Conduction. Metals, Semiconductors and Interconnects. Fig 2.1

Transcription

1 Conduction Metals, Semiconductors and Interconnects Fig 2.1

2 Metal interconnects are used in microelectronics to wire the devices within the chip, the intergraded circuit. Multilevel interconnects are used for implementing the necessary interconnections. Back-end is the interconnect structure of the chip. From backend of the fabrication facility as it is added last. Front-end is the silicon processing and device fabrication. SOURCE: Dr. Don Scansen, Semiconductor Insights, Kanata, Ontario, Canada

3 Classical conduction in solids A classical model of movement of charged particles (usually electrons) is very useful for developing intuition for conduction in metals (and semiconductors) The model is of free electrons as a thermal excited charged ideal gas. We are interested in a model of electron flow due to an electric field (drift). Later we will look at flow due to concentration gradients (Diffusion). A large scale model of a current though a metal or other solid is a resistor through which a constant current will flow when a voltage is applied. Drift may seem intuitively simple but is in fact quite subtle. Modeling conduction (drift) well is very important for devices/interconnects and many material effects. Fig 2.1

4 Definition of current density A current is a flow of charge through an area in a net amount of time. Drift of electrons in a conductor in the presence of an applied electric field. Electrons drift with an average velocity v dx in the x-direction. (E x is the electric field.) We will often use current density J not I where: I = J X Area. Fig 2.1

5 Definition of Drift Velocity v dx = 1 N [v x1 + v x 2 + v x v xn ] v dx = drift velocity in x direction, N = number of conduction electrons, v xi = x direction velocity of ith electron We define a drift velocity v d of an electron gas as the average velocity of all the electrons (usually in one direction the direction of the applied electric field). For a classical metal the electrons we are dealing with are the outermost valence electrons that have been given up to the metal. The inner electrons are bound to the atoms.

6 Current Density and Drift Velocity J x (t) = env dx (t) This allows for a definition of the current density J x = current density in the x direction, e = electronic charge, n = electron concentration, v dx = drift velocity Later we will need QM to more accurately look at: the velocity distribution (the 1/2KT rule and MB distribution is not good for electrons which are fermions) The number of electrons (not all the valence electrons contribute to the conduction).

7 What limits the velocity? Naively, we might apply F=ma to the electrons to determine the velocities of the electrons subject to a force. We have F = qe and also have a = dv/dt. For a constant field we would have v(t) = qe/m t + v 0 and the velocity increase is unlimited and J goes to infinity. We wish to relate the drift velocity to the applied field and in particular find out what limits the velocity in time? Fig 2.1

8 Thermal agitation An electron gas at a temperature T consists of randomly moving particles with a velocity distribution prescribed by T. The electrons are (even without a field) undergoing a great deal of motion. They are constantly interacting with the lattice (bulk) to exchange energy and be in thermal equilibrium with the lattice. Electrons are doing a random walk. Fig 2.1

9 Scattering The key factor is that the crystal is not perfect. The defects discussed before mean that the electrons can not travel freely. A key scattering factor is the lattice vibrations (we will see why when we look at QM). As an electron encounters a defect it scatters losing energy and changing direction. We use a mean time between collisions, a relaxation time and mean free path to characterize this process. It is this scattering which limits the average drift velocity of the electrons. Modeling the scattering is not simple and is important in determining mobility. Specular or Diffuse? Fig 2.1 Scattering parameters energy dependant?

10 (a) A conduction electron in the electron gas moves about randomly in a metal (with a mean speed u) being frequently and randomly scattered by thermal vibrations of the atoms. In the absence of an applied field there is no net drift in any direction. (b) In the presence of an applied field, E x, there is a net drift along the x-direction. This net drift along the force of the field is superimposed on the random motion of the electron. After many scattering events the electron has been displaced by a net distance, x, from its initial position toward the positive terminal Fig 2.2 Each trajectory is curved by the E field. The perturbation in the velocity is small, but it produces a net drift of the electrons.

11 Drift Velocity Δx Δt = v dx Δx = net displacement parallel to the field, Δt = time interval, v dx = drift velocity The motion of a single electron in the presence of an electric field E. During a time Interval t i, the electron traverses a distance s i along x. After p collisions, it has drifted a Distance s = x. Fig 2.4

12 Velocity gained in the x direction at time t from the electric field (E x ) for three electrons. There will be N electrons to consider in the metal. Looked at in 1D we see the scattering breaks up the constant acceleration producing a constant drift for a given field strength. Fig 2.3

13 Definition of Drift Mobility v dx = µ d E x Generally characterize this as linear relationship. But this not really true. Call the linear factor the mobility. v dx = drift velocity, µ d = drift mobility, E x = applied field Drift Mobility and Mean Free Time µ d = eτ m e Mobility is function of the mass of the electron and the mean time between scattering events. µ d = drift mobility, e = electronic charge, τ = mean scattering time (mean time between collisions) = relaxation time, m e = mass of an electron in free space.

14 Unipolar Conductivity σ = enµ d = e2 nτ m e σ = conductivity, e = electronic charge, n = number of electrons per unit volume, µ d = drift velocity, τ = mean scattering (collision) time = relaxation time, m e = mass of an electron in free space. If we know the number of electrons we can define a conductivity. Which gives J = σ E We also use resistivity which is: =1/

15 Temperature dependence of resistivity A vibrating metal atom Temperature dependence As the lattice vibration cause scattering we could expect the scattering to be temperature dependent. Assume the atoms are balls on springs we can model them as simple harmonic τ = Scattering of an electron from the thermal vibrations of the atoms. The electron travels a mean distance l = u between collisions. Since the scattering cross-sectional area is S, in the volume sl there must be at least one scatterer, N s (Su ) = 1. 1 SuN s Fig 2.5 oscillators (we do this with just about everything). Associate 1/2KT with oscillator and the collision cross-section of the atom with tau. Find that ρ T = resistivity of the metal, A = temperature independent constant, T = temperature From Principles ofelectronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005) ρ T = AT

16 Multiple Scattering - Matthiessen s Rule Two different types of scattering processes involving scattering from impurities alone and from thermal vibrations alone. Fig 2.6 More then one scattering mechanism can be present Lattice scattering Defect scattering Impurity scattering As the scattering processes are in parallel we add the tau s in parallel 1/tau = 1/tau1 + 1/tau2 Assuming independence of scattering Find the resistivity add like a series effect. Called Matthiessen s rule ρ = ρ T + ρ I ρ = ρ T + ρ R

17 Definition of Temperature Coefficient of Resistivity α o = 1 ρ o δρ δ T T = α o = TCR (temperature coefficient of resistivity), δρ = change in the resistivity, ρ o = resistivity at reference temperature T o, δt = small increase in temperature, T o = reference temperature ρ = resistivity, ρ o = resistivity at reference temperature, α 0 = TCR (temperature coefficient of resistivity), T = new temperature, T 0 = reference temperature T o Temperature Dependence of Resistivity ρ = ρ ο [1 + α o (T-T o )] Temperature Dependence and scattering This rule is adding the temperature dependent lattice scattering and the essentially independent defect scattering. Simple case is then ρ = ρ 0 + α (T-Τ 0 ) where α is the temperature dependence of the resistivity However, life is complicated and this only works for some materials. Electron density n can be function of T ρ 0 can dominate ( bad material, alloys) At low temperature QM effects dominate lattice scattering Annealing can be used to repair the metal and lower rho So this is useful for some materials over a range of T

18 The resistivity of various metals as a function of temperature above 0 C. Tin melts at 505 K whereas nickel and iron go through a magnetic to non-magnetic (Curie) transformations at about 627 K and 1043 K respectively. The theoretical behavior ( ~ T) is shown for reference. [Data selectively extracted from various sources including sections in Metals Handbook, 10th Edition, Volumes 2 and 3 (ASM, Metals Park, Ohio, 1991)] Fig 2.7

19 Low temperatures it does not work! The resistivity of copper from lowest to highest temperatures (near melting temperature, 1358 K) on a log-log plot. Above about 100 K, T, whereas at low temperatures, T 5 and at the lowest temperatures approaches the residual resistivity R. The inset shows the vs. T behavior below 100 K on a linear plot ( R is too small on this scale). Fig 2.8

20 Alloys are more resistive. Although ordered alloys can have low resistance. Typical temperature dependence of the resistivity of annealed and cold-worked (deformed) copper containing various amounts of Ni in atomic percentage. SOURCE: Data adapted from J.O. Linde, Ann Pkysik, 5, 219 (Germany, 1932) Fig 2.9

21 Increase in ρ is big issue. We often need to use an alloy for metallurgical reasons, but this increases the resistivity greatly.

22 Mixtures and Structures A solid solution of two metals will cause additional scattering. At low concentrations we get a defect scattering that increases linearly with the concentration. Assume random distribution of solute atoms. The ρ peaks at 50% and then decrease as we go to pure metal again. Nordheim s rule is used to predict the ρ. If not random at high concentrations material can act like a pure material and ρ will drop. If a material is structured for example two phases present, or layered materials the material can be anisotropic and calculation of ρ is difficult.

23 (a) Phase diagram of the Cu-Ni alloy system. Above the liquidus line only the liquid phase exists. In the L + S region, the liquid (L) and solid (S) phases coexist whereas below the solidus line, only the solid phase (a solid solution) exists. (b) The resistivity of the Cu-Ni alloy as a Function of Ni content (at.%) at room temperature Nordheim s Rule for Solid Solutions ρ I = CX(1 - X) ρ I = resistivity due to scattering of electrons from impurities C = Nordheim coefficient X = atomic fraction of solute atoms in a solid solution Essentially an empirical model.

24 Annealing allows for ordered alloys to form lowering resistivity. Electrical resistivity vs. composition at room temperature in Cu-Au alloys. The quenched sample (dashed curve) is obtained by quenching the liquid, and the Cu and Au atoms are randomly mixed. The resistivity obeys the Nordheim rule. When the quenched sample is annealed or the liquid is slowly cooled (solid curve), certain compositions (Cu 3 Au and CuAu) result in an ordered crystalline structure in which the Cu and Au atoms are positioned in an ordered fashion in the crystal and the scattering effect is reduced. Fig 2.12

25 Combined Matthiessen and Nordheim Rules ρ = ρ matrix + CX(1 - X) ρ = resistivity of the alloy (solid solution) ρ matrix = resistivity of the matrix due to scattering from thermal vibrations and other defects C = Nordheim coefficient X = atomic fraction of solute atoms in a solid solution

26 Meso-scopic Structure effects The effective resistivity of a material with a layered structure. (a) Along a direction perpendicular to the layers. (b) Along a direction parallel to the plane of the layers. (c) Materials with a dispersed phase in a continuous matrix. Fig 2.13 Structure makes life difficult Anisotropic? Random? Electromagnetic (AC) effects even more difficult.

27 Effective Resistance of Mixtures R eff = L α ρ α A + L βρ β A For a structure which results in resistors in series. R eff = effective resistance L α = total length (thickness) of the α-phase layers ρ α = resistivity of the α-phase layers A = cross-sectional area L β = total length (thickness) of the β-phase layers ρ β = resistivity of the β-phase layers

28 Resistivity-Mixture Rule ρ eff = χ α ρ α + χ β ρ β In series! ρ eff = effective resistivity of mixture, χ α = volume fraction of the α- phase, ρ α = resistivity of the α-phase, χ β = volume fraction of the β- phase, ρ β = resistivity of the β-phase Conductivity-Mixture Rule σ eff = χ α σ α + χ β σ β In parallel! σ eff = effective conductivity of mixture, χ α = volume fraction of the α-phase, σ α = conductivity of the α-phase, χ β = volume fraction of the β-phase, σ β = conductivity of the β-phase

29 Mixture Rule (ρ d > 10ρ c ) ρ eff = ρ c ( χ d ) (1 χ d ) ρ eff = effective resistivity, ρ c = resistivity of continuous phase, χ d = volume fraction of dispersed phase, ρ d = resistivity of dispersed phase Mixture Rule (ρ d < 0.1ρ c ) ρ eff = ρ c (1 χ d ) (1 + 2χ d ) ρ eff = effective resistivity, ρ c = resistivity of the continuous phase, χ d = volume fraction of the dispersed phase, ρ d = resistivity of the dispersed phase Random structures are much more difficult to model. Mixing formulas can be derived, but are of limited use. Other transport mechanisms are sometimes present. Hopping from conductive island to conductive island is how transport in some polymers works. We may have to do a detailed model of the transport. Even more complicated for AC, particularly high frequencies.

30 Macroscopic geometry and meso-scopic structure Size and structure can mix to make things complicated. Thin metal lines can have alternating phases. Embedded nano particles? (a) A two-phase solid. (b) A thin fiber cut out from the solid. Fig 2.14

31 Example of alloy resistivity moving from single phase to two phase. Model is essentially a linear mixture model extrapolating from the limits of Nordheim s rule. Eutectic-forming alloys, e.g., Cu-Ag. (a) The phase diagram for a binary, eutectic-forming alloy. (b) The resistivity versus composition for the binary alloy. Fig 2.15

32 Hall effect B is present The Hall effect can also be explained using classical transport theory In the presence of a B field the electrons feel a force perpendicular to their direction of flight. This is the Lorentz force The force is proportional to the velocity As the electrons are moving on average under the presence of applied field they will be pushed to one side. As they build up a charge distribution will form setting up a E field that pushes the electrons back towards the other side. This field is know as the Hall field. We can use this effect to measure magnetic fields. Fig 2.15

33 Hall Effect A moving charge experiences a Lorentz force in a magnetic field. (a) A positive charge moving in the x direction experiences a force downwards. The Hall effect can also be explained using classical transport theory In the presence of a B field the electrons feel a force perpendicular to their direction of flight. This is the Lorentz force The force is proportional to the velocity (b) A negative charge moving in the -x direction also experiences a force downwards. Fig 2.17

34 Illustration of the Hall effect. The z direction is out of the plane of the paper. The externally applied magnetic field is along the z direction. As the electrons are moving on average under the presence of applied field they will be pushed to one side. As they build up a charge distribution will form setting up a E field that pushes the electrons back towards the other side. This field is know as the Hall field. We can use this effect to measure magnetic fields. Fig 2.16

35 Definition of Hall Coefficient R = H J E R H = Hall coefficient, E y = electric field in the y-direction, J x = current density in the x- direction, B z = magnetic field in the z-direction x y B z Split drain fet. How would it work? We will revisit this for semi-conductors.

36 Applications: Watt Meter Compass was a bit difficult 20 years ago (my phd), but it appears they have solved it. My phone does it! What I could find out: Wattmeter based on the Hall effect. Load voltage and load current have L as subscript; C denotes the current coils for setting up a magnetic field through the Hall-effect sample (semiconductor). Silicon monolithic Hall-effect magnetic sensor with magnetic concentrator realizes 3-axis magnetometer on a silicon chip. Analog circuit, digital logic, power block and interface block are also integrated on a chip.

37 Summary The classical model of conduction in a metal: Free valence electrons Ideal charged thermally excited gas Scattering limits velocity Intrinsic Thermal Linear relationship between E and V (ohms law) Intuitively very useful. Use of various mixing formulas for alloys and structured materials. Disorder increases resistance Order decreases resistance Hall effect Magnetic field causes a perpendicular field to be created. This field can be used to measure the field.