Other Aspects of Conduc1on. Thermal, thin films, semiconductors, etc.

Transcription

1 Other Aspects of Conduc1on Thermal, thin films, semiconductors, etc.

2 Thermal conduc1on- metals In a metal two types of heat transport Electron gas (which is dominant). LaCce vibra1ons (phonons) Heat flow can be described by a diffusion equa1on (hot flows to cold). The temperature of lacce is what we feel as heat, the electrons are usually in thermal equilibrium with the lacce and T E = T l, but not always the case. Very useful to view heat transfer like current flow and define a thermal resistance.

3 In a metal two types of heat transport Electron gas (which is dominant). LaCce vibra1ons (phonons) Heat flow can be described by a diffusion equa1on (hot flows to cold). Fig 2.19 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. From Principles of Electronic Materials and Devices, Third Edi7on, S.O. Kasap ( The temperature of lacce is what we feel as heat, the electrons are usually in thermal equilibrium with the lacce and T E = T l but not always the case. In non- metals with few conduc1on electrons, phonons (lacce vibra1ons) dominant. The balls on spring models is used to model the process of vibra1onal waves traveling through the material.

4 Very useful to view heat transfer like current flow and define a thermal resistance/conduc1vity. Fig 2.20 Heat flow in a metal rod heated at one end. Consider the rate of heat flow, dq/dt, across a thin section δx of the rod. The rate of Heat flow is proportional to the temperature gradient δt/δx and the cross-sectional area A. From Principles of Electronic Materials and Devices, Third Edi7on, S.O. Kasap ( Fourier s Law of Thermal Conduction Q ʹ = dq dt = κa δt δx Ohm s Law of Electrical Conduction I = Aσ δv δx

5 Thermal conduc1on in metals In metals the primary heat conduc1on is by free electrons. This illustrates for metals the connec1on (electron transport) between thermal and electrical conduc1on. Thermal conductivity versus electrical conductivity for various metals (elements and alloys) at 20 C. The solid line represents the WFL law with C WFL W K -2. Fig 2.21 From Principles of Electronic Materials and Devices, Third Edi7on, S.O. Kasap ( Ra1o of the two conduc1vi1es gives the Wiedemann- Franz law κ σt =C WFL = W Ω K 2

6 This law holds (roughly) above 100K. Below that we need to look at QM Thermal conductivity versus temperature for two pure metals (Cu and Al) and two Alloys (brass and Al-14% Mg). SOURCE: Data extracted form I.S. Touloukian, et al., Thermophysical Properties of Matter, vol. 1: Thermal Conductivity, Metallic Elements and Alloys, New York: Plenum, Fig 2.22 From Principles of Electronic Materials and Devices, Third Edi7on, S.O. Kasap (

7 Heat conduc1on in insulators ( and semiconductors) The transfer of heat in insulators (no free electrons) is by lacce vibra1ons. Atoms disturb the next door neighbors and pass along kine1c energy. As with metals we can define a thermal conduc1vity/resis1vity Conduction of heat in insulators involves the generation and propagation of atomic Vibrations through the bonds that couple the atoms (an intuitive figure). Fig 2.23 From Principles of Electronic Materials and Devices, Third Edi7on, S.O. Kasap (

8 Fourier s Law Fourier s Law the heat flow equa1on A rate equa1on that allows determina1on of the conduc1on heat flux from knowledge of the temperature distribu1on in a medium Its most general (vector) form for mul1dimensional conduc1on is: qʹ ʹ = k T Implica1ons: Heat transfer is in the direc1on of decreasing temperature (basis for minus sign). Fourier s Law serves to define the thermal conduc1vity of the medium k qʹ ʹ / T Direc1on of heat transfer is perpendicular to lines of constant temperature (isotherms). q == q from before Heat flux vector may be resolved into orthogonal components.

9 Heat Flux Components Cartesian Coordinates: T( x, y, z) T T T qʹ ʹ = k i k j k k x y z qʹ ʹ x qʹ ʹ y qʹ ʹ z Cylindrical Coordinates: T( r, φ, z) T T T qʹ ʹ = k i k j k k r r φ z qʹ ʹ r q φʹ ʹ qʹ ʹ z T r, φθ, T T T qʹ ʹ = k i k j k k r r θ rsinθ φ qʹ ʹ r q θʹ ʹ q φʹ ʹ Spherical Coordinates: ( ) (2.3) (2.18) (2.21)

10 Heat Equa1on The Heat Equa1on A differen1al equa1on whose solu1on provides the temperature distribu1on in a sta1onary medium. Based on applying conserva1on of energy to a differen1al control volume through which energy transfer is exclusively by conduc1on. Cartesian Coordinates: Heat Flow equa1on (PDE boundary value transient problem) " $ x k T % '+ " $ # x & y k T % '+ " $ # y & z k T % '+ q # z & (2.13) = ρc p T t Net transfer of thermal energy into the control volume (inflow- ouilow) Thermal energy genera1on Change in thermal energy storage

11 Heat Equa1on (Radial Systems) Cylindrical Coordinates: 1 T 1 kr k T k T + q ρc T = p r r r r φ φ z z t (2.20) Spherical Coordinates: 1 2 T 1 T 1 kr k k sin T θ q ρc T = p r r r r sin θ φ φ r sinθ θ θ t (2.33)

12 Boundary Condi1ons Boundary and Ini1al Condi1ons For transient conduc1on, heat equa1on is first order in 1me, requiring T x, t T x,0 specifica1on of an ini1al temperature distribu1on: ( ) ( ) t= 0 = Since heat equa1on is second order in space, two boundary condi1ons must be specified. Some common cases: Constant Surface Temperature: Constant Heat Flux: Applied Flux ( ) T 0, t = Ts Insulated Surface T x= 0= 0 x T k = x ʹ ʹ x = 0 q s Given the heat flow equa1on, geometry, BC s and IC s we can solve for T(x,y,z,t). Convec1on T k x= 0= h T T 0, t x ( ) The BCs can be non- linear making the equa1on non- linear.

13 Analogy to drik (ohm law) Ê = rv r Ĵ =0 Ĵ = qnµ n Ê Ĵ = qnµ n rv qnµ n r 2 V =0 r 2 V =0 appler 2 T = (ˆr) Drik and Heat Flow: Steady State Pure drik Gives Poisson equa1on. Set ρ = 0 and this the same as the heat flow equa1on. Perfect analogy between heat flow and current flow.

14 Oken we use the simplest modeling approach which is to use a thermal resistance Devices for Spice Packaging θ = L Aκ To model as a circuit add thermal capacitance to ground and current sources as thermal power genera1on. Can get quite complicated with thermal coupling and large networks. Gross approxima1on of numerical solu1on of PDE. Fig 2.24

15 Despite the analogy to drik fundamentally the process is characterized by a diffusion equa1on. And we can define a thermal diffusivity. One- Dimensional Conduc1on in a Planar Medium with Constant Proper1es and No Genera1on 2 x T 2 1 T = α t k α ρc p thermal diffu sivity of the medium

16 Very large range of thermal conduc1vi1es. Some semiconductors are not good (ie low and high hea1ng). Note: doped semi- conductors are significantly higher. Fig 2.25 From Principles of Electronic Materials and Devices, Third Edi7on, S.O. Kasap (

17 Semiconductors - conduc1on As you know from 398 (and we shall look at in more depth) semi- conductors have two carriers Electrons nega1ve charge, posi1ve mass Holes posi1ve charge, posi1ve mass (virtual par1cle for accoun1ng) Both are can be modeled as free ideal gas with Boltzmann distribu1on. We define n and p as the concentra1ons of electrons and holes And the current flow is due to both par1cles Hall effect is modified due to the two different carriers

18 Somewhat naive classical picture of holes. Can be intui1vely useful. (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 neighboring 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 From Principles of Electronic Materials and Devices, Third Edi7on, S.O. Kasap (

19 Conductivity of a Semiconductor σ = enµ e + epµ h Two carriers n and p σ = conductivity, e = electronic charge, n = electron concentration, µ e = electron drift mobility, p = hole concentration, µ h = hole drift mobility Drift Velocity and Net Force v e = µ e e F net Two drik currents one for n and one for p. Each given by an equa1on like this. v e = drift velocity of the electrons, µ e = drift mobility of the electrons, e = electronic charge, F net = net force From Principles of Electronic Materials and Devices, Third Edi7on, S.O. Kasap (

20 R H = pµ 2 2 h nµ e e( pµ h + nµ e ) 2 R H = p nb2 e( p + nb) 2 Hall effect for ambipolar conduction as in a semiconductor where there are both electrons and holes. The magnetic field B z 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 From Principles of Electronic Materials and Devices, Third Edi7on, S.O. Kasap ( Hall effect is complicated by the presence of two carriers. If n = p and mobility's are equal will be small.

21 Other types of conduc1on Non- metal also exhibit charge flow from other types of carriers Ionic crystals have charge atoms that can move through vacancies Impuri1es can be ionized Defects can bring about hole and electron transfer These processes are typically inhibited by poten1al barrier, thermal ac1vated and characterized by and ac1va1on energy The total conduc1on is the sum of all the different processes. σ = Σq i n i µ i σ = σ o exp E σ kt

22 Ionic conduc1on Possible contribution to the conductivity of ceramic and glass insulators. (a) Possible mobile charges in a ceramic. (b) An Na + ion in the glass structure diffuses and therefore drifts in the direction of the field. Fig 2.28 From Principles of Electronic Materials and Devices, Third Edi7on, S.O. Kasap (

23 This show the thermal ac1va1on energy of the process involved in conduc1on. Conductivity versus reciprocal temperature for various low-conductivity solids SOURCE: Data selectively combined from numerous sources. Fig 2.29 From Principles of Electronic Materials and Devices, Third Edi7on, S.O. Kasap (

24 High frequency effects Electromagne1c 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 induc1ve effects.

25 This is called the skin effect and is very important for high speed electronics. Inductance of inner part is higher forces current to the edges. Illustration of the skin effect. A hypothetical cut produces a hallow outer cylinder and a solid inner cylinder. Cut is placed where it would give equal current in each section. The two sections are in parallel so that the currents in (b) and (c) sum to that in (a). Or view it as the cancella1on of the inner current due to eddy effects from the 1me varying B in the wire. At very high frequencies a solid conductor can not be used and waveguide is needed. Fig 2.30 From Principles of Electronic Materials and Devices, Third Edi7on, S.O. Kasap (

26 Define a skin depth to model this effect δ = ωσµ r ac = ρ A ρ 2πaδ 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 From Principles of Electronic Materials and Devices, Third Edi7on, S.O. Kasap (

27 Thin film effects Metals for electronics are usually deposited as thin films. Using vapor deposi1on techniques. These methods produce polycrystalline thin films that are far from perfect The resis1vity of the film is dominated by: The characteris1c 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 scasering at surfaces and grain boundaries.

28 Grain boundary scasering (a) Grain boundaries cause scattering of the electron and therefore add to the Resistivity by the Matthiessen s rule. (b) For a very grainy solid, the electron is scattered from grain boundary to grain boundary and the mean free path is approximately equal to the mean grain diameter. Fig 2.32 From Principles of Electronic Materials and Devices, Third Edi7on, S.O. Kasap (

29 Surface scasering (for very thin films or small lines) Conduction in thin films may be controlled by scattering from the surfaces The mean free path of the electron depends on the angle after scattering. Fig 2.33 From Principles of Electronic Materials and Devices, Third Edi7on, S.O. Kasap (

30 (a) ρ film of the Cu polycrystalline films vs. reciprocal mean grain size (diameter), 1/d. Film thickness D = 250 nm nm does not affect the resistivity. The straight line is ρ film = 17.8 n m + (595 n m nm)(1/d), (b) ρ film 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 ρ film 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 From Principles of Electronic Materials and Devices, Third Edi7on, S.O. Kasap (

31 Interconnects Interconnects are metal lines that hook up the devices. Referred to as the backend. Need mul1ple levels (lines and vias) Used to be neglected, but of increasing importance. RC delays dominate chip speeds Reliability is almost all backend

32 Fig 2.36 From Principles of Electronic Materials and Devices, Third Edi7on, S.O. Kasap (

33 Many C s line- line, level- level, line- ground plane, etc. Some1mes need to create transmission lines. Reliability has many factors (electromigra1on, barriers, diffusion) Recently a move to Cu and low K to reduce delays. 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 From Principles of Electronic Materials and Devices, Third Edi7on, S.O. Kasap (

34 a b A big modeling issue is extrac1ng all the C s, R s and L s for the back- end structure. This is needed to circuit model the delays and signal integrity on chips, packages and boards. c (a) A single line interconnect surrounded by dielectric insulation. (b) Interconnects crisscross each other. There are three levels of interconnect: M 1, M, and M + 1 (c) An interconnect has vertical and horizontal capacitances C v and C H. Fig 2.37 From Principles of Electronic Materials and Devices, Third Edi7on, S.O. Kasap (

35 Electro- migra1on and reliability temperature ac1vated! (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, Fig 2.38 From Principles of Electronic Materials and Devices, Third Edi7on, S.O. Kasap (

36 Summary Thermal conduc1on Diffusion process Detailed analysis use differen1al equa1on Thermal resistance/capacitance models Metals dominated by electron transport Insulators by lacce vibra1ons (s1ll diffusion) Semiconductors Holes and electrons (Hall effect) Other effects High frequency effects (skin effect) Thin film effects Extrac1on of R, C and L Electromigra1on