TBD

  - Semiconductor components \\ (approx. 4 blocks, based on previous lectures on [[circuit_design:2_diodes|Diodes]] and [[circuit_design:2_transistors|Transistors]] )
    - 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 =====
<callout>
After this 90-minute block, you can

  * 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.
</callout>

===== 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 [[block12|Block 12]].

===== Conceptual overview =====
<callout icon="fa fa-lightbulb-o" color="blue">
  * 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.
</callout>

<panel type="info" title="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 [[block12|Block 12]].
</panel>

~~PAGEBREAK~~ ~~CLEARFIX~~

===== Core content =====

==== Conductors, semiconductors, and insulators ====

Materials differ strongly in their specific resistance \(\rho\).

<WRAP>
<panel type="default">
<imgcaption fig_band_model|Band model for conductors, semiconductors, and insulators.></imgcaption>
{{drawio>block11_band_model_overview.svg}}
</panel>
</WRAP>

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 |

<panel type="info" title="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.
</panel>

==== 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**.

<callout>
There are two types of mobile charge carriers in semiconductors:

  * **electrons** with negative charge,
  * **holes** with positive effective charge.
</callout>

<panel type="info" title="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.
</panel>

==== Doping: n-type and p-type semiconductors ====

Doping means adding a very small amount of foreign atoms to the semiconductor crystal.

<WRAP group>
<WRAP column half>
<panel type="default">
<imgcaption fig_n_doping|N-doping: donor atoms provide additional mobile electrons.></imgcaption>
{{:circuit_design:ndoping.svg?500}}
</panel>
</WRAP>

<WRAP column half>
<panel type="default">
<imgcaption fig_p_doping|P-doping: acceptor atoms create additional holes.></imgcaption>
{{:circuit_design:pdoping.svg?500}}
</panel>
</WRAP>
</WRAP>

<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 |

<callout>
Doping does **not** mean that the semiconductor becomes strongly charged as a whole.  
The crystal is still approximately electrically neutral.  
Doping mainly changes how many mobile charge carriers are available.
</callout>

==== The pn junction ====

A diode is formed when p-doped and n-doped regions meet.

<WRAP>
<panel type="default">
<imgcaption fig_pn_junction|Diode symbol and pn junction with anode A and cathode K.></imgcaption>
{{:circuit_design:pnjunction.svg?650}}
</panel>
</WRAP>

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**.

<WRAP>
<panel type="default">
<imgcaption fig_pn_depletion|Formation of the depletion region at a pn junction.></imgcaption>
{{:circuit_design:evolutionofpnjunction.svg?650}}
</panel>
</WRAP>

The depletion region behaves like an internal barrier.  
Without an external voltage, it prevents a large current.

<panel type="info" title="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.
</panel>

==== 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 |

<callout type="info" icon="true">
**Mnemonic**

\[
\begin{align*}
\text{Positive Anode, Negative Is Cathode}
\end{align*}
\]

This helps to remember the forward direction of a diode.
</callout>

==== Ideal diode model ====

The simplest model is the ideal diode.

\[
\begin{align*}
\text{forward direction: } u_{\rm AK}=0,\quad i_{\rm D}>0
\end{align*}
\]

\[
\begin{align*}
\text{reverse direction: } i_{\rm D}=0,\quad u_{\rm AK}<0
\end{align*}
\]

<WRAP>
<panel type="default">
<imgcaption fig_ideal_diode_characteristic|Ideal diode characteristic.></imgcaption>
{{drawio>block11_ideal_diode_characteristic.svg}}
</panel>
</WRAP>

<panel type="info" title="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.
</panel>

==== 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 \(\mu{\rm A}\) |

<callout type="warning" icon="true">
The value \(0.7~{\rm V}\) for a silicon diode is not a physical constant.  
It is a useful approximation for typical currents in small signal and basic power circuits.
</callout>

==== Practical diode models for circuit calculation ====

For hand calculations we usually do not use the full exponential equation.

<WRAP>
<panel type="default">
<imgcaption fig_diode_models|Comparison of ideal, constant-voltage, and piecewise-linear diode models.></imgcaption>
{{drawio>block11_diode_models.svg}}
</panel>
</WRAP>

<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

\[
\begin{align*}
r_{\rm D}
=
\frac{{\rm d}u_{\rm D}}{{\rm d}i_{\rm D}}
\approx
\frac{mU_{\rm T}}{I_{\rm D}}.
\end{align*}
\]

<callout type="info" icon="true">
**Unit check**

\[
\begin{align*}
[r_{\rm D}]
=
\frac{[U_{\rm T}]}{[I_{\rm D}]}
=
\frac{{\rm V}}{{\rm A}}
=
\Omega.
\end{align*}
\]
</callout>

==== 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

\[
\begin{align*}
I_{\rm D}
\approx
\frac{U_{\rm E}-U_{\rm TO}}{R}.
\end{align*}
\]

<callout type="danger" icon="true">
Never connect a normal diode or LED directly to an ideal voltage source in forward direction.  
The diode current must be limited.
</callout>

==== 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

\[
\begin{align*}
u_{\rm Z}
\approx
U_{\rm Z}+r_{\rm Z}i_{\rm Z}.
\end{align*}
\]

<panel type="info" title="Z-diode preview">
Z-diodes are useful for voltage limitation and voltage stabilization.  
The practical circuits are treated in [[block12|Block 12]].
</panel>

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 | \(\approx 3.2~{\rm V}\) |

<callout type="warning" icon="true">
LEDs usually tolerate only small reverse voltages.  
Do not operate an LED in reverse direction unless the datasheet explicitly allows it.
</callout>

~~PAGEBREAK~~ ~~CLEARFIX~~

===== Exercises =====

#@TaskTitle_HTML@##@Lvl_HTML@#~~#@ee2_taskctr#~~.1  Quick check: doping and charge carriers
#@TaskText_HTML@#

Complete the table.

^ Doping type ^ Typical dopant atom ^ Main mobile charge carrier ^ Dopant name ^
| n-type | ? | ? | ? |
| p-type | ? | ? | ? |

#@ResultBegin_HTML~ExerciseDoping~@#

^ 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.

#@ResultEnd_HTML@#
#@TaskEnd_HTML@#


#@TaskTitle_HTML@##@Lvl_HTML@#~~#@ee2_taskctr#~~.1  Quick check: diode polarity
#@TaskText_HTML@#

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?

#@ResultBegin_HTML~ExercisePolarity~@#

\[
\begin{align*}
u_{\rm AK}
=
U_{\rm A}-U_{\rm K}
=
4.8~{\rm V}-4.1~{\rm V}
=
0.7~{\rm V}.
\end{align*}
\]

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.

#@ResultEnd_HTML@#
#@TaskEnd_HTML@#


#@TaskTitle_HTML@##@Lvl_HTML@#~~#@ee2_taskctr#~~.1  Quick check: current with the constant-voltage model
#@TaskText_HTML@#

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 \(I_{\rm D}\).

#@ResultBegin_HTML~ExerciseSeriesResistor~@#

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

\[
\begin{align*}
I_{\rm D}
=
\frac{U_R}{R}
=
\frac{4.3~{\rm V}}{1.0~{\rm k}\Omega}
=
4.3~{\rm mA}.
\end{align*}
\]

#@ResultEnd_HTML@#
#@TaskEnd_HTML@#


#@TaskTitle_HTML@##@Lvl_HTML@#~~#@ee2_taskctr#~~.1  Quick check: differential diode resistance
#@TaskText_HTML@#

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

\[
\begin{align*}
r_{\rm D}\approx \frac{mU_{\rm T}}{I_{\rm D}}.
\end{align*}
\]

#@ResultBegin_HTML~ExerciseDifferentialResistance~@#

\[
\begin{align*}
r_{\rm D}
&\approx
\frac{mU_{\rm T}}{I_{\rm D}}
\\
&=
\frac{1\cdot 26~{\rm mV}}{10~{\rm mA}}
\\
&=
2.6~\Omega.
\end{align*}
\]

This is a small-signal resistance around the operating point.  
It is not the same as the large-signal ratio \(\frac{U_{\rm D}}{I_{\rm D}}\).

#@ResultEnd_HTML@#
#@TaskEnd_HTML@#


#@TaskTitle_HTML@##@Lvl_HTML@#~~#@ee2_taskctr#~~.1  Longer exercise: operating point with a piecewise-linear diode
#@TaskText_HTML@#

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 \(U_{\rm D}=0.65~{\rm V}\).

#@ResultBegin_HTML~ExercisePiecewiseLinearDiode~@#

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 \(r_{\rm F}\ll R\).

#@ResultEnd_HTML@#
#@TaskEnd_HTML@#


===== 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 =====

<WRAP group>
<WRAP column half>
<panel type="info" title="PhET: Semiconductors">
Use this simulation to explore doping and the formation of a diode.

{{url>https://phet.colorado.edu/en/simulations/semiconductor 700,500 noborder}}
</panel>
</WRAP>

<WRAP column half>
<panel type="info" title="Falstad: Diode I/V curve">
Use this simulation to compare a resistor characteristic with the nonlinear diode characteristic.

{{url>https://www.falstad.com/circuit/e-diodecurve.html 700,500 noborder}}
</panel>
</WRAP>
</WRAP>

~~PAGEBREAK~~ ~~CLEARFIX~~