Equipment

1. Set of zener diodes (5 pieces)
2. Resistor 20 Ohm
3. 2 multimeters
4. 3 pairs of «Banana-Alligator» wires
5. 1 pair of «Alligator-Alligator» wires
6. DC voltage source
7. Heated calorimeter (electric kettle in thermal insulation)
8. Thermometer
9. Distilled water
10. 2 buckets
11. Laptop

Theoretical introduction

Zener diode is a semiconductor diode made in such a way that a breakdown is observed when connected in the opposite direction. On the CVC it looks like a sharp increase in current at module of voltage greater than some $Us$. If you connect a non-ideal voltage source with some $E>U_s$ to a zener diode, the voltage across the stabilizer will still be $U_s$. That means, it can be used as a stabilizer.

Let's look at the principle of semiconductor operation using silicon as an example. In pure silicon, all four electrons from the outer shell are in a bound state with the atom (valence electrons). Let's denote their energy as zero. In order for some valence electron to become free (i.e., freely propagating in silicon), it must be “excited” so that its energy becomes higher than $E_G$. And electrons CANNOT have an intermediate energy: either zero or $E=E_G+K$, where $K$ is the kinetic energy . The $E_G$ energy is called the energy of the forbidden zone.

The $E_G$ energy turns out to be much larger than thermal energy $k_BT$ for room temperature, so in pure silicon all electrons are valence ones, and therefore pure silicon is weakly conductive.

Semiconductors are modified with impurities. For example, by adding phosphorus atoms to the silicon lattice, we create free electrons (the number of valence electrons remains the same). The electrons are the charge carriers in such a semiconductor and have a negative charge, and so such a semiconductor is an $n$-semiconductor (negative).

We can add boron atoms, and then we take away the valence electrons. But we can look at it another way: we will consider that the number of valence electrons remains the same, but in place of some (where in reality there are no valence electrons) we put positive charges, which we will call holes. Holes are free charge carriers and have a positive charge and therefore such a semiconductor is a $p$-semiconductor (positive).

A semiconductor diode is a piece of semiconductor that has two halves modified differently, one half is an $n$-semiconductor and the other half is a $p$-semiconductor. All non-trivial properties of a semiconductor diode (e.g. nonlinear CVC) are determined by processes in the transition region between the $p$-area and the $n$-area.

ATTENTION! Direct diode connection is a connection where current flows from the $p$-area to the $n$-area. The $n$-area on the stabilizer is marked with a black ring.

Then the reverse connection, which will be studied in the whole problem, is the connection of the positive voltage to the side with the black ring.
 

Breakdown in a stabilizer can occur by three mechanisms:

• Avalanche mechanism. It occurs if the charge carriers during the movement through the semiconductor begin to have kinetic energy sufficient for the formation of new electron-hole pairs. In this case, the number of carriers (and the conductivity caused by them) increases significantly.
• Tunnel mechanism. It is a consequence of quantum tunneling effect.
• Thermal mechanism. Under strong heating associated with current flow, the $pn$-junction is destroyed.

The avalanche and tunnel mechanisms work together. The ratio between the contribution of these mechanisms to the breakdown characteristics of a particular zener diode depends on its zener voltage.

Part А. Avalanche breakdown.

Let us consider in detail the mechanism of avalanche breakdown. In the region of the $pn$-junction, the electric field is much larger than in the remaining volume. Practically all the voltage falls exactly on the $pn$-junction, and the charge carriers move accelerated. We will neglect the action of all forces except the interaction with the electric field.

If some particle with kinetic energy $K>E_G$ moves inside the semiconductor, it can “hit” the atom with some probability and spend its kinetic energy to excite the valence electron. Then instead of a valence electron we have a free electron and a hole.

If an electron or a hole is such a particle, then as a result of the “impact” we have three charge carriers instead of one charge carrier.

A1 When traveling through a conductor, an electron regularly “collides” with atoms of the lattice and the average distance between “collisions” is called the electron's free path length $\lambda$.

Find the electron's kinetic energy $K$, which the electron has before colliding with an atom, if after the previous collision its energy was zero. Express the answer in terms of the electric field strength in the medium $E$, elementary charge $e$.

Write the condition for the field $E$ under which the electron will create a new electron-hole pair at each collision.

There is no analog of the same mechanism for holes, since holes are an effective, not a real particle.

The effective mass of the electron, which must be used in the equations of motion (e.g., Newton's second law), is denoted by $m_e^*$.

A2 Using the result of A1, express the mean free path $\lambda$ through the elementary charge $e$, the effective electron mass $m^*_n$, $U_s$ the zener voltage of the zener diode, $d$ the width of the $pn$ junction, and the band gap of silicon $E_G$.

A3 In the answer sheet, draw a qualitative plot of the CVC of the zener diode in the reverse direction under avalanche breakdown.

Note. CVC - $I(U)$ dependence, not vice versa!

Part В. Tunnel breakdown.

Now let us consider in detail the mechanism of tunneling breakdown. It is a consequence of quantum nature of particles. If a classical particle, when hitting an energy barrier, is reflected from it, a quantum particle has a small chance to tunnel through this barrier.

As the zener diode voltage increases, the energy level of free electrons in the $p$-semiconductor (relative to $n$-semiconductor) increases and tunneling is more efficient. When heated, the effective mass of electrons changes and the breakdown voltage will decrease.

The number of electrons $Z$ that tunnel through the $pn$-junction per unit time:

$$Z = \frac{V e^2 E^2 }{18 \pi \hbar^2 } \sqrt{\frac{m_r}{E_G}} \cdot \text{exp}\left( - \frac{\pi \sqrt{m_r E_G^3} }{2 \hbar eE} \right)$$
where $V$ is the volume of the $pn$-transition, $E$ is the electric field inside the $pn$-transition, $\hbar$ is the reduced Planck constant, $E_G$ is the width of the forbidden zone, $m_r = \sqrt{\frac{m_p^* m_n^*}{m_p^*+ m_n^*}}$ is the reduced effective mass of carriers.

В1 Using the formula proposed in the condition, construct a characteristic CVC plot for $pn$-junction at tunneling breakdown.

Note. CVC - $I(U)$ dependence, not vice versa!

Part С. Experiment.

To check the theoretical models, you are asked to examine 5 given zener diodes at different temperatures. Pay attention that the zener diodes have flags with signatures D1-D5.

Important note!

• In all points of this part of the problem the study of the zener diode at the reverse connection is implied.
• In all points of this part of the problem, the positive voltage is the voltage at which the current flows in the reverse direction. The positive direction of current is the reverse direction.
• The zener diode voltage $U_s$ is the voltage at which the current is 60 mA unless otherwise specified.
• Never apply a current greater than 120 mA to the stabilizer!!!

The following circuit is proposed for measurements. Issued resistor with nominal value 
20 Ohm is denoted by r, and the zener diode is denoted by D. The voltage on voltmeter $V_1$ is denoted as $U_0$, on voltmeter $V_2$ is denoted as $U_r$.

Pay attention to the polarity of the connection.

С1 Justify why this circuit is optimal for measurements and why it is not possible to connect voltmeter 1 directly to the zener diode.

С2 Express the current $I_d$ through the zener diode and the voltage $U_d$ on it through $U_0,~U_r,~r$.

C3 Measure the zener voltages $U_s$ of zener diodes at room temperature, write them in the answer sheet under the number of the corresponding zener diode.

Note. Like any element, a zener diode heats up when current is passed through it, so measurements should be made with the stabilizer submerged in water.

Let's go directly to the measurement of the current-voltage characteristic.

Directions for data processing.

The data will be processed and stored using MS Excel. You can record all measured values immediately in the spreadsheets. In order for the jury to check your work correctly, name the files and record the data in the tables strictly as specified in the tasks!!!

On the desktop of your computer there is a folder called « M2 ». It contains the files «check.bat», «example.xlsx» and the folder «First name Second Name».

You need to rename this folder to contain your first and last name, otherwise your work will not be checked!

After you have renamed the folder, open it. It contains subfolders with item numbers and the file «Report.docx».

Do not change the names of the subfolders and the name of the file «Report.docx».

In a subfolder of the item, all direct measurements and further solution should be saved in DIFFERENT files. The solution files should contain comments on what values are calculated in EVERY column. These same comments MUST be written on the answer sheets for the corresponding item.

Solution files should be named «SolX.xlsx» where $X$ - is the name of the item being solved, and saved in the folder of that item. Solution files saved in the wrong folders will not be checked.

Thus, if you are solving item C4, you must create the file «SolX.xlsx» in the folder «C4».

The file «example.xlsx» is a template for recording measurements. All measured data must be formatted according to this template. Files with wrong format will not be marked! The measurement files must be named «MesX.xlsx» and saved in the folder of this item. The files with measurements must not contain anything except the original measurements: number of zener diode (only digit), temperature of water $T$ in which the zener diode is immersed, measurements of $U_0$ and $U_r$. All measurement files that are not designed according to the template or contain any unnecessary data will not be marked!!!!
Thus, if you solve item C4, you must copy the file «example.xlsx» and rename it to «MesC4.xlsx» in the folder «C4».

Below is an example of a completed measurement file and the correct solution file below.

You can run the file «check.bat» to verify that the original measurements were saved correctly. After execution, this file should give you the following lines in the console:

If the file produces any errors provided by it or any console errors, it means that not all files meet the design requirements and they need to be corrected. If the file does not output any series, it means that it is also not correctly recorded and saved!

Let's go directly to the measurements.

Order of Measurement.

• Rename the folder to your last and first name!
• Copy the template (not move!) to the folder of the item you are going to measure. Rename it to «MesX.xslx». 
• Without changing any of the filled cells in the «MesX.xlsx» file, and only by filling in the blank cells according to the condition, take and enter the required measurements into the file $U0$, $Ur$, $T$.
• Create a file «SolX.xslx» and copy the necessary data into it, then do the necessary recalculations, leaving comments on your actions in the table and in the answer sheets!
• Save the necessary graphs to the file «Report.docx», in the cell provided for your graph for this item and the zener diode.

C4 Take the CVC of all given zener diodes at room temperature in the current range from $0~mA$ to $100~mA$. The CVCs must contain at least 15 points and they must cover the current range evenly. Construct CVC plots and paste the plots into «Report.docx».

The result of this item must be the file «MesC4.xlsx» containing the original measurements and only them!, the file «SolC4.xlsx» in which the necessary recalculation are made, comments to it and plots of CVCs are made and images of the plots saved in «Report.docx».

Не трудно заметить, что, начиная с некоторого напряжения, ВАХ диода меняет свой характерный вид. Это связано со сменой преобладающего механизма пробоя.

C5 Используя теоретическую часть определите какой из механизмов пробоя в каких стабилитронах преобладает. Оцените напряжение смены механизма пробоя. Запишите полученные результаты в листы ответов.

Для начала исследуем туннельный пробой:

С6 С помощью формулы, предложенной в части А, линеаризуйте зависимость полученную для диода с туннельным механизмом пробоя и наименьшем напряжением открытия. Постройте линеаризованный график и вставьте его в $\textit{Отчет}$. Используя константы в выданной таблице контант, определите ширину $d$ $p-n$ перехода и запишите её в листы ответов.

$\textbf{Примечание.}$ Эффективные массы электрона и дырки даны при комнатной температуре и идеальных условиях, в реальности эти величины могут сильно отличаться от данных в таблице.

Результатом этого пункта должна быть таблица $\textit{С6_sol.xlsx}$ с решением и изображения графиков сохраненные в $\textit{Отчет}$.

К сожалению, как уже было сказано, эффективные массы дырки и электрона очень сильно зависят от многих факторов, в том числе от температуры. Зависимость от температуры возникает в следствие того, что концентрация носителей заряда зависит от температуры, из-за чего электронный газ внутри полупроводника ведет себя по-разному при разных температурах, что выражается в изменении эффективных масс. В следующих пунктах мы попробуем получить зависимость приведённой массы от температуры.

Для этого воспользуемся тем, что ВАХ туннельного стабилитрона зависит от температуры.

C7 При 4-ех других температурах снимите зависимость ВАХ того же стабилитрона для которого вы выполняли пункт С6 также в диапазоне от 0 до 100 мА. ВАХи также должны содержать не менее 15 точек и они должны равномерно покрывать диапазон токов.

Результатом этого пункта должны быть файлы $\textit{С7_mes.xlsx}$, $\textit{С7_sol.xlsx}$ и графики ВАХ перенесенные в $\textit{Отчет}$.

С8 Аналогично пункту $\textbf{С6}$ линеаризуйте данные зависимости и получите приведенную массу для каждой температуры и занесите её в таблицу в \textit{Отчете}. Постройте график зависимости эффективной приведенной массы от температуры и занесите его в $\textit{Отчет}$. Все линеаризованные графики также должны быть перенесены в него.


Результатом этого пункта должен быть файл $\textit{С8_sol.xlsx}$, графики перенесенные в $\textit{Отчет}$ и таблица заполненная в нём.

Теперь рассмотрим лавинные стабилитроны.

$\textbf{Примечание.}$ Во всех последующих пунктах задачи напряжение стабилизации (=напряжение пробоя $U_B$) — напряжение излома (его явно видно на графике ВАХ и ток при нём $<10-15$ мА).

C9 Для лавинного стабилитрона с наибольшим напряжением пробоя найдите длину свободного пробега, используя пункт B3 константы в таблице и $d$ которое вы получили в пункте $\textbf{C6}$ (Если вы не сделали этот пункт, то здесь и далее примите $d = 10$ нм).

Результатом этого пункта должно быть решение в листе ответов.

Также как и для туннельных стабилитронов, напряжение пробоя лавинного — зависит от температуры. Используя этот факт, мы можем найти зависимость длины свободного пробега электрона в полупроводнике от температуры.

C10 Используя тот же стабилитрон, для 8-ми других температур, снимите ВАХ стабилитрона $\textbf{в окрестности излома}$. Каждый ВАХ должен содержать не менее 5 точек и они обязательно должны лежать в окрестности точки излома.

Результатом этого пукта должны быть файл с измерениями $\textit{С10_mes.xlsx}$, файл с решением $\textit{С10_sol.xlsx}$ и графики ВАХ перенесенные в $\textit{Отчет}$.

C11 Для каждой температуры найдите напряжения пробоя и занесите их в таблицу в $\textit{Отчете}$ (включая напряжение пробоя при комнатной температуре). Найдите длину свободного пробега $\lambda$ электрона и занесите их в ту же таблицу. Постройте график зависимости длины свободного пробега от температуры и также занесите его в $\textit{Отчет}$.

Результатом этого пункта является файл $\textit{С11_sol.xlsx}$, заполненные таблица и поле для графика в $\textit{Отчете}$.

Таблица констант.