Lecture Notes – 08ECE 531 Semiconductor Devices

Dr. Andre Zeumault

Outline

1 Overview 1

2 Qualitative Derivation of the Ideal Diode Equation 12.1 Equilibrium (VA = 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Forward Bias (VA > 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 Reverse Bias, (VA < 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.4 Diode Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.5 A Closer Look at Reverse Bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Strategy for Quantitative Derivation 63.1 Quasi-neutral Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Depletion Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.3.1 At the Contacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.4 At the Edges of the Depletion Region . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Quantitative Derivation 9

1 Overview

In this lecture, we derive the steady-state current-voltage relationship (aka IV characteristic) of apn junction diode based on ideal assumptions. In so doing, we derive the so-called ideal diodeequation. Despite its simplicity, and marginal utility in practice, the formalism provides deepinsight into the physical operation of the diode as a starting point for more detailed analysis.Topics to cover include:

• Qualitative Derivation: common-sense derivation of the diode equation from anticipatedbehavior and predictions from the energy band diagram.

• Strategy for Quantitative Derivation: an outline of the approach to obtaining a quanti-tative solution.

• Quantitative Derivation: step-wise derivation of the ideal diode equation.

At the end of this lecture, we will understand the quantitative operation of an ideal pn junction(diode). In the next lecture, we will discuss sources of non-ideal behavior.

2 Qualitative Derivation of the Ideal Diode Equation

The ideal diode equation can be derived without writing down a single equation using the energyband diagram and knowledge of energy dependence of the carrier concentration.

2.1 Equilibrium (VA = 0)

The equilibrium band diagram is shown in Figure 1. An energy barrier exists, limiting the diffusionof electrons (closed circles) from the n-side to the p-side. This same energy barrier is mirrored inthe valence band, limiting the diffusion of holes (open circles) from the p-side to the n-side. Theheight of this energy barrier is exactly specified by the built-in potential of the diode, φbi, convertedinto energy units.

∆E = Ec,p − Ec,n = eφbi

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Lecture Notes – 08ECE 531 Semiconductor Devices

Dr. Andre Zeumault

Figure 1: Equilibrium band diagram of a pn junction. The closed circles represent electrons and theopen circles represent holes. The arrows indicate direction of travel. Holes/Electrons with energybelow the barrier are reflected. This is indicated by the curved arrow. Scanned image from Pierret.

A small concentration of electrons/holes having sufficient thermal energy can diffuse across thedepletion region which, since the semiconductor is at thermal equilibrium, exactly balances theconcentration of electrons/holes drifting across the depletion region in the opposite direction.

Jn = Jn,drift + Jn,diff = 0 =⇒ Jn,drift = −Jn,diffJp = Jp,drift + Jp,diff = 0 =⇒ Jp,drift = −Jp,diff

Why do we draw electrons and holes using such pyramidal representations? Electrons are dis-tributed in energy approximately exponentially, according to the Boltzmann approximation. Asa consequence, the concentration of electrons having energy greater than or equal to this energybarrier is small. Recall that electrons/holes only have approximately 3

2kbT of thermal energy inthe bands, corresponding to ≈ 0.04 eV at room temperature. Visually, the magnitude of this en-ergy is less than 5% of the bandgap of Silicon (1.12 eV). Thus, if drawn appropriately to scale, itshould be clear when inspecting such energy band diagrams that even a small barrier can result ina substantially large suppression of electronic current.

2.2 Forward Bias (VA > 0)

A forward biased diode is shown in Figure 2. The effect of applying a voltage is to lower the energybarrier to diffusion in linear proportion to the applied voltage. Based on the direction of diffusiveflux, this is referred to as majority carrier injection, since majority carriers are injected intothe depletion region from their respective regions.

∆E = eφbi − eVA VA ≤ φbi

As a result, a larger concentration of electrons on the n-side (and holes on the p-side) can cross thedepletion region in the opposite direction. This produces a net positive current flowing from thep-side to the n-side. The electron concentration (and hole concentration) depend exponentially onthe difference between the Fermi level and the intrinsic level.

n = nieEF−Ei(x)

kbT

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Lecture Notes – 08ECE 531 Semiconductor Devices

Dr. Andre Zeumault

Figure 2: Band diagram of a pn junction under forward bias conditions. The closed circles representelectrons and the open circles represent holes. Scanned image from Pierret.

The height of the potential barrier is a function of the intrinsic level, which decreases linearlywith increasing applied potential. Therefore, based on these dependencies, it is expected that theelectron concentration varies exponentially with the applied potential.

n ∝ eVA

2.3 Reverse Bias, (VA < 0)

The effect of applying a reverse bias VR = −VA > 0 is to increase the height of the potential barrierlimiting electron/hole diffusion across the depletion region (Figure 3).

∆E = eφbi + eVR

This practically eliminates all electron/hole diffusion across the depletion region for reverse biasescorresponding to an effective barrier height of only a couple times the thermal energy (≈ 2kbTto 3kbT ). Any additional increase in barrier height above this value does not further reduce themagnitude of diffusion current. Instead, a current is measured flowing in the opposite direction(from n-side to p-side). This current is associated with minority carriers (aka minority carrierextraction) drifting across the depletion region, and is expected to be small or negligible incomparison to the forward current. It is expected to be roughly constant (or saturated) withvoltage, for any reasonable reverse bias exceeding a few kbT .

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Lecture Notes – 08ECE 531 Semiconductor Devices

Dr. Andre Zeumault

Figure 3: Band diagram of a pn junction under reverse bias conditions. The closed circles representelectrons and the open circles represent holes. Scanned image from Pierret.

Figure 4: Anticipated IV characteristics based on qualitative arguments. Scanned image fromPierret.

2.4 Diode Equation

Combining results for forward and reverse bias, we can write down an anticipated expression forthe diode current.

I = I0eVAV0 − I0

= I0

(e

VAV0 − 1

)If we substitute the value of kbT for V0, the resulting expression is identical to the ideal diodeequation and is plotted in Figure 4. Thus, we can summarize the operation of a diode as follows:

A diode rectifies current by blocking diffusion of majority carriers across the SCR.

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Lecture Notes – 08ECE 531 Semiconductor Devices

Dr. Andre Zeumault

Figure 5: An overview of the various conduction processes occuring under reverse bias. Scannedimage from Pierret.

2.5 A Closer Look at Reverse Bias

An issue with the preceding analysis is that the processes of majority carrier injection (carrierbuildup) and minority carrier extraction (carrier reduction) do not imply a steady-state solutionfor the carrier concentrations in the quasi-neutral regions. Instead, they imply that the carrierconcentration should either be continuously increasing (under forward bias) or decreasing (underreverse bias).What are we missing? A close examination of Figure 5 provides the answer. Because we draw thequasi-Fermi levels as constant in the quasi-neutral regions, these regions must exist in a state ofequilibrium.

Steady-state quasi-equilibrium conditions are maintained in the semiconductor bulk.

Since, technically speaking, we are not in equilibrium, we use the term quasi-equilibrium in thiscase to account for the fact that voltages are applied. We expressed the notion of equilibrium earlierin terms of rates of generation and recombination but this is just a matter of bookkeeping to keeptrack of where electrons/holes are going and conserve their total number, and is therefore generallyapplicable to the extraction and injection processes as well. In particular, the extraction of minoritycarriers which occurs at the boundaries of the depletion region must be balanced by another process– thermal generation. This is illustrated in Figure 5. As shown, the extraction of an electron fromthe p-side and the extraction of a hole from the n-side is balanced by thermal generation – creatingan electron and a hole. The thermally generated electron on the p-side replenishes those lostdue to drift across the depletion region, thus maintaining steady-state equilibrium. However, thegeneration process also creates a hole in the valence band. This hole leads to a concentrationgradient of holes in the valence band in the quasi-neutral region that produces a diffusion currentfrom right to left. These holes recombine at the metal contact at the left, thus maintaining steady-state hole concentration in the valence band on the p-side. Similarly, the thermally generated holeon the n-side replenishes those lost due to drift across the depletion region to maintain equilibrium.The additional electron created in the generation process produces a diffusion current also flowingright to left. These electrons are extracted at the metal contact at the right, which flow through

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Lecture Notes – 08ECE 531 Semiconductor Devices

Dr. Andre Zeumault

Figure 6: Regions corresponding to ideal diode analysis.

the circuit, thus maintaining steady-state electron concentration in the conduction band on then-side.

3 Strategy for Quantitative Derivation

The assumptions for deriving the ideal diode equation are as follows:

• Steady-state conditions

• Step-junction approximation

• Boltzmann approximation

• Current flow is one dimensional (taken to be the x-direction)

• Low level injection

• No light is present

The relevant equations we will be solving are the following:

Total Current: I = AJ

Total Current Density: J = Jn + Jp

Electron Current Density: Jn = eµnnE + eDndn

dx

Electron Current Density: Jp = eµppE − eDpdp

dx

The physical regions in which we seek solutions are referred to in Figure 6.

3.1 Quasi-neutral Regions

The minority carrier diffusion equations in the quasi-neutral regions are given by:

0 = Dnd2∆npdx2

− ∆npτp

· · · x ≤ −xp

0 = Dpd2∆pndx2

− ∆pnτn

· · · x ≥ xn

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Lecture Notes – 08ECE 531 Semiconductor Devices

Dr. Andre Zeumault

Since, in the step-junction approximation, the electric field is confined to the depletion region.Therefore, E = 0 within the quasi-neutral regions. Also, since we’ve assumed low-level injectionapplies, dn0

dx = dp0dx = 0 Therefore, the current is given by:

Jn = eDnd∆npdx

· · · x ≤ −xp

Jp = −eDpd∆pndx

· · · x ≥ xn

These expressions are straightforward to solve analytically, but they are restricted to distinct regions– Jn on the p-side and Jp on the n-side. Since the total current is the sum of these two at thesame position in space, it is not possible to determine what it is in this case. Instead, we need toknow the values of Jn and Jp at the same position. We now turn to the depletion region for sucha condition.

3.2 Depletion Region

We can write down the steady-state current-continuity expressions in any region of the device asfollows:

0 =1

e

dJndx

+

(∂n

∂t

)R−G

0 = −1

e

dJpdx

+

(∂p

∂t

)R−G

Within the depletion region, we assume (without justification) that we can neglect thermal genera-tion and recombination terms. Physically, this can be likened to reducing the width of the depletionregion to zero. The resulting expressions:

0 =1

e

dJndx

· · · − xp ≤ x ≤ xn

0 = −1

e

dJpdx

· · · − xp ≤ x ≤ xn

essentially state that the current density is constant within the depletion region. Since the currentdensity is constant within the depletion region, and we know the value of the current density atthe quasi-neutral region boundary, we can write:

Jn(−xp ≤ x ≤ xn) = Jn(−xp)Jp(−xp ≤ x ≤ xn) = Jp(xn)

This allows us to compute the total current density as follows:

J = Jn(−xp) + Jp(xn)

In short, to obtain the ideal diode equation, we evaluate the current densities at the depletionregion edges and add the edge current densities together, and multiply by the area of the diode toobtain the current.

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Lecture Notes – 08ECE 531 Semiconductor Devices

Dr. Andre Zeumault

3.3 Boundary Conditions

The minority carrier diffusion equations require boundary conditions in order to solve. In particular,the excess minority carrier concentrations must be known at the edges of the semiconductor – atthe contacts as well as the depletion region boundary.

3.3.1 At the Contacts

We approximate the contacts as being located infinitely far from the depletion region. In such acase, the boundary conditions for the excess minority carriers are as follows:

∆np(−∞) = 0

∆pn(∞) = 0

3.4 At the Edges of the Depletion Region

Recall the definitions of the electron and hole concentrations under non-equilibrium.

n = nieEF,n−Ei

kbT

p = nieEi−EF,p

kbT

The product of n and p are given by:

np = n2i eEF,n−EF,p

kbT = n2i eeVAkbT

This is referred to as the law of the junction and provides the ability to establish useful relationsbetween carrier concentrations at the boundary of the depletion region.

n(−xp)p(−xp) = n(−xp)N−A = n2i e

eVAkbT

n(−xp) =n2iN−

A

eeVAkbT

n(xn)p(xn) = N+Dp(xn) = n2i e

eVAkbT

p(xn) =n2iN+

D

eeVAkbT

From these expressions, we can write the expression for the excess carrier concentrations:

∆n(−xp) =n2iN−

A

(e

eVAkbT − 1

)∆p(xn) =

n2iN+

D

(e

eVAkbT − 1

)

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Lecture Notes – 08ECE 531 Semiconductor Devices

Dr. Andre Zeumault

4 Quantitative Derivation

Since the details of the depletion region are irrelevant for the ideal diode equation we can, concep-tually speaking, simply collapse the depletion region to a point located at x = 0. The depletionregion boundaries are therefore located at x = 0 and the contacts at ±∞. The minority carrierdiffusion equations are therefore:

0 = Dnd2∆npdx2

− ∆npτp

· · · x ≤ 0

0 = Dpd2∆pndx2

− ∆pnτn

· · · x ≥ 0

With boundary conditions for excess holes:

∆pn(∞) = 0

∆pn(0) =n2iN+

D

(e

eVAkbT − 1

)And boundary conditions for excess electrons:

∆xp(∞) = 0

∆xp(0) =n2iN−

A

(e

eVAkbT − 1

)The general solution for the excess minority carrier concentrations are:

∆pn(x) = A1e− x

Lp +A2exLp · · · x ≥ 0

∆np(x) = A1e− x

Ln +A2ex

Ln · · · x ≤ 0

Where Lp and Ln are the minority carrier diffusion lengths, given by:

Lp =√Dpτp

Ln =√Dnτn

The solutions are given by:

∆pn(x) =n2iN+

D

(e

eVAkbT − 1

)e− x

Lp · · · x ≥ 0

∆np(x) =n2iN−

A

(e

eVAkbT − 1

)e

xLn · · · x ≤ 0

It follows that the current densities are given by:

Jp(x) = −eDpd∆pndx

=eDp

Lp

n2iN+

D

(e

eVAkbT − 1

)e− x

Lp · · · x ≥ 0

Jn(x) = eDnd∆npdx

=eDn

Ln

n2iN−

A

(e

eVAkbT − 1

)e

xLn · · · x ≤ 0

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Lecture Notes – 08ECE 531 Semiconductor Devices

Dr. Andre Zeumault

Figure 7: Majority and minority carrier concentrations under forward bias.

Evaluating these expressions at the depletion region boundary (x = 0) yields the following:

Jp(0) =eDp

Lp

n2iN+

D

(e

eVAkbT − 1

)Jn(0) =

eDn

Ln

n2iN−

A

(e

eVAkbT − 1

)The current is therefore given by:

I = AJ

= A (Jp(0) + Jn(0))

= eA

(Dp

Lp

n2iN+

D

+Dn

Ln

n2iN−

A

)(e

eVAkbT − 1

)= I0

(e

eVAkbT − 1

)Where the quantity I0 is the reverse saturation current defined as:

I0 = eA

(Dp

Lp

n2iN+

D

+Dn

Ln

n2iN−

A

)Various results obtained can be visualized in Figure 7, showing the minority carrier and major-

ity carrier concentrations under forward bias conditions, Figure ??, showing the minority carrierconcentration under reverse bias conditions, and Figure 9, showing the spatial dependence of thecurrent density components.

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Lecture Notes – 08ECE 531 Semiconductor Devices

Dr. Andre Zeumault

Figure 8: Minority carrier concentrations under reverse bias.

Figure 9: Spatial dependence of the current density of electrons, Jn and holes, Jp.

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