3 10-12 10-9 Degenerately Doped Si Te Intrinsic GaAs 10-6 Alloys Intrinsic Si 10-3 100 Graphite NiCr Ag 103 106 Conductivity (½ m)-1 Range of conductivites exhibited by various materials Fig 2.25 109 1012

High frequency effects Electromagnetic effects can be a big factor in the resistance of a metal film or line. At high frequencies the current is push to the edge of the conductor by inductive effects. This called the skin effect and is very important for high speed electronics. At very high frequencies a solid conductor can not be used and waveguide is needed. Fig 2.29

ωl R δ = S k in d e p th 2a At high frequencies, the core region exhibits more inductive impedance than the surface region, and the current flows in the surface region of a conductor defined approximately by the skin depth, δ. Fig 2.31

Skin Depth for Conduction δ= 1 1 ωσµ 2 δ = skin depth, ω = angular frequency of current, σ = conductivity, µ = magnetic permeability of the medium HF Resistance per Unit Length Due to Skin Effect ρ ρ rac = A 2πaδ rac = ac resistance, ρ = resistivity, A = cross-sectional area, a = radius, δ = skin depth

Thin film effects Metals for electronics are usually deposited as thin films. Using vapor deposition techniques. These methods produce polycrystalline thin films that are far from perfect The resistivity of the film is dominated by: Grain boundary scattering Surface scattering (for very thin films or small lines) The characteristic parameter is the mean free path of the electron Small grains, thin film or small line will restrict the mean free path. Difficult due to specular and non-specular scattering at surfaces and grain boundaries. Fig 2.32

G r a in 1 G ra in 2 G r a in B o u n d a ry (a) (b) (a ) Gra in bounda rie s ca use sca tte ring of the e le ctron a nd the re fore a dd to the re sistivity by Ma tthie sse n's rule. (b) For a ve ry gra iny solid, the e le ctron is sca tte re d from gra in bounda ry to gra in bounda ry a nd the me a n fre e pa th is a pproxima te ly e qua l to the me a n gra in dia me te r. Fig 2.32

Jx D Conduction in thin films may be controlled by scattering from the surfaces. Fig 2.33

+ y S c a tte r in g 2 = D θ = D /c o s θ x + x S c a tte r in g 1 y The mean free path of the electron depends on the angle θ after scattering. Fig 2.34

300 35 (b ) (a ) A s d e p o s ite d A n n e a le d a t 1 0 0 C A n n e a le d a t 1 5 0 C 30 100 25 50 20 ρ b u lk = 1 6. 7 n m 10 15 0 0.0 5 0.0 1 0.0 1 5 0.0 2 0.0 2 5 5 10 50 100 500 F ilm th ic k n e s s ( n m ) 1 /d (1 /n a n o m e te r) (a) ρfilm of the Cu polycrystalline films vs. reciprocal mean grain size (diameter), 1/d. Film thickness D = 250 nm - 900 nm does not affect the resistivity. The straight line is ρfilm = 17.8 n m + (595 n m nm)(1/d), (b) ρfilm of the Cu thin polycrystalline films vs. film thickness D. In this case, annealing (heat treating) the films to reduce the polycrystallinity does not significantly affect the resistivity because ρfilm is controlled mainly by surface scattering. SOURCE: Data extracted from (a) S. Riedel et al, Microelec. Engin. 33, 165, 1997 and (b). W. Lim et al, Appl. Surf. Sci., 217, 95, 2003) Fig 2.35

Interconnects Interconnects are metal lines that hook up the devices. Referred to as the backend. Need multiple levels (lines and vias) Used to be neglected, but of increasing importance. RC delays dominate chip speeds Reliability is almost all backend Many C s line-line, level-level, line-ground plane, etc. Sometimes need to create transmission lines. Reliability has many factors (electromigration, barriers, diffusion) Recently a move to Cu and low K to reduce delays. Fig 2.35

M7 M6 Low permittivity dielectric M5 M3 Cu interconnects M2 M4 M3 M2 M1 M1 Silicon Metal interconnects wiring devices on a silicon crystal. Three different metallization levels M1, M2, and M3 are used. The dielectric between the interconnects has been etched away to expose the interconnect structure. Cross section of a chip with 7 levels of metallization, M1 to M7. The image is obtained with a scanning electron microscope (SEM). SOURCE: Courtesy of IBM SOURCE: Courtesy of Mark Bohr, Intel. Fig 2.36

Three levels of interconnects in a flash memory chip. Different levels are connected through vias. SOURCE: Courtesy of Dr. Don Scansen, Semiconductor Insights, Kanata, Ontario, Canada

(a ) (b ) Void and failure H illo c k G ra in b o u n d a ry E le c tr o n Void H illo c k C u rre n t In terc o n n e c t In t erf ace Hot C o ld (c ) Hillocks C o ld G ra in b o u n d ary Current (a) Electrons bombard the metal ions and force them to slowly migrate (b) Formation of voids and hillocks in a polycrystalline metal interconnect by the electromigration of metal ions along grain boundaries and interfaces. (c) Accelerated tests on 3 mm CVD (chemical vapor deposited) Cu line. T = 200 oc, J = 6 MA cm-2: void formation and fatal failure (break), and hillock formation. SOURCE: Courtesy of L. Arnaud et al, Microelectronics Reliability, 40, 86, 2000. Fig 2.38