metals, semiconductors and interconnectstjs/4700_pdf/lecphys3.pdfconduction metals, semiconductors...

Conduction

Metals, Semiconductors and Interconnects

Fig 2.1

From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)


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.

|SOURCE: Dr. Don Scansen, Semiconductor Insights, Kanata, Ontario, Canada

“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.

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


Fig 2.1


Drift of electrons in a conductor in the presence of an applied electric field. Electrons drift with an average velocity vdx in the x-direction. (Ex is the electric field.)

•  A current is a flow of charge through an area in a net amount of time.

•  We will often use current density J not I where: I = J X Area.

Definition of current density

Definition of Drift Velocity

vdx =1N[vx1 + vx2 + vx3 + ⋅ ⋅ ⋅ + vxN ]

vdx = drift velocity in x direction, N = number of conduction electrons, vxi = x direction velocity of ith electron

•  We define a drift velocity vd 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.

Current Density and Drift Velocity

Jx = current density in the x direction, e = electronic charge, n = electron concentration, vdx = drift velocity

Jx (t) = envdx(t)

•  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).

This allows for a definition of the current density

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 + v0 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


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


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.

Fig 2.1

Modeling the scattering is not simple and is important in determining mobility. Specular or Diffuse? Scattering parameters energy dependant?

Fig 2.2


(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, Ex, 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

Each trajectory is curved by the E field. The perturbation in the velocity is small, but it produces a net drift of the electrons.

Fig 2.4


The motion of a single electron in the presence of an electric field E. During a time Interval ti, the electron traverses a distance si along x. After p collisions, it has drifted a Distance s = x.

Drift Velocity

Δx = net displacement

parallel to the field, Δt = time interval, vdx = drift velocity

ΔxΔt

= vdx

Fig 2.3


Velocity gained in the x direction at time t from the electric field (Ex) 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.


Definition of Drift Mobility

vdx = drift velocity, µd = drift mobility, Ex = applied field

vdx = µdEx

µd = drift mobility, e = electronic charge, τ = mean scattering time (mean time between collisions) = relaxation time, me = mass of an electron in free space.

Drift Mobility and Mean Free Time

€

µd =eτme

Generally characterize this as linear relationship. But this not really true. Call the linear factor the mobility.

Mobility is function of the mass of the electron and the mean time between scattering events.


Unipolar Conductivity

σ = conductivity, e = electronic charge, n = number of electrons per unit volume, µd = drift velocity, τ = mean scattering (collision) time = relaxation time, me = mass of an electron in free space.

σ = enµd =e2nτme

If we know the number of electrons we can define a conductivity. Which gives J = σ E We also use resistivity which is:

⇢ = 1/�

Fig 2.5

From Principles ofElectronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)

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, Ns (Su ) = 1.

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 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

Temperature dependence of resistivity

τ =1

SuNs

ρT = resistivity of the metal, A = temperature independent constant, T

= temperature

ρT = AT

Fig 2.6


Two different types of scattering processes involving scattering from impurities alone and from thermal vibrations alone.

•  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

Multiple Scattering - Matthiessen’s Rule

ρ = ρT + ρI

ρ = ρT + ρR


Definition of Temperature Coefficient of Resistivity

αo = TCR (temperature coefficient of resistivity), δρ = change in the resistivity, ρo = resistivity at reference temperature To , δT

= small increase in temperature, To = reference temperature

oTToo T =

⎥⎦

⎤⎢⎣

⎡=δδρ

ρα

1

ρ = resistivity, ρo = resistivity at reference temperature, α0 = TCR (temperature

coefficient of resistivity), T = new temperature, T0 = reference temperature

Temperature Dependence of Resistivity

ρ = ρο [1 + αo(T-To)]

Temperature Dependence and scattering

•  This rule is adding the temperature dependent lattice scattering and the essentially independent defect scattering.

–  Simple case is then ρ = ρ0 + α (T-Τ0) 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 –  ρ0 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


Fig 2.7


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 2.8


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).

Low temperatures it does not work!

Fig 2.9


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)

Alloys are more resistive. Although ordered alloys can have low resistance.


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

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.


(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

ρI = CX(1 - X)

Essentially an empirical model.

ρI = resistivity due to scattering of electrons from impurities

C = Nordheim coefficient

X = atomic fraction of solute atoms in a solid solution

Fig 2.12


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 (Cu3Au 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.

Annealing allows for ordered alloys to form lowering resistivity.


Combined Matthiessen and Nordheim Rules

ρ = resistivity of the alloy (solid solution) ρmatrix = 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

ρ = ρmatrix + CX(1 - X)

Fig 2.13


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.

Structure makes life difficult Anisotropic? Random? Electromagnetic (AC) effects even more difficult.

Meso-scopic Structure effects


Effective Resistance of Mixtures

Reff = effective resistance Lα = total length (thickness) of the α-phase layers

ρα = resistivity of the α-phase layers A = cross-sectional area

Lβ = total length (thickness) of the β-phase layers ρβ = resistivity of the β-phase layers

Reff =LαραA

+LβρβA

For a structure which results in resistors in series.


Resistivity-Mixture Rule

ρeff = effective resistivity of mixture, χα = volume fraction of the α-phase, ρα = resistivity of the α-phase, χβ = volume fraction of the β-

phase, ρβ = resistivity of the β-phase

ρeff = χαρα + χβρβ

Conductivity-Mixture Rule

σeff = effective conductivity of mixture, χα = volume fraction of the α-phase, σα = conductivity of the α-phase, χβ = volume fraction of

the β-phase, σβ = conductivity of the β-phase

σeff = χασα + χβσβ

In series!

In parallel!


Mixture Rule (ρd > 10ρc )

ρeff = effective resistivity, ρc = resistivity of continuous phase, χd = volume fraction of dispersed phase, ρd =

resistivity of dispersed phase

ρeff = ρc(1 + 1

2χd )

(1− χ d )

Mixture Rule (ρd < 0.1ρc )

ρeff = effective resistivity, ρc = resistivity of the continuous phase, χd = volume fraction of the dispersed

phase, ρd = resistivity of the dispersed phase

ρeff = ρc(1 − χd )(1 + 2χd )

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.

Fig 2.14


(a)  A two-phase solid. (b) A thin fiber cut out from the solid.

Size and structure can mix to make things complicated. Thin metal lines can have alternating phases. Embedded nano particles?

Macroscopic geometry and meso-scopic structure

Fig 2.15


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.

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.

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


Fig 2.17


A moving charge experiences a Lorentz force in a magnetic field.

(a)  A positive charge moving in the x direction experiences a force downwards.

(b) A negative charge moving in the -x direction also 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

Hall Effect

Fig 2.16


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.

RH = Hall coefficient, Ey = electric field in the y-direction, Jx = current density in the x-direction, Bz = magnetic field in the z-direction

Definition of Hall Coefficient

zx

yH BJ

ER =

Split drain fet. How would it work? We will revisit this for semi-conductors.

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).

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: “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.”

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.


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