Lecture Notes – 04ECE 531 Semiconductor Devices

Dr. Andre Zeumault

Outline

1 Overview 1

2 Drift 12.1 Definition and Visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Drift vs Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 Drift Current Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.4 Drift Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.5 Charge Carrier Mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.6 Drift Current Density at Low-Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Diffusion 63.1 Definition and Visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Diffusion of Electrons and Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4 Total Current Density 84.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

5 Equilibrium Considerations 85.1 Zero Current at Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85.2 Constant Fermi Energy at Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . 95.3 Einstein’s Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

6 Conductivity and Resistivity 10

1 Overview

In this lecture, we discuss the foundations of carrier motion in electronic bands at equilibrium and/orunder perturbing situations which can be assumed to be very near equilibrium. Key concepts tobe introduced include:

• Drift: Motion of a charged particle in an electric field.

• Diffusion: Thermally-activated motion of a particle due to a concentration gradient.

• Conductivity: Proportionality constant that relates electrical current density to electricfield.

In the next lecture, we will discuss the salient aspects of a material related to transport and theconcept of charge carrier mobility as a performance metric for charge transport in semiconductordevices, especially field-effect devices. Later, in subsequent lectures, we will complete our discussionof semiconductor fundamentals by detailing the semiconductor response under non-equilibriumconditions.

2 Drift

2.1 Definition and Visualization

Drift is defined as the motion of a delocalized charged particle in an electric field (see Figure 1).All charged particles respond to electric fields through the Coulombic force. We can understanddrift using classical physics. From Newton’s law, a charged particle will accelerate in an electricfield.

~F = q ~E (Coulombic Force)

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Lecture Notes – 04ECE 531 Semiconductor Devices

Dr. Andre Zeumault

Figure 1: Simple depiction of the drift process. Positively charged particles (e.g. holes) move inthe direction of the applied electric field. Negatively charged particles (e.g. electrons) move in thedirection opposite to the applied electric field.

~a =q ~E

m∗COND(Acceleration)

Depending on whether the particle is postively charged (q > 0) or negatively charged (q < 0),the particle will either move with or against the applied electric field ( ~E ) with acceleration ~a.An important observation arises: the acceleration leads to a seemingly unbounded increase in theparticle velocity. Is there a speed limit to an electron/hole in a semiconductor? If so, why? Whatdefines this speed limit?.

In a semiconductor, scattering processes ensure that an accelerating charge carrier does notacquire an arbitrary amount of energy from the electric field. We will discuss scattering in the nextlecture. For now, we acknowledge that if scattering processes did not exist, charge carriers wouldacquire arbitrary energy from the applied electric field and therefore be out of equilibrium witheach other and with the phonons of the crystal. We therefore should think of drift as an averageresult of a collection of charges rather than the motion of an individual charge. We can visualizethis conceptually in Figure 2. As a charge carrier moves, it will experience a sequence of randomcollisions. When an electric field is applied, the electric field does not modify the sequence of randomscattering events, rather, it adds an average velocity component to the collection of charges in thedirection of the applied electric field. The fact that we can approximate the motion of a collection ofcharges as one classical particle allows us to use so-called semi-classical transport equations, withouthaving to know the individual trajectories of each charged particle. This approximation is sufficientas an entry point into devices. For more detailed calculations, we consider all possible scatteringevents for a single charged particle and average over the entire collection (ensemble). Validityof these approaches is typically accounted for by a combination of experiment and comparison tocomputations of single-particle trajectories usually through a stochastic method (e.g. Monte-Carlo).

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Lecture Notes – 04ECE 531 Semiconductor Devices

Dr. Andre Zeumault

Figure 2: Motion of a hole in an electric field. The bracketed quantity, 〈vd,h〉 indicates the ensembleaveraged drift velocity for holes, which is parallel to the electric field.

Figure 3: Motion of a hole in response to an electric field.

2.2 Drift vs Polarization

When considering drift, it is important that the charges under consideration be delocalized, meaningthey have the freedom to move under the application of an electric field. If not, and the chargesare localized, there will be a polarization that is out of phase with the applied electric field. Thisdistinction is subtle, but important. Even though polarization and conduction are the result of thesame force – the Coulombic force. Polarization in electronic insulators (i.e. dielectrics) gives rise toa dielectric response that is useful for things like energy storage (e.g. capacitors, supercapacitors)but not for electrical conductivity. By contrast, we think of good electrical conductors as beinginfinitely polarizable because carriers are treated as spatially delocalized, but metals aren’t verygood charge storage materials.

2.3 Drift Current Density

Consider a hole moving under the influence of an electric field, E in the x-direction as shown inFigure 3. We are interested in the current due to this charge density. Computing the current

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Lecture Notes – 04ECE 531 Semiconductor Devices

Dr. Andre Zeumault

due to drift is a simple matter of bookkeeping, where we are taking consideration of one importantconstant – the charge exists and will continue to exist. As long as it doesn’t get annihilated somehow(we will talk about this later...) then we just need to describe its rate of displacement.

Using Figure 3 as a reference, a concentration of holes, p will move a total distance of 〈vd,p〉 dtin a time interval dt. Likewise, the volume traversed through a cross-sectional area of size dA inthis same time interval is therefore 〈vd,p〉 dtdA.

The total number of holes that pass through this volume in this amount of time is given by

p 〈vd,p〉 dtdA

Since each hole has a charge q = +e, the total amount of charge that passes through this volumeis

ep 〈vd,p〉 dtdA

Since current is the amount of charge passing through a volume per unit time, we recognize thecurrent as,

ep 〈vd,p〉 dA

Similarly, we can define the current density, through cross-sectional area dA as,

ep 〈vd,p〉

We therefore define the drift current density due to holes as,

~Jdrift,p = ep 〈vd,p〉

Similarly, for electrons we define the drift current density due to electrons as,

~Jdrift,n = −ep 〈vd,n〉

Note the sign change in the expression for electrons. Why is this? (hint: recall the signconvention for current)

2.4 Drift Velocity

The expression for current density, so far, does not explicitly depend on the electric field. However,we know that it must, since the electric field is the driving force causing charges to move. To moveforward and establish this dependence, we need to provide a somewhat empirical definition for thedrift velocity in a semiconductor (Figure 4).

〈vd,p〉 =µp ~E[

1 +(µp|E |vsat

)β]1/β (holes)

〈vd,n〉 = − µn ~E[1 +

(µn|E |vsat

)β]1/β (electrons)

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Lecture Notes – 04ECE 531 Semiconductor Devices

Dr. Andre Zeumault

Figure 4: Drift velocity versus electric field for a variety of semiconductor materials at roomtemperature. Neamen 4th edition.

At low electric fields, the velocity increases linearly with electric field,

〈vd,p〉 = µp ~E (µp|E | << vsat,p)

〈vd,n〉 = −µn ~E (µn|E | << vsat,n)

At high electric fields, the velocity saturates with electric field. The saturation velocity vsat,n/pis a scalar quantity and is a material-specific parameter. See Figure 4 for a sense of the differentsaturation velocities encountered in some semiconductor materials.

〈vd,p〉 = vsat,p (µp|E | >> vsat,p)

〈vd,n〉 = −vsat,n (µn|E | >> vsat,n)

2.5 Charge Carrier Mobility

The expressions for drift velocity depend on a new quantity, µ referred to as the charge carriermobility or simply mobility. Mobility is a measure of a charge carriers ability to move in responseto an electric field. We will talk about mobility at greater length in the next lecture. For now, wetake it as a definition.

µ0 = µp (for holes)

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Lecture Notes – 04ECE 531 Semiconductor Devices

Dr. Andre Zeumault

Figure 5: A simple representation of classical diffusion.

µ0 = µn (for electrons)

2.6 Drift Current Density at Low-Fields

Taking the low field version of the drift velocity for electrons and holes, we can express the driftcurrent density as a function of electric field.

~Jdrift,p = epµp ~E (holes)

~Jdrift,n = enµn ~E (electrons)

The total drift current density is therefore given as the sum of that due to electrons and holes:

~Jdrift = ~Jdrift,p + ~Jdrift,n

= (enµn + epµp) ~E

3 Diffusion

3.1 Definition and Visualization

Diffusion is the process whereby a particle tends to redistribute spatially due to its thermal energyand the presence of a concentration gradient. Diffusion is a classical process, characteristic ofparticles and does not depend on the charge state of the particle. Electrons and holes diffuse aswell, provided that a concentration gradient exists.

To visualize the diffusion process using a simple cartoon, consider a collection of gaseous particlessuch that there is a larger concentration of particles on one side as opposed to another (see Figure 5).Initially, a concentration gradient is maintained due to the presence of an impermeable membrane,confining all of the particles to one side of the container. At some time, the membrane is broken, andparticles are allowed to diffuse from left to right. After some time the particles will be distributeduniformly throughout the volume of the container.

In this case, a diffusion flux will exist from left to right as the particles spread out to fill the

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Lecture Notes – 04ECE 531 Semiconductor Devices

Dr. Andre Zeumault

Figure 6: Illustration showing electron/hole diffusion and the direction of electronic current flowproduced as a result.

volume of the container. Since these are gaseous particles, assumed to be electrically neutral, noelectrical current is produced as a result of their motion.

3.2 Diffusion of Electrons and Holes

For electrons and holes, the process is similar to diffusing gas particles, except that, the direction ofcurrent flow that is produced depends on the polarity of the charge. To illustrate, see Figure 6. Aconcentration gradient for electrons and holes is depicted as a function and cartoon, decreasing inconcentration from left to right. We have seen from the previous cartoon (Figure 5) that diffusionoccurs in the direction of the concentration gradient. Based on our current convention, the electroniccurrent due to electrons will be in the negative x direction whereas the current due to holes will bein the positive x direction.

Fick’s law relates the flux of a diffusing species to a concentration gradient through a propor-tionality factor known as the diffusion constant. In one spatial dimension, we have:

~Φdiff,p = −Dp∂p

∂x(holes)

~Φdiff,n = −Dn∂n

∂x(electrons)

This accounts for the flow of diffusing electrons/holes, but is not exactly the electronic current.To get to the electronic current, we use the following relations,

~Jdiff,n = −e~Φdiff,n (electrons)

~Jdiff,p = e~Φdiff,p (holes)

Using an appropriate scaling of Fick’s law by the electronic charge shown above, the concen-

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Lecture Notes – 04ECE 531 Semiconductor Devices

Dr. Andre Zeumault

tration gradients for electrons and holes can be related to the electronic diffusion currents mathe-matically, as:

~Jdiff,n = eDn∂n

∂x(electrons)

~Jdiff,p = −eDp∂p

∂x(holes)

The proportionality constants, Dn, Dp are the diffusion constants for electrons and holes, re-spectively. In general, these are temperature dependent, and follow an Arrhenius dependence withactivation energies EA,n, EA,p.

Dn = Dn,0e−

EA,nkbT (electrons)

Dp = Dp,0e−

EA,pkbT (holes)

Thus, at sufficiently low temperatures, sufficiently low diffusion constants or large enough acti-vation energies, diffusion cannot occur, despite how large of a concentration gradient may exist.

4 Total Current Density

4.1 Definitions

The total current density due to electrons and holes is given by the sum of the parts – drift anddiffusion current.

~Jn = ~Jdrift,n + ~Jdiff,n

= enµn ~E + eDn∂n

∂x(electron current)

~Jp = ~Jdrift,p + ~Jdiff,p

= epµp ~E − eDp∂p

∂x(hole current)

The total electronic current is given by the sum of contributions due to electrons and holes.

~J = ~Jn + ~Jp

= (enµn + epµp) ~E + eDn∂n

∂x− eDp

∂p

∂x(electronic current)

5 Equilibrium Considerations

5.1 Zero Current at Equilibrium

Under equilibrium conditions, no current flows. This can be shown by setting the expression forthe total current density to 0. In doing so, it is clear that the current density is equal to 0 when

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Lecture Notes – 04ECE 531 Semiconductor Devices

Dr. Andre Zeumault

there are no electric fields or concentration gradients. Additionally, as will be clear when we discussdiodes, if the current flow due to drift balances out the current flow due to diffusion, no net currentwill flow. This is precisely the equilibrium condition of a diode and we will show this later in thecourse.

5.2 Constant Fermi Energy at Equilibrium

Using the Boltzmann approximation, the expressions for the total current density, and the fact thatthe current density is zero at equilibrium, it can be shown that the Fermi energy must be spatiallyinvariant.

n = nieEF−Ei

kbT

p = nieEi−EF

kbT

In one dimension, the first spatial derivative of these expressions is given by:

dn

dx=

n

kbT

(dEFdx− dEi

dx

)dp

dx= − p

kbT

(dEFdx− dEi

dx

)Using the definition of electric field, E = 1

edEidx , these can be simplified to the following:

dn

dx=

n

kbT

dEFdx− enE

kbTdp

dx= − p

kbT

dEFdx

+epE

kbT

Substituting these expressions into the expression for total current density, it can be shown thatthe current density satisfies the following simple expression:

~J =σ

e

dEFdx

This expression can be generalized to 3D by using the gradient operator:

~J =σ

e∇EF

From this result, it is clear that at equilibrium ( ~J = 0), the Fermi energy is constant withrespect to position. Thus, energy band diagrams at equilibrium must always be drawn with aFermi energy that is horizontal.

EF = constant (At Equilibrium)

5.3 Einstein’s Relations

We can derive some useful relationships between the diffusion constant and mobility of a chargecarrier, provided we can assume non-degenerate doping levels (Boltzmann Approximation) andthat equilibrium conditions apply.

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Lecture Notes – 04ECE 531 Semiconductor Devices

Dr. Andre Zeumault

Recall that the expression for the electron concentration for a non-degenerate semiconductor is,

n = nieEF−Ei

kbT

The first spatial derivative of this is given by,

∂n

∂x=

n

kbT

(dEFdx− dEi

dx

)At equilibrium, the Fermi energy does not vary with space, dEF

dx = 0 therefore we have thefollowing:

∂n

∂x= − n

kbT

dEidx

By definition, the electric field is the gradient of the potential energy. Since Ei is a reference tothe electronic energy levels in a semiconductor, relative to the band edges, we have:

~E =1

e

dEidx

Using the above relationship, the concentration gradient can be expressed as:

∂n

∂x= − en

kbT~E

Substituting this into the expression for the total electronic current yields the following:

0 = enµn ~E − eDnen

kbT~E

After rearranging terms, one arrives at the following expression:

Dn

µn=kbT

e

The similar process can be done for hole currents, yielding the following:

Dp

µp=kbT

e

6 Conductivity and Resistivity

For a uniformly doped semiconductor, the conductivity, σ is defined as the proportionality con-stant between current density and the electric field.

~J = (enµn + epµp) ~E

= σ ~E

Thus, the conductivity is given by:

σ = enµn + epµp (conductivity)

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Lecture Notes – 04ECE 531 Semiconductor Devices

Dr. Andre Zeumault

Figure 7: Resistivity of silicon at room temperature, for two different dopant types. Neamen 4thedition.

Resistivity, is defined as the inverse of the conductivity. The ability to modify the conductivityof semiconductors by several orders of magnitude is the main property that makes semiconductorsspecial. For reference, the resistivity of silicon is shown at room temperature, covering severalorders of magnitude of dopant density (Figure 7).

ρ =1

σ

=1

enµn + epµp(resistivity)

.The conductivity or resistivity can be used interchangeably when discussing semiconductors.

In the next lecture, we will discuss the temperature dependence of the conductivity as it pertainsto scattering mechanisms of interest.

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