Table of Contents
TBD
-
- Fundamentals (conductors, semiconductors, insulators, doping, band model, intrinsic conductivity)
- Diodes (real characteristic curve, operating point, equivalent circuit)
- Zener diode
- LED
- Protective circuit with diodes
- Rectifier circuits (single-phase rectifier, center tap circuit, bridge rectifier, smoothing capacitor)
- Bipolar transistor (structure, designations, characteristic curve, characteristic values)
- Transistor as a switch (circuit, switching times and behavior)
- MOSFET (structure, comparison with bipolar transistor)
- Optional: Transistor as an amplifier
Block 11 — Semiconductor Fundamentals and Diodes
Learning objectives
- distinguish conductors, semiconductors, and insulators using the band model.
- explain intrinsic conduction, electron conduction, and hole conduction.
- explain how n-doping and p-doping change the number of mobile charge carriers.
- describe the formation of a pn junction and the depletion region.
- decide whether a diode is forward-biased or reverse-biased from \(u_{\rm AK}\).
- compare the ideal diode model, the constant-voltage model, and the piecewise-linear diode model.
- use the diode equation
\[ \begin{align*} i_{\rm D}=I_{\rm S}(T)\left({\rm e}^{\frac{u_{\rm AK}}{mU_{\rm T}}}-1\right) \end{align*} \] at a qualitative level.
- calculate simple diode operating points with a series resistor.
- identify basic diode types such as universal diodes, Z-diodes, and LEDs.
90-minute plan
- Warm-up (10 min):
- Why does a diode conduct in one direction but not in the other?
- Recall from EEE1: voltage, current direction, power, and resistors.
- Recall from EEE2: transient overvoltages at inductive loads will later need diode protection.
- Core concepts (55 min):
- Conductors, semiconductors, insulators, and the band gap.
- Intrinsic conduction, electron conduction, hole conduction.
- Doping: n-type and p-type material.
- pn junction, depletion region, diffusion voltage.
- Diode operation in forward and reverse direction.
- Ideal and real diode characteristics.
- Practical diode models for circuit calculations.
- Practice (20 min):
- Determine diode polarity and conduction state.
- Calculate current with a constant-voltage diode model.
- Estimate differential resistance at a given operating point.
- Compare ideal and real diode assumptions.
- Wrap-up (5 min):
- Key messages: pn junction, forward/reverse bias, current limiting, diode models.
- Preview: rectifiers, smoothing, protection circuits, LEDs, and Z-diode stabilizers in Block 12.
Conceptual overview
- A semiconductor is neither a good conductor nor a perfect insulator. Its conductivity can be controlled by material, temperature, light, and doping.
- A diode is a pn junction with two terminals:
- anode A on the p-side,
- cathode K on the n-side.
- In forward direction, the external voltage reduces the depletion region and current can flow.
- In reverse direction, the depletion region becomes wider and only a very small leakage current flows, until breakdown occurs.
- A diode is nonlinear. It is not a resistor.
- In circuits, diode current must usually be limited by another component, often a resistor.
Scope of this block
This block explains why diodes behave as they do and how we model them.
Diode applications such as
- rectifiers,
- smoothing capacitors,
- freewheeling diodes,
- input protection circuits,
- LED circuits,
- Z-diode voltage stabilizers
are continued in Block 12.
Core content
Conductors, semiconductors, and insulators
Materials differ strongly in their specific resistance \(\rho\).
In the band model, two energy ranges are especially important:
- the valence band, where electrons are bound,
- the conduction band, where electrons can move through the crystal.
The energy gap between them is called the band gap \(E_{\rm g}\).
<tabcaption tab_band_gap|Qualitative band model>
| Material type | Band model | Electrical behavior |
|---|---|---|
| conductor | conduction band available or overlapping | many mobile charge carriers |
| semiconductor | small band gap, typically a few \({\rm eV}\) | conductivity can be controlled |
| insulator | large band gap | almost no mobile charge carriers |
Physical picture
A semiconductor can be imagined as a parking garage with two floors.
- The lower floor is almost full: the valence band.
- The upper floor allows movement: the conduction band.
- The band gap is the energy needed to move an electron to the upper floor.
Doping adds useful “parking spots” or “missing spots” so that charge transport becomes much easier.
Intrinsic conduction, electrons, and holes
In a pure semiconductor, some electrons can gain enough energy to leave their bonds. Then
- the electron becomes mobile in the conduction band,
- a missing electron remains in the valence band,
- this missing electron behaves like a positive mobile charge carrier.
The missing electron is called a hole.
- electrons with negative charge,
- holes with positive effective charge.
Analogy: empty seat in a lecture hall
Imagine a fully occupied row of seats. If one student moves to the right into an empty seat, the empty seat appears to move to the left.
The empty seat is not a real object, but it behaves as if it moves. A hole in a semiconductor is similar: it is a missing electron, but it behaves like a positive moving charge carrier.
Doping: n-type and p-type semiconductors
Doping means adding a very small amount of foreign atoms to the semiconductor crystal.
<tabcaption tab_doping|Doping of silicon>
| Doping type | Typical dopant atoms | Main mobile charge carriers | Name of dopant |
|---|---|---|---|
| n-type | phosphorus, arsenic, antimony | electrons | donors |
| p-type | boron, aluminium, indium | holes | acceptors |
The pn junction
A diode is formed when p-doped and n-doped regions meet.
At the junction:
- electrons diffuse from the n-side into the p-side,
- holes diffuse from the p-side into the n-side,
- electrons and holes recombine,
- a region with almost no mobile charge carriers forms.
This region is called the depletion region or space-charge region.
The depletion region behaves like an internal barrier. Without an external voltage, it prevents a large current.
Analogy: a door with a spring
The depletion region is like a spring-loaded door.
- In one direction, you push against the spring and can open the door.
- In the other direction, the spring pushes the door more firmly closed.
The diode behaves similarly: one polarity reduces the barrier, the other polarity increases it.
Forward and reverse operation
We define the diode voltage
\[ \begin{align*} u_{\rm AK}=u_{\rm A}-u_{\rm K}. \end{align*} \]
- \(u_{\rm AK}>0\): anode is more positive than cathode.
- \(u_{\rm AK}<0\): anode is more negative than cathode.
<tabcaption tab_diode_bias|Diode operation depending on \(u_{\rm AK}\)>
| Condition | Name | Effect on depletion region | Current |
|---|---|---|---|
| \(u_{\rm AK}>0\) | forward bias | depletion region becomes smaller | large current possible |
| \(u_{\rm AK}<0\) | reverse bias | depletion region becomes larger | only small leakage current, until breakdown |
\[ \begin{align*} \text{Positive Anode, Negative Is Cathode} \end{align*} \]
This helps to remember the forward direction of a diode.
Ideal diode model
The simplest model is the ideal diode.
\[ \begin{align*} \text{forward direction: } u_{\rm AK}=0,\quad i_{\rm D}>0 \end{align*} \]
\[ \begin{align*} \text{reverse direction: } i_{\rm D}=0,\quad u_{\rm AK}<0 \end{align*} \]
Engineering meaning
The ideal diode is useful for a first decision:
- Is the diode conducting?
- Is the diode blocking?
- Which path can current take?
It is too simple for accurate voltage and current calculations.
Real diode characteristic
A real diode has an exponential current-voltage characteristic.
\[ \begin{align*} \boxed{ i_{\rm D} = {\color{red}{I_{\rm S}(T)}} \left( {\rm e}^{\frac{{\color{blue}{u_{\rm AK}}}}{{\color{green}{mU_{\rm T}}}}} -1 \right) } \end{align*} \]
with
\[ \begin{align*} U_{\rm T}=\frac{kT}{e}. \end{align*} \]
<tabcaption tab_diode_equation_symbols|Symbols in the diode equation>
| Symbol | Meaning |
|---|---|
| \(I_{\rm S}(T)\) | reverse saturation current, strongly temperature-dependent |
| \(m\) | emission coefficient, typically \(1\ldots 2\) |
| \(U_{\rm T}\) | thermal voltage |
| \(k\) | Boltzmann constant |
| \(e\) | elementary charge |
| \(T\) | absolute temperature in \({\rm K}\) |
At room temperature, \(U_{\rm T}\) is approximately
\[ \begin{align*} U_{\rm T}\approx 26~{\rm mV}. \end{align*} \]
Typical values at \(25^\circ{\rm C}\):
<tabcaption tab_typical_diode_values|Typical diode values>
| Diode material | Approximate threshold voltage \(U_{\rm TO}\) | Reverse saturation current \(I_{\rm S}\) |
|---|---|---|
| silicon | \(\approx 0.7~{\rm V}\) | some \({\rm pA}\) |
| germanium | \(\approx 0.3~{\rm V}\) | some \(\mu{\rm A}\) |
Practical diode models for circuit calculation
For hand calculations we usually do not use the full exponential equation.
<tabcaption tab_diode_models|Diode models for circuit calculations>
| Model | Forward direction | Reverse direction | Use |
|---|---|---|---|
| ideal diode | \(u_{\rm AK}=0\) | \(i_{\rm D}=0\) | switching logic, first estimate |
| constant-voltage model | \(u_{\rm AK}\approx U_{\rm TO}\) | \(i_{\rm D}\approx 0\) | quick current calculations |
| piecewise-linear model | \(u_{\rm AK}\approx U_{\rm TO}+r_{\rm F}i_{\rm D}\) | \(i_{\rm D}\approx 0\) | more accurate operating point |
The differential forward resistance is
\[ \begin{align*} r_{\rm F} = \frac{\Delta U_{\rm F}}{\Delta I_{\rm F}}. \end{align*} \]
For large forward voltages compared with \(U_{\rm T}\), the diode equation leads approximately to
\[ \begin{align*} r_{\rm D} = \frac{{\rm d}u_{\rm D}}{{\rm d}i_{\rm D}} \approx \frac{mU_{\rm T}}{I_{\rm D}}. \end{align*} \]
\[ \begin{align*} [r_{\rm D}] = \frac{[U_{\rm T}]}{[I_{\rm D}]} = \frac{{\rm V}}{{\rm A}} = \Omega. \end{align*} \]
Operating point with a series resistor
A diode must usually be operated with a current-limiting element.
For the circuit
\[ \begin{align*} U_{\rm E} \rightarrow R \rightarrow D \end{align*} \]
the loop equation is
\[ \begin{align*} U_{\rm E} = U_R+U_{\rm D}. \end{align*} \]
With the constant-voltage model,
\[ \begin{align*} U_{\rm D}\approx U_{\rm TO}. \end{align*} \]
Therefore
\[ \begin{align*} I_{\rm D} \approx \frac{U_{\rm E}-U_{\rm TO}}{R}. \end{align*} \]
Z-diodes and LEDs as diode types
A Z-diode is operated in reverse breakdown. In its operating range, the diode voltage is approximately constant:
\[ \begin{align*} u_{\rm Z}\approx U_{\rm Z}. \end{align*} \]
The piecewise-linear model is
\[ \begin{align*} u_{\rm Z} \approx U_{\rm Z}+r_{\rm Z}i_{\rm Z}. \end{align*} \]
Z-diode preview
Z-diodes are useful for voltage limitation and voltage stabilization. The practical circuits are treated in Block 12.
An LED is a diode that emits light in forward direction. The required forward voltage depends on the semiconductor material and therefore on the color.
<tabcaption tab_led_forward_voltage|Typical LED forward voltages>
| LED color | Typical \(U_{\rm TO}\) |
|---|---|
| infrared | \(\approx 1.3~{\rm V}\) |
| red | \(\approx 1.6~{\rm V}\) |
| yellow | \(\approx 1.7~{\rm V}\) |
| green | \(\approx 1.8~{\rm V}\) |
| blue | \(\approx 3.2~{\rm V}\) |
Exercises
Exercise E1.1 Quick check: doping and charge carriers
Complete the table.
| Doping type | Typical dopant atom | Main mobile charge carrier | Dopant name |
|---|---|---|---|
| n-type | ? | ? | ? |
| p-type | ? | ? | ? |
| Doping type | Typical dopant atom | Main mobile charge carrier | Dopant name |
|---|---|---|---|
| n-type | phosphorus, arsenic, or antimony | electrons | donor |
| p-type | boron, aluminium, or indium | holes | acceptor |
N-type material has additional mobile electrons. P-type material has additional mobile holes.
The semiconductor as a whole remains approximately electrically neutral.
Exercise E2.1 Quick check: diode polarity
A diode has the anode voltage
\[ \begin{align*} U_{\rm A}=4.8~{\rm V} \end{align*} \]
and the cathode voltage
\[ \begin{align*} U_{\rm K}=4.1~{\rm V}. \end{align*} \]
- Calculate \(u_{\rm AK}\).
- Is the diode forward-biased or reverse-biased?
- For a silicon diode, is a noticeable current likely?
\[ \begin{align*} u_{\rm AK} = U_{\rm A}-U_{\rm K} = 4.8~{\rm V}-4.1~{\rm V} = 0.7~{\rm V}. \end{align*} \]
Since
\[ \begin{align*} u_{\rm AK}>0, \end{align*} \]
the diode is forward-biased.
For a silicon diode, \(0.7~{\rm V}\) is a typical forward voltage in the mA range. Therefore a noticeable current is likely.
Exercise E3.1 Quick check: current with the constant-voltage model
A silicon diode is connected in series with a resistor.
\[ \begin{align*} U_{\rm E}=5.0~{\rm V}, \qquad R=1.0~{\rm k}\Omega. \end{align*} \]
Use the constant-voltage model
\[ \begin{align*} U_{\rm D}\approx 0.7~{\rm V}. \end{align*} \]
Calculate the diode current \(I_{\rm D}\).
The voltage across the resistor is
\[ \begin{align*} U_R = U_{\rm E}-U_{\rm D} = 5.0~{\rm V}-0.7~{\rm V} = 4.3~{\rm V}. \end{align*} \]
Therefore
\[ \begin{align*} I_{\rm D} = \frac{U_R}{R} = \frac{4.3~{\rm V}}{1.0~{\rm k}\Omega} = 4.3~{\rm mA}. \end{align*} \]
Exercise E4.1 Quick check: differential diode resistance
A diode operates at
\[ \begin{align*} I_{\rm D}=10~{\rm mA}. \end{align*} \]
Assume
\[ \begin{align*} m=1, \qquad U_{\rm T}=26~{\rm mV}. \end{align*} \]
Estimate the differential diode resistance
\[ \begin{align*} r_{\rm D}\approx \frac{mU_{\rm T}}{I_{\rm D}}. \end{align*} \]
\[ \begin{align*} r_{\rm D} &\approx \frac{mU_{\rm T}}{I_{\rm D}} \\ &= \frac{1\cdot 26~{\rm mV}}{10~{\rm mA}} \\ &= 2.6~\Omega. \end{align*} \]
This is a small-signal resistance around the operating point. It is not the same as the large-signal ratio \(\frac{U_{\rm D}}{I_{\rm D}}\).
Exercise E5.1 Longer exercise: operating point with a piecewise-linear diode
A diode is connected in series with a resistor.
\[ \begin{align*} U_{\rm E}=12~{\rm V}, \qquad R=560~\Omega. \end{align*} \]
For the diode, use the piecewise-linear forward model
\[ \begin{align*} U_{\rm D} = U_{\rm TO}+r_{\rm F}I_{\rm D} \end{align*} \]
with
\[ \begin{align*} U_{\rm TO}=0.65~{\rm V}, \qquad r_{\rm F}=5.0~\Omega. \end{align*} \]
- Draw the loop equation.
- Calculate \(I_{\rm D}\).
- Calculate \(U_{\rm D}\).
- Calculate the diode power \(P_{\rm D}\).
- Compare briefly with the constant-voltage model \(U_{\rm D}=0.65~{\rm V}\).
The loop equation is
\[ \begin{align*} U_{\rm E} = RI_{\rm D} + U_{\rm D}. \end{align*} \]
Insert the piecewise-linear diode model:
\[ \begin{align*} U_{\rm E} = RI_{\rm D} + U_{\rm TO} + r_{\rm F}I_{\rm D}. \end{align*} \]
Thus
\[ \begin{align*} I_{\rm D} = \frac{U_{\rm E}-U_{\rm TO}}{R+r_{\rm F}}. \end{align*} \]
Insert the values:
\[ \begin{align*} I_{\rm D} &= \frac{12~{\rm V}-0.65~{\rm V}}{560~\Omega+5.0~\Omega} \\ &= \frac{11.35~{\rm V}}{565~\Omega} \\ &= 20.1~{\rm mA}. \end{align*} \]
The diode voltage is
\[ \begin{align*} U_{\rm D} &= U_{\rm TO}+r_{\rm F}I_{\rm D} \\ &= 0.65~{\rm V} + 5.0~\Omega\cdot 20.1~{\rm mA} \\ &= 0.65~{\rm V}+0.101~{\rm V} \\ &= 0.751~{\rm V}. \end{align*} \]
The diode power is
\[ \begin{align*} P_{\rm D} = U_{\rm D}I_{\rm D} = 0.751~{\rm V}\cdot 20.1~{\rm mA} = 15.1~{\rm mW}. \end{align*} \]
With the constant-voltage model,
\[ \begin{align*} I_{\rm D} = \frac{12~{\rm V}-0.65~{\rm V}}{560~\Omega} = 20.3~{\rm mA}. \end{align*} \]
The difference is small here because \(r_{\rm F}\ll R\).
Common pitfalls
- Thinking a diode is just a resistor: A diode is nonlinear. The ratio \(U/I\) is not constant.
- Forgetting current limitation: A forward-biased diode needs a current-limiting component.
- Treating \(0.7~{\rm V}\) as exact: The forward voltage depends on current, temperature, and semiconductor material.
- Mixing anode and cathode: Current flows easily from anode to cathode when the diode is forward-biased.
- Ignoring reverse limits: Real diodes have maximum reverse voltage. LEDs often tolerate only small reverse voltages.
- Confusing hole movement with electron movement: Holes are missing electrons, but they behave like positive mobile charge carriers.
- Using the exponential diode equation without unit care: \(U_{\rm T}\) must be in volts and \(T\) in kelvin.
Embedded resources
PhET: Semiconductors
Use this simulation to explore doping and the formation of a diode.
Falstad: Diode I/V curve
Use this simulation to compare a resistor characteristic with the nonlinear diode characteristic.