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Department of Engineering Science and Physics College of Staten Island PHY315 Advanced Physics Laboratory Lab Manuals

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Content Notes about this Lab Course

Safety First!

Lab Reports

Lab Works

1. Basic Electrical Parameters of Semiconductors: Sheet Resistivity, Resistivity and Conduction Type.

2. Measuring Hall-Effect: Conductivity type, Concentration of Charge Carriers and Their Mobility.

3. Temperature Activation of Charge Carriers in Semiconductors: Measuring Bandgap Energy of Semiconductors.

4. Simulation of Ion Doping of Solids.

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Notes about this Lab Course The experiments developed for this lab course PHY315 have been designed to expand on the material covered in lectures and to experimentally demonstrate the validity of physical principles. The manual describes 8 experiments and instructors will select the most important topics for experimental confirmation. The students would better understand the fundamentals of physics if they were able to experimentally see for themselves that the principles presented in the text are real, or pretty close to reality. It also important that through measuring and comparing the experimental results with analytical solutions and simulations the students become aware of the limitations of the analytical description of real physical processes.

The present lab course “Advanced Physics Laboratory PHY315” is a newly developed one. Therefore the feedback from student is extremely important for improving the design of experiments. The students are asked to give a brief assessment of the laboratory work in the lab reports. After each lab work, the students are invited to comment on ways to improve the experiment just completed. Students are also encouraged to propose topics they feel would benefit through experimental confirmation of principles discussed in class.

Safety First! Safety is always an important topic whenever laboratory work is being considered, and it is certainly true in the case of PHY315 labs. Safety is important. Although the laboratory experiments use low voltages and low currents. However, the lab equipment is powered by the standard 110 V, 60 Hz line voltage. Be careful with the line voltages. Do not touch exposed prongs on the equipment plugs when connecting the equipment to the lines. Take care when using power supplies, which may be low voltage but can supply currents in the ampere range. Shorting such a supply can lead to a serious burn as current arc can ignite flammable materials.

Some equipment is heavy enough to be generally stable on the bench. Be sure to keep the equipment away from the edges of the benches to avoid having a piece of equipment fall off the bench. Besides endangering people who might be struck, falling equipment endangers everyone in vicinity by stressing the power cords, possibly causing a line short or live fault on the equipment, not to mention damage to the expensive lab equipment. In general, electronic equipment does not survive harsh treatment.

General Laboratory Rules 1. No eating, or drinking in the lab.

2. No use of cell phones in the lab.

3. Lab computers are for experiment use only. No web surfing, e-mail reading, or computer games allowed.

4. When finished using lab computer, put keyboard and mouse in the original place.

5. After the experiment is finished, the used equipment must be arranged in the way you found it.

6. Some equipment is required to be signed out and checked back in.

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7. When leaving, clean up after yourself and leave your working lab desk in the state you found it.

8. If you need any assistance, ask your lab teacher, lab technician, or call 718 982 2812.

Thank you for your co-operation!

Lab Reports Scientists and engineers are most effective if they can clearly communicate their ideas and developments to others, both their colleagues and their managers. For this reason, writing and documenting are essential aspects of an engineer’s/researcher's job. On average, experimentalists spend most of their time documenting their work and communicating the results to others. Many science and engineering students do not realize the importance of this documentation and communication process and have difficulties in their first job because of lack of skill in documenting their work. Scientists and engineers in the workplace are evaluated on their communication skills, which include both the quality and quantity of their publications and technical reports.

In this lab class the students are required to prepare lab reports for each lab work. The lab report is the main document in which the student communicate the results of her/his experimental work, processing the experimental data and the conclusions made.

The lab report is as important as the work itself done in the lab. Unless you can communicate the results of your work, the work has little usefulness. Furthermore, the lab report reinforces the material that was learned in the lecture class. Development of both oral and written technical communication skills is one of the most important things you can learn as an undergraduate student.

General requirements Lab report should conform to the following guidelines.

- Each report should be a self-contained document and should present all the information regarding the pre-experiment preparations, experimental work, measurements, obtained data, data processing, discussion and conclusions.

- Each report is to be typed using a word processor.

- Figures, drawings, charts, and tables should be added where they are needed and should contain understandable labels, including units for the axes of the graphs.

- When plotting B vs. A, B is the dependent variable and is plotted on the y-axis, while A is the independent variable and is plotted on the x-axis. The figure should refer to the main text and should not stand alone. Except for raw data, all figures should be computer-drawn using any suitable plotting programs such as Origin or Excel.

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- All pages should be consecutively numbered. Sign your report on the cover page. This signature shows that you take responsibility for what is contained in the report. Reports are due in a week after the experimental lab work has been finished.

Lab report content 1. Title page – the first separate page showing the student name, course and section numbers, lab title, date of the performance of experiment and the names of teammates.

2. Objectives - a short description (a few sentences) of the purpose and aim of the experiment.

3. Physical principles - a short description (up to one page) of the basic physical principles, definitions and relevant formulae describing the experiment and used for the processing of the obtained experimental data.

4. List of the experimental equipment used.

5. Experimental procedure - description of the major steps of performance of the experiment.

6. Laboratory data Sheet with the original experimental data preferentially arranged in tabular form. IMPORTANT: The lab data sheet of every student must be signed by the instructor at the end of the lab work session. The lab report without signed lab data sheet is not accepted. 7. Computations and graphs. This part of the report contains computations of the physical values and their experimental errors with indication of the corresponding units. Standard graph paper can be used for the graphical presentation of the experimental and calculated data.

8. Discussion. In this section, students discuss and analyze the obtained experimental data, results of calculations, graphs and the experimental errors. Discussion is the most creative and as such the important part of the lab report. Quality of the Discussion largely determines the grade of the lab report. 9. Conclusion, in which a statement is made as to whether the aim of the lab work has been achieved. In most cases, this statement is supported by the numerical data obtained.

10. Answers to the questions given at the end of every lab manual.

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LAB WORKS

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1. Basic Electrical Parameters of Semiconductors: Sheet Resistivity, Resistivity and Conduction Type.

1.1 Objectives 1. Familiarizing with experimental techniques used for the measurements of electrical properties of semiconductors.

2. Measurements of conductivity of doped low-ohmic semiconductors using Van der Pauw method.

3. Measurements of conductivity of high-ohmic intrinsic and compensated semiconductors by resistance method.

4. Determination of conductivity type of semiconductors by “Hot-Probe” method.

1.2 Principles 1.2.1 The van der Pauw Method The Van der Pauw method is a technique commonly used to measure resistivity ρ, sheet resistance RS, concentration of majority charge carriers n, p (electrons, or holes) and their sign as well as mobility µ of charge carriers of semiconductor. The power of the van der Pauw method lies in its ability to accurately measure the properties of a thing sample of arbitrary shape. The method was first propounded by Leo J. van der Pauw in 1958.

In order to use the van der Pauw method, the thickness of sample t must be much less than its width and length. It is preferable that the sample is symmetrical (Fig. 1.1).

Fig. 1.1. Two common types of van der Pauw samples: clover leaf and square. Each sample has four symmetrical electrical contacts.

The correct measurements require that four ohmic contacts are attached on the boundary of the sample. The contacts must be much smaller than the distance between them. In addition, the

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leads connected to the contacts must be made of the same batch of wire to minimize thermoelectric effects. For the same reason, all four contacts should be of the same material.

The contacts are numbered from 1 to 4 in a counter-clockwise order, beginning at the top-left contact. The voltages measured between contacts and the currents flowing between them are defined as following. For instance, the current I12 is a positive DC current injected into contact 1 and taken out of contact 2, and is measured in amperes. The voltage V34 is a DC voltage measured between contacts 3 and 4 (i.e. V4-V3) with no externally applied magnetic field, measured in volts.

1.2.2. Resistivity measurements The average resistivity ρ of a sample can be calculated as a product of its sheet resistance RS and thickness t:

ρ = RS⋅t. (1)

RS can be found measuring two Van der Pauw resistances along two perpendicular sides of square sample. Van der Pauw resistance is a ratio of the voltage applied across one edge of sample over the current generated along the opposite edge of the sample. For instance, R41,23 is measured as voltage V23 over current I41 (Fig. 1.2).

𝑹𝑹𝟒𝟒𝟒𝟒,𝟐𝟐𝟐𝟐 = 𝑽𝑽𝟐𝟐𝟐𝟐𝑰𝑰𝟒𝟒𝟒𝟒

(2)

Fig. 1.2. Example of measuring Van der Pauw resistance. Voltage is applied along one side, while the generated current is measured along the opposite side. Picture shows the measurement of resistance R41,23. Two “perpendicular” resistances which can be used for resistivity measurements are, for instance, R12,34 and R23,41.

The sheet resistance can be determined from two of these resistances - one measured along a vertical edge, such as Rvertical = R12,34, and a corresponding one measured along a horizontal edge, such as Rhorizontal = R23,41. The actual sheet resistance is related to these resistances by the van der Pauw equation:

𝑒𝑒𝑒𝑒𝑒𝑒 �− 𝜋𝜋𝑅𝑅𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑅𝑅𝑆𝑆

� + 𝑒𝑒𝑒𝑒𝑒𝑒 �−𝜋𝜋𝑅𝑅ℎ𝑜𝑜𝑣𝑣𝑣𝑣𝑜𝑜𝑜𝑜𝑜𝑜𝑣𝑣𝑣𝑣𝑣𝑣𝑅𝑅𝑆𝑆

� = 1 (3)

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For symmetrical samples, e.g. square samples used in this lab work, Rvertical = Rhorizontal = R and the equation (3) has simple solution:

𝑹𝑹𝑺𝑺 = 𝜋𝜋𝑹𝑹ln2

(4)

In order to obtain a more precise value for the resistances, four Van der Pauw resistances for all four sides of the sample are measured and R is taken as average of these four resistances.

1.2.3. Resistance method Van der Pauw method is usually applied to doped semiconductors of moderate and low resistivity not exceeding 10 kΩ.cm. Resistivity of highly resistive semiconductors, e.g. undoped intrinsic silicon, can be conveniently determined measuring resistance of a rectangular sample (slab) made of this material. For a sample of length l, width w and thickness t, its resistance R is:

𝑹𝑹 = 𝝆𝝆 𝒍𝒍𝒘𝒘𝒘𝒘

(5)

Thus the resistivity ρ can be found as:

𝝆𝝆 = 𝑹𝑹𝒘𝒘𝒘𝒘𝒍𝒍

(6)

The most correct measurements of resistance are performed by four-probe method. Current in the sample is generated applying voltage to the outer contacts on the slab while voltage is measured between the internal contacts. The length l in the Eq. 5 is the distance between the internal contacts.

The advantage of the four-probe method, as compared with the two-probe method, is the exclusion of resistances of the contacts from the measurements. The contact resistances are usually unknown and, when comparable with the resistance of the sample, they may cause considerable experimental error.

1.2.4 “Hot-Probe” method When a piece of semiconductor is heated non-uniformly, the charge carriers diffuse from hot region in cold one generating this way a voltage (thermo-voltage) between these regions (Fig. 1.3).

Fig. 1.3. Generation of thermo-voltage between hot and cold ends of a semiconductor slab.

Since sign of thermo-voltage corresponds to the sign of the diffusing charge carriers, the measurement of the voltage induced between hot and cold contacts on a piece of semiconductor is a simple way to determine its conductivity type. This measurement known as the "hot-probe" method provides a simple way to distinguish between n-type and p-type semiconductors and can

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be easily demonstrated using a soldering iron and a standard multi-meter. The experiment is performed by contacting a semiconductor sample with a "hot" probe (tip of heated soldering iron) and a "cold" probe (regular contact). Both probes are wired to a sensitive volt-meter. The hot probe is connected to the positive terminal of the meter, while the cold probe is connected to the negative terminal. The experimental set-up is shown in Fig. 1.4.

Fig. 1.4. Principle of experimental set-up of the "hot-probe" experiment.

1.3. Experimental Equipment - Two Van der Pauw samples of doped silicon (n-type and p-type) with holders

- 1 bar sample of intrinsic silicon with holder

- Various free standing samples of silicon

- DC Power Supply

- Digital ammeter

- Digital voltmeter

- Soldering Iron

- Micrometer

1.4. Procedure 2.4.1. Van der Pauw measurements 1. Assemble the measuring set-up as shown in Fig. 1.5 below.

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Fig. 1.5. Schematics of the experimental circuit for the measurement of Van der Pauw resistance.

2. Measure Van der Pauw resistances along all sides of the n-type samples for both directions of current.

3. Calculate average Van der Pauw resistance and sheet resistivity of the sample using Eq. 4.

4. Measure thickness of the samples using micrometer.

5. Calculate resistivity of the sample using Eq.1.

6. Repeat the measurements on the p-type Van der Pauw sample.

1.4.2. Resistance method 1. Assemble the measuring set-up as shown in Fig. 1.6.

Fig. 1.6. Schematics of the experimental circuit for the electrical measurements using resistance method.

2. Apply voltage to the outer contacts of one of the intrinsic samples and measure current flowing through the sample.

3. Measure voltage between the internal contacts.

4. Calculate resistance dividing the measured voltage by the measured current.

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5. Measure thickness of the sample, its width and the distance (length) between the internal contacts using micrometer or caliper.

6. Calculate resistivity of the sample using Eq. 6.

1.4.3. “Hot-Probe” measurements 1. Connect digital voltmeter to the metal substrate and to the tip the soldering iron.

2. Place one of the samples on the substrate.

3. Switch on soldering iron and wait until it is hot.

4. Touch sample with the hot tip and immediately take reading of the voltmeter. Note the sign of the reading.

5. Repeat the measurements on another free-standing samples.

1.5. Procedure 1. Discuss the processes of measurements and the obtained results.

2. Compare values of resistivity measured by Van der Pauw and resistance methods.

3. Using standard values of mobility of electrons and holes in silicon, calculate concentrations of major charge carriers in the measured samples.

4. Discuss the results of your calculations.

5. Compare the data obtained by the van der Pauw and resistance methods.

1.6. Questions 1. Which of the methods used in this lab work is the most accurate? Why?

2. Why is the four-point measurement of resistance more accurate than the two-point one?

3. Predict the magnitude of mobility in the silicon samples you have measured.

4. How does thermo-voltage depend on concentration of charge carriers and mobility?

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2. Hall-Effect in Semiconductors. Conduction type, Concentration of Charge Carriers and Their Mobility

2.1. Objectives 1. Familiarizing with the measurements of Hall-effect on semiconductors.

2. Measurement of concentration of majority charge carriers and their mobility in silicon.

2.2. Principles Charge carrier concentration and mobility can be found measuring Hall-effect on Van der Pauw sample placed in magnetic field. Magnetic field makes charge carriers to deviate from their straight drift between the contacts. This deviation results in generation of Hall voltage VH in the direction perpendicular to the direction of current flow (Fig. 2.1).

Fig. 2.1. (a) Principle of measurement of Hall-effect. (b) Example of measurement of Hall-voltage V42. Current is applied between contacts 3 and 1 (I31), while voltage is measured between contacts 4 and 2.

In order to measure Hall-voltage on Van der Pauw sample, two sets of measurements are made: one with magnetic field in the positive z-direction, and one with it in the negative z-direction. The voltages recorded with positive field will have a subscript P (such as, V13, P = V3, P - V1, P), whereas those recorded with negative field will have a subscript N (such as V13, N = V3, N - V1, N). For all of the measurements, the magnitude of the injected currents should be kept the same. The magnitude of the magnetic field must be the same in both directions also. Hall voltage is then calculated as:

𝑽𝑽𝑯𝑯 = 𝑽𝑽𝟒𝟒𝟐𝟐+𝑽𝑽𝟐𝟐𝟒𝟒+𝑽𝑽𝟐𝟐𝟒𝟒+𝑽𝑽𝟒𝟒𝟐𝟐𝟖𝟖

; (6)

V13 = V13, P − V13, N

V24 = V24, P − V24, N

V31 = V31, P − V31, N

V42 = V42, P − V42, N

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The polarity of the Hall voltage indicates the type of material the sample is made of. If it is positive, the material is p-type, and if it is negative, the material is n-type.

When Hall-voltage VH is known, sheet concentration of charge carriers nS can be found as:

𝒏𝒏𝑺𝑺 = 𝑰𝑰𝑰𝑰𝒆𝒆|𝑽𝑽𝑯𝑯|

, (7)

where I is the current flowing along the sample and e is the electron charge.

Density of the majority charge carriers nm and their mobility µm can be found as:

𝒏𝒏𝒎𝒎 = 𝒏𝒏𝑺𝑺𝒘𝒘

= 𝑰𝑰𝑰𝑰𝒆𝒆|𝑽𝑽𝑯𝑯|𝒘𝒘

, (8)

µ𝒎𝒎 = 𝟒𝟒𝒆𝒆𝒏𝒏𝑺𝑺𝑹𝑹𝑺𝑺

.

2.3. Experimental Equipment - Digital voltmeter

- Digital ammeter

- Holder with silicon van der Pauw samples

- DC Power Supply

- Permanent Co-Sm magnet of magnetic field strength B = 0.5 T

2.4. Procedure 1. Assemble the measurement set-up as shown in Fig. 2.2.

Fig. 2.2. Schematics of the experimental circuit for the Hall-effect measurements using Van der Pauw method.

2. Place magnet over the silicon sample.

3. Apply ~5 mA current through one diagonal of the sample. This is the current I shown in formula (7) and (8). Measure voltage over the other perpendicular diagonal. Flip the magnet and repeat the measurement. Calculate difference between the measured voltages. The obtained value is the voltage V13 in formula (6).

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4. Reverse the direction of current and repeat the procedure 3. The obtained value is the voltage V31 in formula (6).

5. Repeat the procedures 4 and 5 applying current to the other diagonal and measuring voltage over the perpendicular one. The obtained values are the voltages V24 and V42 correspondingly in formula (6).

6. Measure thickness of the silicon sample t.

2.5. Calculations and Discussion 1. Retrieve the data on Van der Pauw resistance R, sheet resistance RS and resistivity ρ of the extrinsic silicon samples you measured in lab work 2.

2. Calculate Hall-voltage VH using the voltages measured in procedures 4-6 and formulae (6).

3. Using the values of R, VH, t, I and B calculate density of the majority charge carriers nm, mobility of majority charge carriers µm, sheet resistance RS and sheet carrier density nS. Determine type of the majority charge carriers: electrons, or holes.

4. Compare the obtained values with that known for the sample you measured and with that known for silicon used in electronic industry.

5. Discuss your measurements and the obtained results.

2.6. Questions 1. Is it possible to measure Hall-effect on intrinsic semiconductor? Explain your answer.

2. Is it possible to measure Hall-effect on compensated semiconductor? Explain your answer.

3. How temperature influences Hall-effect measurements?

3. Predict Hall-voltage on your samples when they would be measured at a temperature of 100°C.

4. Is it possible to measure Hall-effect on your samples at liquid nitrogen temperature? Support your answer with calculations of charge carrier concentration.

Appendix (a) The intrinsic carrier density in silicon at 300 K equals:

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(b) Table of the intrinsic carrier density in semiconductors at different temperatures.

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3. Temperature Activation of Charge Carriers in Semiconductors. Bandgap Energy and Temperature Dependence of Mobility.

3.1. Objectives 1. Experimental measurements of conductance of intrinsic and doped semiconductors as a function of temperature,

2. Familiarizing with basic methods of measurement of bandgap energy of semiconductor and activation energies of donors and acceptors.

3.2. Principles 4.2.1. Measuring Bandgap energy Conductance S of a piece of an undoped (intrinsic) semiconductor is proportional to the concentration of intrinsic charge carriers ni. The value of ni is determined by the thermal activation of electrons and holes over the bandgap:

𝒏𝒏𝒊𝒊 = �𝑵𝑵𝑪𝑪𝑵𝑵𝑽𝑽𝒆𝒆𝒆𝒆𝒆𝒆 �−𝑬𝑬𝒈𝒈𝟐𝟐𝟐𝟐𝑰𝑰𝑻𝑻

�, (1)

where NC and NV are densities of states in conduction and valence bands correspondingly, Eg is the bandgap energy, kB is the Boltzmann constant and T is absolute temperature.

Since NC and NV do not depend much on temperature, the temperature dependence of ni and consequently the temperature dependence of conductance S of intrinsic semiconductor is primarily determined by the exponential term in equation (1):

𝑺𝑺 = 𝑪𝑪𝒆𝒆𝒆𝒆𝒆𝒆 �− 𝑬𝑬𝒈𝒈𝟐𝟐𝟐𝟐𝑰𝑰𝑻𝑻

�, (2)

where C is a constant. By measuring conductance S1 and S2 at two different temperatures T1 and T2 one has a simple way to experimentally determine the value of the bandgap of semiconductor:

𝑆𝑆1 = 𝐶𝐶𝑒𝑒𝑒𝑒𝑒𝑒 �− 𝐸𝐸𝑔𝑔2𝑘𝑘𝐵𝐵𝑇𝑇1

�, (3)

𝑆𝑆2 = 𝐶𝐶𝑒𝑒𝑒𝑒𝑒𝑒 �−𝐸𝐸𝑔𝑔

2𝑘𝑘𝐵𝐵𝑇𝑇2�

𝑆𝑆1𝑆𝑆2

= 𝑒𝑒𝑒𝑒𝑒𝑒 �−𝐸𝐸𝑔𝑔

2𝑘𝑘𝐵𝐵�

1𝑇𝑇1−

1𝑇𝑇2�� =

𝐼𝐼1𝐼𝐼2

If a constant voltage is used for the measurements of conductance, the ratio of conductances S1/S2 equals to the ratio of corresponding currents I1/I2. Then the formula for calculation of the bandgap energy is:

𝐸𝐸𝑔𝑔 =2𝑘𝑘𝐵𝐵𝑙𝑙𝑙𝑙�

𝐼𝐼1𝐼𝐼2�

1𝑇𝑇2− 1𝑇𝑇1

. (4)

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3.2.2. Temperature dependence of mobility Conductance of doped semiconductors is primarily determined by the concentration of majority charge carriers – electrons, n, or holes p. At room temperature and elevated temperatures, these concentrations approach ND and NA respectively: n ≈ ND and p ≈ NA. Since concentration of intrinsic charge carriers is much less than that delivered by dopants, conductance of doped semiconductors can be found as (5):

𝑺𝑺𝒏𝒏 = 𝑺𝑺𝟎𝟎𝒏𝒏𝒏𝒏𝝁𝝁𝒏𝒏, 𝑺𝑺𝒆𝒆 = 𝑺𝑺𝟎𝟎𝒆𝒆𝒆𝒆𝝁𝝁𝒆𝒆, (5)

where S0n and S0p are constant geometrical factors of the samples measured and μ is mobility. In these formulae, the only parameter strongly dependent on temperature is the mobility. In the temperature range 100 to 400K, mobility drops with temperature: μ ~ T-3/2 (Fig. 3.1). Thus, measuring ratio of conductances S1 and S2 at different temperatures T1 and T2, one can verify the temperature dependence of mobility in doped semiconductors.

Fig. 3.1. Temperature dependence of mobility of electrons in n-type silicon (a) and holes in p-type silicon (b) for doping levels of 1012 cm-3, 1016 cm-3 and 1017 cm-3.

100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400300

400

500600700800900

1000

2000

3000

4000

50006000700080009000

10000

20000

Electrons a ND1e12cm-3 ND1e16cm-3 ND1e17cm-3

Mob

ility,

cm

2 /Vs

Temperature, K100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400

100

200

300

400500600700800900

1000

2000

3000

400050006000700080009000

10000

Holes b NA1e12cm-3 NA1e16cm-3 NA1e17cm-3

Mob

ility,

cm

2 /Vs

Temperature, K

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Fig. 3.2. Temperature dependence of electron concentration (a) and conductivity (b) of As-doped n-type silicon. Note that the electron concentration always grows with temperature, whereas the conductivity of low-doped silicon may considerably decrease with temperature from 100 to 400K.

3.3. Experimental Equipment - Two holders with Wan der Pauw samples of n-type and p-type silicon

- One holder with bar sample of intrinsic silicon

- DC Power Supply

- Digital voltmeter

- Digital ammeter

- Hot plate

- Thermocouple

- 1 beaker with boiling water

- 1 beaker with dry ice

3.4. Procedure 1. Assemble the measuring circuit with the bar sample of intrinsic silicon as shown in Fig. 1.6.

2. While performing the measurements, place sample with thermocouple inside a test-tube submerged in the boiling water to slowly heat it to about 100°C. It may not reach exactly boiling point because it is isolated inside the test-tube.

3. After reaching maximum temperature, take out the sample and allow it to cool to room temperature.

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4. Once the sample is back at room temperature, cool it in beaker with dry ice to about -60°C.

5. Once again remove the sample and allow it to warm up to room temperature.

6. Export the graph to Excel and process data as needed. The combination of heating and cooling curves in one graph will remove any error from hysteresis.

7. Repeat the procedures 1 to 8 for n-type silicon and p-type silicon.

3.5. Calculations and Discussion

1. Using the data obtained from the intrinsic silicon sample. Process data to plot ln(𝐼𝐼) 𝑣𝑣𝑣𝑣 1𝑇𝑇 and

find the slope. Formula (2) describes this relationship and can be used to calculate bandgap energy of silicon.

2. Compare the obtained value of the bandgap energy. Theoretically, the slope must be constant. If it deviates considerably calculate the magnitude of the deviation. Compare this value of bandgap energy with the known value of Eg = 1.12 eV for silicon.

3. Discuss the process of measurements, your results and possible reasons of deviation of the measured values of bandgap energy from the known one.

4. Using the data obtained from doped silicon samples, process the data to plot lg(I) versus lg(T) and find the slope. This slope is negative and its value should be about -1.5. Discuss the obtained values in terms of conductivity and compare them with the temperature change of mobility shown in Fig. 1 and with the temperature change of conductivity shown in Fig. 2.

3.6. Questions 1. Using your experimentally obtained values of Eg, predict the conductance of the bar sample at the temperature of liquid nitrogen (-195°C) and maximum working range of devices (120°C).

2. At what temperature the conductance of the intrinsic silicon sample would be equal to the room temperature conductance of the doped silicon samples?

3. Is it possible within this lab work to measure conductivity and sheet resistance of the silicon samples?

4. Is it possible within this lab work to measure concentration of donors and acceptors in the doped silicon samples?

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4. Simulation of Ion Doping of Semiconductors 4.1. Objectives - To give students hand-on experience of numerical simulation of ion doping used for fabrication of semiconductor electronic structures.

- To familiarize students with SRIM software used for numerical simulation of ion doping.

- To perform numerical simulation of ion doping of planar bipolar transistor.

4.2. Principles 4.2.1. Parameters of ion-doped layer Ion implantation is the main doping method used for fabrication of in microelectronic devices. Over all, it is the most precise and controllable method of impurity doping of solids. In ion implantation, impurity toms are introduced into semiconductor substrate by ionizing them (creating ions), accelerating the ions to energies ranging from kiloelectronvolt (keV) to megaelectronvolt (MeV), and then literally shooting these ions onto the substrate surface (Fig. 4.1).

Fig. 4.1. Principle of local doping of semiconductor using ion implantation and masking technique. Openings in the mask define the ion-doped areas. Mask must be thick enough to protect the masked areas from doping.

Ions penetrate into semiconductor substrate to a certain doping depth Ri. This way a buried ion doped layer is created. Distribution of density of the implanted ions N(x) through the depth x is not uniform. It is approximately described by a Gaussian function (1):

Si substrate

P+ ion doping Ion-doped region

Mask Mask

Ions stopped by mask

Accelerated ions

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𝑁𝑁(𝑒𝑒) = 𝑁𝑁𝑣𝑣∆𝑅𝑅√2𝜋𝜋

𝑒𝑒−(𝑥𝑥−𝑅𝑅𝑑𝑑)2/2∆𝑅𝑅𝑝𝑝 (1)

Thus the distribution of ions through the depth on the implanted layer is described by a broad peak, the parameters of which are the maximum concentration Ni located below the surface at a depth Rp (the projected range) and the spread ∆Rp (implantation straggle) (Fig. 4.2).

Fig. 4.2. Depth distribution of boron ions implanted into silicon with equal dose 1015 cm-2, but at different energies. Depth of the doped layer and its width (2∆Rp) increase with the implantation energy. Projected range Rp and straggling ∆Rp are shown for 400 keV ions.

The doping depth Ri primarily depends on the mass of the implanted ions, their energy and the chemical composition of the substrate. It is roughly proportional to the ion energy and inversely proportional to the ion mass. The ion-doped layer is buried under the substrate surface. The average depth of the doped layer is Rp, and its effective width is 2∆Rp.

In order to dope selected areas, masking technique is used. Mask covers the areas which must remain undoped. The openings in the mask define the areas of ion doping (Fig. 4.1). The mask must be thick enough to stop the ions completely and prevent from doping in the masked areas.

Using ion implantation, layers doped with donors (e.g. phosphorous ions, P+) and acceptors (e.g. boron ions, B+) can be created. Using multiple implantations with appropriate energies through corresponding masks a multilayer doped structures can be made. Fig. 4.3 shows an example of three layer ion-doped structure of bipolar transistor.

Fig. 4.3. Structure of planar bipolar transistor made by two implantations of B+ ions and one implantation of P+ ions.

Rp

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4.2.2. Numerical simulation of ion doping with SRIM computer code Stopping and Range of Ions in Matter (SRIM) is a group of computer programs which calculate interaction of ions with matter. The essential program of these is Transport of Ions in Matter (TRIM). SRIM is very popular in the ion implantation research and technology community. The programs were developed by J. F. Ziegler and J. P. Biersack around 1983 and are being continuously upgraded. SRIM is based on a Monte Carlo simulation method, namely the binary collision approximation with a random selection of the impact parameter of the next colliding ion. The main output data of the SRIM simulation used in this lab work are three-dimensional distribution of the implanted ions and the implantation induced damage.

The output data of simulation can be viewed in plots (while the calculation is proceeding) and also in detailed numerical files. The plots are especially useful to see if the calculation is proceeding as expected, but are usually limited in resolution. Most of the data files can be requested in the Setup Window for TRIM (menus at the bottom of the window), or can be requested during the calculation. All calculated averages are made over the entire calculation.

Simulation of fabrication of planar bipolar p-n-p structure Simulation of ion doping of p-n-p structure starts with the calculation of the depth distribution of boron acceptors in the deepest p-type collector layer. Energy of the boron ions and the ion dose are chosen so that they ensure formation of the collector layer at the required depth with the required concentration of acceptors. An example is shown in Fig. 4.4. Collector layer is formed at a depth range from 260 to 500 nm by implantation of 100 keV B+ ions (Fig. 4.4a). The maximum acceptor concentration of 5.7×1017 cm-3 is achieved at a depth of 350 nm (collector depth RC).

The second step is the simulation of formation of the phosphorous-doped n-type base layer. The energy of P+ ions and their dose have to be adjusted so that the distribution of the implanted phosphorous overlaps with the boron distribution in the collector layer only partially. The maximum concentration in the phosphorous-doped layer must correspond to the required density of donors in the base layer. In the depth range of overlapping, boron acceptors and phosphorous donors compensate each other. At the depth RCB, where the boron and phosphorous concentrations are equal, complete compensation occurs. At this depth the collector-base p-n junction is formed. In Fig. 4.5a the base layer is formed at depths from 10 to 250 nm by implantation of 130 keV P+ ions. The maximum donor concentration in the base layer is about 6.8×1017 cm-3 at a depth of 180 nm (base depth RB). Collector-base junction is formed at a depth of 260 nm (RCB).

Once RCB is determined, the simulation of the boron ion doping of the emitter layer is performed. The ion energy and dose are to be adjusted so that the emitter boron-doped layer has the required acceptor concentration and forms the emitter-base p-n junction at the required depth REB. The emitter layer in Fig. 4.4b is formed by 25 keV B+ ion implantation. The maximum acceptor concentration in the emitter layer is about 1.2×1018 cm-3 at a depth of 110 nm (emitter depth RE) The emitter-base junction is formed at a depth of 160 nm (REB).

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0 100 200 300 400 500 6000.0

2.0x1017

4.0x1017

6.0x1017

Bor

on C

once

ntra

tion,

cm

-3

Depth (nm)

B 100 keV

0 100 200 300 400 500 6000.0

4.0x1017

8.0x1017

1.2x1018

Bor

on C

once

ntra

tion,

cm

-3

Depth (nm)

B 25 keV

Fig. 4.4. Depth distribution of ion-implanted boron. (a) Implantation of 100 keV boron ions. (b) Implantation of 25 keV boron ions.

0 100 200 300 400 500 6000.0

2.0x1017

4.0x1017

6.0x1017

8.0x1017

1.0x1018

Phos

phor

ous

conc

entra

tion,

cm

-3

Depth (nm)

P 130 keV

0 100 200 300 400 500 6000.0

2.0x1017

4.0x1017

6.0x1017

8.0x1017

1.0x1018

1.2x1018

Boron

Phosphorous

Dop

ant C

once

ntra

tion,

cm

-3

Depth (nm)

Boron

Fig. 4.5. (a) Depth distribution of ion-implanted phosphorous. (b) Distribution of implanted boron and phosphorous plotted on one graph. There is considerable overlapping of the distribution profiles.

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0 100 200 300 400 500 6000.0

2.0x1017

4.0x1017

6.0x1017

8.0x1017

1.0x1018

pnp

BC junction

EB junction

Con

cent

ratio

n, c

m-3

Depth, nm

Fig. 4.6. Distribution of non-compensated boron acceptors (blue) and non-compensated phosphorous donors (red). Position of p-n junctions are shown with arrows.

The ion dose which is required to achieve maximum concentration Nmax is calculated using formula:

𝑁𝑁(𝑒𝑒) = 𝑁𝑁𝑣𝑣∆𝑅𝑅√2𝜋𝜋

𝑒𝑒−(𝑥𝑥−𝑅𝑅𝑑𝑑)2/2∆𝑅𝑅𝑝𝑝 (2)

4.3. Procedure 1. Open SRIM simulation program. Open “Stopping/Range Table” option. Generate table of projected ranges and stragglings for boron ions. Determine ion energy EC corresponding to the chosen collector depth, e.g. RC = 400 nm. This depth corresponds to the projected range RpC of the boron ions in the collector layer.

2. Perform simulation of implantation of silicon with boron ions of energy EC. Obtain value of straggling ∆RpC for the collector layer. Save the simulation data.

3. Calculate difference RCB = RpC - ∆RpC. This is an approximate depth of the CB junction.

4. In the “Stopping/Range Table” option, generate table of projected ranges and stragglings for phosphorous ions. Determine energy EB of P+ ions, for which RpB + ∆RpB ≈ RCB. This is the energy of phosphorous ions implanted into base layer.

5. Perform simulation of implantation of silicon with phosphorous ions of energy EB. Obtain value of straggling ∆RpC for the base layer. Save the simulation data.

6. Calculate difference REB = RpB - ∆RpB. This is an approximate depth of the EB junction.

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7. In the table of projected ranges and stragglings for B+ ions find the energy of boron ions EE, for which RpE + ∆RpE ≈ REB. This is the energy for boron ions implanted into emitter layer.

8. Perform two separate simulations of implantation in silicon of boron ions with the energies EC and EE. Perform simulation of implantation in silicon of phosphorous ions with the energy EB. Save the simulation data.

9. Plot the obtained three simulation profiles on one graph in coordinates “Ion Concentration” versus “Depth”.

10. Adjust each simulation profile so that the maximum concentrations correspond to the chosen values: e.g. NCmax = 3×1017 cm-3, NBmax = 8×1016 cm-3, and NEmax = 2×1018 cm-3.

11. Sum up the boron concentration profiles and subtract the phosphorous concentration profile. The depths where the total concentration is zero (complete compensation) are the junction depths.

12. Using the values of NC, NB and NE calculate the ion doses DC, DB and DE, which are required to achieve these concentrations.

4.4. Calculations and Discussion 1. Discuss the obtained distributions of acceptors and donors over the depth of the transistor structure.

2. Compare the nominal depths of the CB and EB junctions found from Stopping/Range Tables with those obtained from the simulation profiles of implanted ions.

3. Calculate average concentrations of acceptors and donors in collector, base and emitter layers.

4. Using the data obtained, predict conductivity of the collector, base and emitter layers.

4.5 Questions 1. Predict maximum voltage which can be applied to the terminals of your simulated transistor.

2. How you would change the parameters of ion doping in order to:

a) reduce the width of the base layer?

b) decrease the resistance of collector and emitter?