Properties of Solids

Department of Physics

Umeå University 2015-04-02

Properties of Solids

Hassan Alhasnawi ([email protected])

Weixin Chen ([email protected])

Benan Aksoy ([email protected])

Solid State Physics

Supervisor

Hamid Reza Barzegar Goltapehei

[email protected]

mailto:[email protected]

Abstract

As a mandatory part of the course Solid State Physics, three separate laboratories practical were

conducted. The goal of the first experiment was to get a better understanding of the electrical

conductivity in a metal (𝑃𝑡), a semiconductor (𝐼𝑛𝑆𝑏) and a superconductor (𝑌𝐵𝑎2𝐶𝑢3𝑂7). The goal

of the second experiment was to get a better understanding of the Hall effect in a semiconductor

(𝐼𝑛𝑆𝑏) and lastly, the third experiment with the goal of determining the band gaps in the

semiconductors 𝑆𝑖 and 𝐺𝑒.

The electrical conductivity was studied by decreasing the temperature in the three different materials

with a vacuum cooling system to a temperature of 10 𝐾. The temperature was then increased and a

current was fed through the samples in order to measure the resistance of the samples with a “4-pole

resistance measurement”. These measurements where registered with a monitoring program “Elledn”

and the measurement series were conducted in a temperature interval ranging from 10 𝐾 to 300 𝐾.

The band gaps were determined by exposing the samples to light of different wavelengths and

determining at what wavelength the material absorbs the light. The transmittance of the material is

determined by placing the sample between the light source and a photoresistor, which absorbed the

transmitted light and returned a signal proportional to the amount of absorbed photons. The

transmittance was measured with and without the sample for every wavelength in an interval ranging

from 800 𝑛𝑚 to 1500 𝑛𝑚 for silicon (𝑆𝑖), with an increase of 5 𝑛𝑚 per measurement and 1500 𝑛𝑚

to 1900 𝑛𝑚 for germanium (𝐺𝑒), with an increase of 10 𝑛𝑚 per measurement. The absorption could

be obtained using the measured values. By plotting the obtained absorption of the different samples as

a function of the photon energy, two linear fits could be applied in each case, where the intercept was

the band gap energy.

The Hall effect was studied by decreasing the temperature of the sample to a temperature of 83 𝐾.

Two measurement series was performed, ranging from 83 𝐾 to 300 𝐾 with an increase of 10 𝐾 per

measurement. The measured quantities in the first series where the thermocouple voltage, the voltage

over the 10 Ω resistor and the Ohmic voltage when the current is moving in either direction. The

measured quantities in this series where the thermocouple voltage, the voltage over the 10 Ω resistor

and the Hall voltage.

The obtained band gap energy for the 𝑆𝑖-sample was 1,13 𝑒𝑉 and for the 𝐺𝑒-sample 0,68 𝑒𝑉 in the

band gap experiment.

In the Hall effect experiment, the thermal band gap energy was determined to be 0,277 𝑒𝑉, the carrier

density increased with increasing temperature and the carrier mobility decreased with decreasing

temperature. The proportionality of the mobility to the temperature was calculated to 𝜇 ∝ 𝑇−1,25.

The conclusion made in this report was that the free electron model and the complementarities from

the nearly free electron model successfully predicts the outcome of the results.

Table of Contents

1. Introduction ..................................................................................................................................... 1

2. Theory ............................................................................................................................................. 1

2.1. Electrical Conductivity ............................................................................................................ 1

2.2. Band Gap in Semiconductor .................................................................................................... 2

2.3. Metals, Insulators and Semiconductors ................................................................................... 3

2.4. The Hall Effect ........................................................................................................................ 3

3. Experimental ................................................................................................................................... 5

3.1. Electrical Conductivity ............................................................................................................ 5

3.2. Band Gap in Semiconductor .................................................................................................... 6

3.3. The Hall Effect ........................................................................................................................ 7

4. Results ............................................................................................................................................. 8

4.1. Electrical conductivity ............................................................................................................. 8

4.2. Band Gap in Semiconductor .................................................................................................. 10

4.3. The Hall Effect ...................................................................................................................... 12

5. Discussion ..................................................................................................................................... 15

5.1. Electrical Conductivity .......................................................................................................... 15

5.2. Band Gap in Semiconductor .................................................................................................. 16

5.3. The Hall Effect ...................................................................................................................... 16

6. Conclusion ..................................................................................................................................... 17

Bibliography .......................................................................................................................................... 18


Weixin Chen ([email protected]) Properties of Solids

Benan Aksoy ([email protected]) 2015-04-02

1

1. Introduction

Why does the electrical conductivity of solids vary with temperature? What is the band gap energy and

why does it vary for different semiconductors? Why does an electric field arise when a solid is

subjected to a magnetic field and a transverse current density? In this report, the free electron model

will be used and complemented with the nearly free electron model to give an explanation to these

different properties, along with experimental verifications of these theories.

Three different laboratories where conducted in order to verify the theories. The goal of the first

laboratory was to study the temperature dependence the electrical conductivity in a metal, a

semiconductor and a superconductor. In the second laboratory, the goal was to determine the band gap

energies of the semiconductors silicon and germanium, by measuring their transmission of light as a

function of the wavelength. In the final laboratory, the task was to determine the band gap as well as

the temperature dependence of the charge carriers mobility and concentration in indium antimonide.

2. Theory

Two models will be introduced in order to explain the properties of different crystalline solids, such as

the electrical conductivity, Hall effect and the band gap.

2.1. Electrical Conductivity

In the case where only a dc electric field is present, the equation of motion for an electron has the

steady state solution

𝒗 = −𝑒𝜏

𝑚𝑒𝑬, (1)

where 𝒗 is the drift velocity, 𝑒 is the elementary charge, 𝜏 is scattering time, 𝑚𝑒 is the mass of an

electron and 𝑬 is an electric field. The proportionality constant in Eq. 1, the mobility 𝜇𝑒, is defined as

𝜇𝑒 =𝑒𝜏

𝑚𝑒. (2)

It is also necessary to define the electric current density 𝒋, using Ohm’s law, as

𝒋 =𝑛𝑒2𝜏

𝑚𝑒𝑬, (3)

where 𝑛 =𝑁𝐴

𝑉 is the electron density and 𝑁𝐴 is Avogadro’s constant. Using Eq. 2, the proportionality

constant from Eq. 3, which is defined as the electrical conductivity becomes

𝜎 = 𝑛𝑒𝜇𝑒. (4)

The scattering time 𝜏 in a metal is assumed to have two independent contributions. The first

contribution being from a temperature-dependent electron-phonon scattering rate and the second being

a constant scattering rate, independent of temperature, that occurs due to electron collisions with

impurity atoms, vacancies and other structural defects. Thus, the total scattering time can be described

as




2

1

𝜏=

1

𝜏𝑝ℎ(𝑇)+

1

𝜏0, (5)

where 𝜏𝑝ℎ(𝑇) is the temperature-dependent contribution from electron-phonon scattering that

approaches infinity as the temperature approaches zero. 𝜏0 is the constant, temperature-independent

scattering time that is present at all temperatures.

From Eq. 3, Eq. 4, Eq. 5 and the knowledge that the resistivity, 𝜌 is inversely proportional to the

electrical conductivity, the expected resistivity for a metal is

𝜌 =𝑚𝑒

𝑛𝑒2𝜏𝑝ℎ(𝑇)+

𝑚𝑒

𝑛𝑒2𝜏0, (6)

where the expected proportionalities for different temperatures is

𝜌 ∝

𝜌0, 𝑇 ≪ Θ𝐷

𝑇, 𝑇 ≫ Θ𝐷,

(7)

where Θ𝐷 is the Debye temperature.[1]

2.2. Band Gap in Semiconductor

The band gap is an energy interval, where no electron states are available. It can also be described as

the required energy for an electron to be raised from the valance band to the conduction band. The

conduction band is the energy range where the electrons can move freely within the lattice of a

material. If the zero potential is chosen to be at the top of the valence band, the energy of an electron

for different values of 𝑘 in the valence band can be described as

𝜀 = −ℏ2𝑘2

2𝑚ℎ, (8)

where ℏ =ℎ

2𝜋, where ℎ is Planck’s constant, 𝑘 is the wavenumber and 𝑚ℎ denotes the mass of a hole.

Similarly, the energy of an electron in the conduction band can be described as

𝜀 = 𝐸𝐺 +ℏ2𝑘2

2𝑚𝑒, (9)

where 𝐸𝐺 is the band gap energy. In the case where the maximum of Eq. 8 and the minimum of Eq. 9

occur at the same value of 𝑘, the band gap is referred to as a direct band gap. For a direct gap, it is

necessary for the electron to obtain energy from a photon

𝜀𝑝ℎ = ℎ𝑓, (10)

where 𝑓 is the frequency, in order to move the conduction band. On the other hand, when the

maximum and minimum occur at different values of 𝑘, the band gap is referred to as an indirect band

gap. For an electron to be raised to the conduction band in an indirect band gap, it is necessary for it to

obtain the required energy from a photon as well as the required momentum from a phonon.

Furthermore the phonon has an energy and thus, although comparably low to that of the photon. Thus,

in order for the energy balance to be satisfied, the photon energy absorbed from the electron must not

equal the band gap energy, due to the fact that the phonon will transfer an energy to the electron as

well. This can be observed in the dependency of the absorption to the photon energy, where photons

are absorbed when their energy is close to that of the band gap, but not when they are equal.[2]




3

2.3. Metals, Insulators and Semiconductors

Crystalline solids depict different capabilities of electric conduction. The solids can be classified into

metals, insulators and semiconductors, using the nearly free electron model and the introduction of

holes. Expected is that for an odd number of valence electrons in a primitive unit cell, the solid will

attain metallic properties. For an even number of valence electrons in a primitive unit cell, the solid

will possess insulating properties if the band gap is large and semiconducting-behavior if the band gap

is small.[3]

When an energy band is completely filled with electrons, no current will be transported. This is due to

the fact that when the electrons experience a shift in a certain point in k-space, the remainder of the

electrons will experience the same shift. This will cause no change in the electron distribution and thus

no current will be produced.[4]

A change in the behavior of the resistivity has been observed when in many materials, when they are

subjected to low temperatures. At a temperature, specific for each material, the finite resistivity drops

to a non-measurably small resistivity. The temperature where the drop in resistivity occurs is referred

to as the transition temperature and the phenomenon is referred to as superconductivity.[5]

2.4. The Hall Effect

The Hall effect is the phenomena that describes the transverse electric field arising in some solid

materials when they are subjected to a magnetic field and a current density is passed through it

simultaneously. The electric field, 𝑬𝑯 (also known as the Hall field) can be described by the relation

𝑬𝑯 = 𝑅𝐻𝑩 × 𝒋, (11)

where 𝑅𝐻 is the hall coefficient and 𝑩 is the magnetic field. Figure 1Error! Reference source not

found. illustrates the orientation of the current density, the magnetic field, the electric field, as well as

the Lorentz force −𝑒𝒗 × 𝑩 and the force −𝑒𝑬𝑯 caused due to the magnetic field and the Hall field

respectively.

Figure 1. An electric field 𝑬𝑯 arising in a metal, due to it being subjected to a magnetic field 𝑩 and a

current density 𝒋. The Lorentz force −𝑒𝒗 × 𝑩 occurring due to the magnetic field is balanced by the

force occurring due to the Hall field.[6]




4

For semiconducting material, there are two types of carriers, electrons and holes. In steady state,

with 𝑣𝑦 = 0, the equation of motion for electrons and holes, and Eq. 10 yield a Hall coefficient

𝑅𝐻 = −1

𝑛𝑒, (12)

when 𝑛 ≫ 𝑝, where 𝑝 is the hole concentration. Similarly, for 𝑝 ≫ 𝑛, the Hall coefficient becomes

𝑅𝐻 =1

𝑝𝑒. (13)

By combining Eq. 4 and Eq. 11 or Eq. 12, the carrier mobility is expected to be

𝜇 = |𝑅𝐻|𝜎.[7] (14)

The temperature dependence of the carrier mobility can be determined by knowing that the mean free

path, 𝑙, is

𝑙 ∝1

𝑇, (15)

the drift velocity is

𝑣 ∝ 𝑇

1

2 (16)

and that the carrier mobility can be rewritten as

𝜇 =𝑙

𝑚𝑒𝑣. (17)

Thus the expected proportionality of the carrier mobility to the temperature is

𝜇 ∝ 𝑇−3

2. (18)

As for the carrier density, it is expected to increase slightly at low temperatures, due to the energy

being high enough to excite the electrons from the valence band to the acceptor level and from the

donor level to the conduction band. At intermediate temperatures, the carrier density will be constant,

due to all electron states in the acceptor level being filled, and all the electrons in the donor level being

excited. At higher temperatures the electrons will receive high enough energies in order to be excited

from the valence band to the conduction band, therefore the carrier density will increase again as the

temperature increase.[8]

In the case of an ideal semiconductor, every electron that is excited leaves a hole behind in the valence

band. This is referred to as an intrinsic semiconductor. The carrier density as a function of the

temperature can for an intrinsic semiconductor be determined by

𝑛𝑖(𝑇) =

1

4(

2𝑘𝐵𝑇

𝜋ℏ2 )

3

2∙ (𝑚𝑐𝑚𝑣)

3

4 ∙ 𝑒−

𝐸𝐺2𝑘𝐵𝑇,

(19)

where 𝑛𝑖 denotes the carrier density for the intrinsic case, 𝑚𝑐 is the effective mass of an electron and

𝑚𝑣 is the effective mass of a hole. The expected proportionality of the carrier density to the

temperature can be determined by using Eq. 19 to

𝑛𝑖 ∝ 𝑇

3

2 ∙ 𝑒−

𝐸𝐺2𝑘𝐵𝑇.[8][9]

(20)




5

The proportionality of the electrical conductivity to the temperature is obtained by applying the

proportionalities in Eq. 18 and Eq. 20 in Eq. 4 to obtain

𝜎 ∝ 𝑒

−𝐸𝐺

2𝑘𝐵𝑇. (21)

The temperature dependence of the band gap energy is quadratic at low temperatures and linear at high

temperatures. It is sufficient to assume that

𝐸𝐺(𝑇) = 𝐸𝐺(1 − 𝛼𝑇), (22)

where 𝛼 is an arbitrary constant. By applying Eq. 22 in Eq. 21 and applying the natural logarithm to it,

the obtained relation is

ln(𝜎) = 𝐶 −𝐸𝐺

2𝑘𝐵𝑇, (23)

where 𝐶 is a constant. This is however valid for higher temperatures due to the above stated argument

that the band gap energy is linear at high temperatures.[9]

3. Experimental

Three laboratories practical where conducted in order to verify the theories. In this section, the method

of these will be presented.


The temperature dependence of the electrical conductivity for the metal platinum (𝑃𝑡), the

semiconductor indium antimonide (𝐼𝑛𝑆𝑏) and the superconductor yttrium barium copper oxide

(𝑌𝐵𝑎2𝐶𝑢3𝑂7) where studied, by lowering the temperature to 10 𝐾 and measuring the resistance

through the different sample, for a given voltage and current.

Before the samples was cooled down, a coarse vacuum pump was used in order to lower the pressure

to approximately 10−1𝑏𝑎𝑟 before a turbo-molecular pump was started to further decrease the pressure

to 10−3𝑏𝑎𝑟 in the container. This is executed in order to decrease the amount of particles within the

container which can decrease the cooling rate.

The cryogenic equipment, used to decrease the temperature of the samples, was turned on when the

desired pressure was reached. The cryogenic equipment consisted of a compressor, a two-stage

cooling head and flexible pressure tubes.

The samples where connected to a copper plate in order to obtain a uniform temperature, further

lowering the temperature gradient through the samples by applying a conducting paste between the

sample and the copper plate. The copper plate is connected to the cooling head, with a indium foil

placed between in order to increase the surface contact between the materials, thus increasing the heat

conduction.

The temperature was measured with two silicon diodes of model DT-500 P/GR and monitored with

the program “Elledn”. In order to further decrease the error from the temperature gradient, the silicon

diodes was situated on the copper plate, and on the cooling head. Thus an average of the two

measurements could be obtained. However, the silicon diode on the cooling head was the only one

used during the experiment due to the fact that the absolute values of the measurements being of lesser

interest, but rather the general behavior of the samples being the important part of the study.




6

In order to measure the resistance, a 4-pole resistance measurement system consisting of a multimeter

was used, with a current measurement in series and a voltage measurement in parallel for each sample.

The program “Elledn” registered the resistance for the corresponding temperature automatically.

When a temperature of 10 𝐾 was reached, a series of resistance measurements where conducted in an

interval ranging from 10 𝐾 to 300 𝐾, with an increase of 5 𝐾 per measurement in the range 10 𝐾 to

80 𝐾, 2 𝐾 per measurement in the range 80 𝐾 to 100 𝐾 and 10 𝐾 per measurement in the range

100 𝐾 to 300 𝐾.


It is not easy to measure the band gap energy directly. However, the wavelength of the incident

photons that excite the electron to the conduction band is measurable. The semiconductors can only

absorb light with specific wavelength in order to excite the electrons from the valence band to the

conduction band. If the wavelength is known, the band gap can be determined by using Eq. 10.

A light source consisting of a light bulb was used in order to emit light upon samples of

semiconducting materials, in this case germanium and silicon. The light source emitted light in all

wavelengths and was directed towards the opening of a monochromator, in order to transmit light of a

specific wavelength. The wavelength of the transmitted light could be varied by changing the angle of

the grating within the monochromator.

The sample of semiconducting material was exposed to the transmitted light and in turn a

photoconductive resistor made of lead sulfide absorbed the light transmitted through the sample. In

order to avoid overheating of the photoconductive resistor, a chopper was situated between the light

source and the sample, thus decreasing the exposure time of the detector to light. An oscilloscope was

connected to the photoconductive resistor, in order to measure the absorbed light, thus obtaining a

measurement of the transmittance in the sample. The oscilloscope registered a voltage, proportional to

the absorbed light on the photoconductive resistor.

A measurement without the sample was conducted for the same wavelength but without the sample, in

order to measure the total emitted light. The absorption, 𝑎 of the sample could be obtained by the

relation

𝑎 = 1 − (𝑉𝑤𝑖𝑡ℎ 𝑠𝑎𝑚𝑝𝑙𝑒

𝑉𝑤𝑖𝑡ℎ𝑜𝑢𝑡 𝑠𝑎𝑚𝑝𝑙𝑒)

where 𝑉 is the voltage and the index denotes the measured voltage with and without the sample.

This procedure was repeated in an interval ranging from 800 𝑛𝑚 to 1500 𝑛𝑚 for silicon (𝑆𝑖), with an

increase of 5 𝑛𝑚 per measurement and 1500 𝑛𝑚 to 1900 𝑛𝑚 for germanium (𝐺𝑒), with an increase of

10 𝑛𝑚 per measurement. A schematic illustration of the laboratory equipment is presented in Figure 2,

oscilloscope excluded.

Figure 2. A schematic illustration of the laboratory equipment used for determining the band gap in

semiconducting material, oscilloscope excluded.[10]




7


The Hall effect in a semiconductor was studied by executing two series of measurements. The first

involved varying the temperature of a sample of 𝐼𝑛𝑆𝑏 exposed to a current and the second involved

the same as the aforementioned, including the appliance of a constant magnetic field transverse to the

current.

The orientation of the connection wires in the sample, as well as the dimensions of the sample is

illustrated in Figure 3. The dimensions where noted from the manual corresponding to the laboration

equipment. In order to apply a current through the sample, a constant current generator, able to change

the direction of the current was connected to it using wires soldered on either side of the sample. The

current was calculated by measuring the voltage over a 10 Ω resistor, connected in series with the

circuit. The wires through which the current is passing are supposed to be connected at the same

height on the sample. This is not the case, due to imprecise installation. Thus, a temperature gradient

between the connection points is present. The Ohmic voltage is measured in both directions in order to

eliminate the measurement error that occurs due to the temperature gradient in the sample. The Ohmic

voltage is thus assumed to be the average of the two measurements. The voltage over the sample was

measured with a digital voltmeter, connected with wires that are soldered on one side of the sample.

The temperature was measured with a Chrome-Alumel thermocouple, with one of the junctions

situated closely to the sample and the other in an ice bath as a reference point. All the measurements

where registered with one single digital voltmeter and a switch that allowed for the use of one single

voltmeter to measure the voltage from all sources consecutively.

Figure 3. An illustration depicting the orientation of a sample used in a Hall effect experiment along

with the connection wires, as well as the dimensions. The connection wires C and D are used to

connect a current source to the sample, and the connection of measuring equipment in order to

measure the Hall voltage and the current through the sample. Connections A and B are used to

measure the Ohmic voltage over the sample.

The temperature of the sample was initially decreased to 83 𝐾 by inserting it in a holder filled with

liquid nitrogen. A measurement series was performed, ranging from 83 𝐾 to 300 𝐾 with an increase

of 10 𝐾 per measurement. The temperature was increased by slightly raising the sample from the

holder. The measured quantities where the thermocouple voltage, the voltage over the 10 Ω resistor

and the Ohmic voltage when the current is moving in either direction.




8

A magnetic field was applied to the sample in the second measurement series by using an

electromagnet supported by a stabilized DC-voltage aggregate. The current passing through the

magnet coils was set 0, 75 𝐴 and kept constant throughout the experiment. The temperature was

decreased to 83 𝐾 using a holder filled with liquid nitrogen. A different holder was used in this part in

order to fit the sample within the magnet. The measurements where performed within the same

interval and with a similar increase as the first measurement series. The measured quantities in this

series where the thermocouple voltage, the voltage over the 10 Ω resistor and the Hall voltage. The

Hall voltage was measured over the current direction twice in order to eliminate the error due to the

temperature gradient. This was executed by rotating the sample within the magnet until the maximum

Hall voltage was achieved and subsequently rotating it again roughly 180° in order to measure the

Hall voltage at the opposite side. The magnetic field was measured with a magnetic probe.

4. Results

The obtained results from the laboratories will be presented in this section.

4.1. Electrical conductivity

The registered values of the resistance were plotted as a function of the temperature for the 𝑃𝑡-sample.

The graph is presented in Figure 4.

Figure 4. The measured values of the resistance in a 𝑃𝑡-sample illustrated graphically as a function of

the temperature.




9

Similarly, the resistance for the 𝐼𝑛𝑆𝑏-sample is plotted against the temperature in Figure 5.

Figure 5. The measured values of the resistance in an indium 𝐼𝑛𝑆𝑏-sample illustrated graphically as a

function of the temperature.

The 𝑌𝐵𝑎2𝐶𝑢3𝑂7 -sample was studied in a similar way as the 𝑃𝑡- and 𝐼𝑛𝑆𝑏-sample, as can be seen in

Figure 6.

Figure 6. The measured values of the resistance in an 𝑌𝐵𝑎2𝐶𝑢3𝑂7 -sample illustrated graphically as

a function of the temperature.

The superconducting transition temperature is 𝑇𝑐 ≈ 90 ± 10 𝐾, which can be compared to a measured

value equaling 𝑇𝑐 = 90 𝐾 and 𝑇𝑐 = 92 𝐾.[11][12]




10


The wavelength of the photons incident on the 𝐺𝑒-sample was calculated using Eq. 10. The calculated

absorption from the measured voltages was plotted as a function of these energies, as can be seen in

Figure 7.

0,7 0,8

0,4

0,6

0,8

1,0

Ge

Linear Fit of Absorption (1)


Ab

so

rption

Energy (eV)

Equation y = a + b*x

Weight No Weighting

Residual Sum of Squares

7,77302E-5 1,52724E-4

Pearson's r 0,99654 0,998

Adj. R-Square 0,99222 0,99551

Value Standard Error

Absorption (1) Intercept -1,61624 0,06365

Absorption (1) Slope 3,22764 0,0952



Figure 7. The light-absorption in a sample of 𝐺𝑒 as a function of the energy of the incident photons.

Two linear regressions can be applied in the regions 0,65 < 𝜀𝑝ℎ < 0,68 𝑒𝑉 and 0,69 < 𝜀𝑝ℎ <

0,73 𝑒𝑉 in order to determine the band gap energy.

As stated in “2.2. Band Gaps”, the transferred energy from the photons to the electrons will be either

higher or lower than the band gap energy. Applying two linear regressions in intervals were linearity is

observed will yield two linear functions, where the intercept is the band gap energy. The obtained

band gap energy for germanium is 𝐸𝐺,𝐺𝑒 = 0,68 𝑒𝑉, which can be compared a tabulated value

𝐸𝐺,𝐺𝑒−𝑡𝑎𝑏𝑙𝑒 = 0,664 𝑒𝑉. [13]




11

A similar approach as the one used for germanium was applied for the 𝑆𝑖-sample. The absorption as a

function of the photon energy is presented in Figure 8.

1,0 1,2

0,4

0,6

0,8

1,0

Absorp

tion

Energy (eV)

Si




Weight No Weighting


0,0022 5,20652E-4

Pearson's r 0,98344 0,94459

Adj. R-Square 0,96059 0,8707






Figure 8. The light-absorption in a sample of 𝐺𝑒 as a function of the energy of the incident photons.

Two linear regressions can be applied in the regions 1,06 < 𝜀𝑝ℎ < 1,13 𝑒𝑉 and 1,14 < 𝜀𝑝ℎ <

1,20 𝑒𝑉 in order to determine the band gap energy.

The calculated band gap energy was 𝐸𝐺,𝑆𝑖 = 1,13 𝑒𝑉 which can be compared to the tabulated value

𝐸𝐺,𝑆𝑖−𝑡𝑎𝑏𝑙𝑒 = 1,124 𝑒𝑉.[13]




12


With the measured data for the Ohmic voltage and the voltage over the 10 Ω resistor, the electrical

conductivity could be calculated by applying Ohm’s law, as well as the fact that the electrical

conductivity is inversely proportional to the resistivity. In order to obtain the band gap energy, the

natural logarithm of the electrical conductivity is plotted as a function of the reciprocal of the

temperature, as is presented in Figure 9. In accordance with Eq. 23, the slope of the obtained curve for

high temperatures should equal −𝐸𝑔

2𝑘𝐵.

0,002 0,004 0,006 0,008 0,010 0,012

5

10

15

20

Conductivity

Theoretical slope

Linear Fit of Sheet1 ln(sigma)

Linear Fit of Sheet1 E_g/(2*K_B*T)

ln(s

igm

a)

(ohm

^-1m

^-1)

1/T (K^-1)


Weight No Weighting

Residual Sum of Squares0,01077 2,50463E-29

Pearson's r -0,99939 -1

Adj. R-Square 0,99864 1


ln(sigma) Intercept 19,88042 0,07924

ln(sigma) Slope -1609,57721 18,76006

E_g/(2*K_B*T) Intercept 20 6,45948E-16

E_g/(2*K_B*T) Slope -1015,43461 9,85704E-14

Figure 9. The natural logarithm of the electrical conductivity as a function of the reciprocal of the

temperature, with a linear regression (blue line) in the temperature range 203 < 𝑇 < 300 𝐾. The

slope of the linear regression is used to calculate the band gap energy. The slope for a tabulated value

of the band gap energy at 300 𝐾 is plotted as a comparison (red line).[13]

The obtained of the band gap energy at 203 < 𝑇 < 300 𝐾 was 𝐸𝐺,𝐼𝑛𝑆𝑏 = 0,277 𝑒𝑉. This can be

compared to a tabulated value 𝐸𝐺,𝐼𝑛𝑆𝑏−𝑡𝑎𝑏𝑙𝑒 = 0,175 𝑒𝑉.[13]




13

By applying Eq. 11 to the acquired data, 𝑅𝐻 could be calculated. Once the Hall coefficient was

obtained, the charge carrier density could be calculated, using Eq. 12. The charge carrier density is

plotted against the reciprocal of the temperature in Figure 10.

0,005 0,010

0,00E+000

5,00E+020

1,00E+021

1,50E+021

2,00E+021

2,50E+021 Carrier density

n (

m^3

)

1/T (K^-1)

Figure 10. The charge carrier density as a function of the reciprocal of the temperature.

The natural logarithm of the charge carrier density was also plotted against the reciprocal of the

temperature, this is presented in Figure 11.

0,005 0,010

44

46

48

50

Carrier density

ln(n

) (m

^3)

1/T (K^-1)

Figure 11. The natural logarithm of the carrier density plotted as a function of the reciprocal of the

temperature.




14

By using Eq. 14, the charge carrier mobility could be calculated. This is plotted as a function of the

reciprocal of the temperature in Figure 12.

0,005 0,010

2000

4000

6000

Carrier mobility

Mu (

m^2V

^-1

s^-1

)

1/T (K^-1)

Figure 12. The charge carrier mobility as a function of the reciprocal of the temperature.

The natural logarithm of the charge carrier mobility is plotted as a function of the natural logarithm of

the temperature in Figure 13. This is used to verify the proportion between these two variables that,

according to Eq. 18 should be equal to −3

2 when the natural logarithm is applied on the data.

4,5 5,0 5,5 6,0

8

10

Carrier mobility

Linear Fit of Sheet1 ln(Mu)

ln(M

u)

(m^2

V^-

1s^-

1)

ln(T) (K)


Weight No Weighting


0,00538

Pearson's r -0,98755

Adj. R-Square 0,97217


ln(Mu) Intercept 14,85024 0,38636

ln(Mu) Slope -1,24573 0,07015

Figure 13. The natural logarithm of the charge carrier mobility as a function of the natural logarithm

of the temperature, with a linear regression applied for the temperature range 203 < 𝑇 < 300 𝐾 used

to verify the theoretical proportionality of the mobility.

The experimentally obtained slope was 𝑘 = −1,25 which implies that the proportionality of the charge

carrier mobility to the temperature is 𝜇 ∝ 𝑇−1,25.




15

5. Discussion

The results of the three laboratories will be discussed in this section.


From Figure 4, one can see that the resistance is independent of the temperature for temperatures much

lower than the Debye temperature and proportional to the temperature for temperatures much higher

than the Debye temperature in accordance with Eq. 7. This is in favor of the conclusion that the

sample in Figure 4 sample is a metal.

The resistance approaches infinity as the temperature approaches 0 𝐾 in Figure 5. This verifies the

theory in “2.3. Metals, Insulators and Semiconductors” that when the valence band is completely

filled, no current will be present. At the temperature interval 0 < 𝑇 < 50 𝐾, the resistivity decrease

indicates an extrinsic region, which means that the semiconductor is doped. This is due to the fact that

the electrons does not possess sufficient thermal energy to be excited from the valence band to the

conduction band at such low temperatures. However, the electrons may have enough energy to be

excited from the valence band to the acceptor level or from the donor level to the conduction band.

In the temperature range 50 < 𝑇 < 150 𝐾 , the acceptor level is filled with electrons and all electrons

in the donor level are excited, thus implying that the number of charge carriers is constant. As the

temperature increases without any increase in the number of charge carriers, the resistance will

increase, due to the carrier mobility’s scattering with thermally excited lattice vibrations. This is

consistent with the obtained result in Figure 11, where the charge carrier density is constant in

approximately the same region.

Finally, an increase is observed in the region 150 < 𝑇 < 300 𝐾, suggesting intrinsic behavior. This

can be explained by the electrons possessing enough thermal energy in order to be excited from the

valence band to the conduction band. This is coherent with the behavior noted in Figure 11, where the

charge carrier density increase in approximately the same region.

A clear transition is observed in Figure 6 from a finite resistance to what seems to be zero resistance.

This is the behavior of a superconductor as it transcends to its superconducting state. The observed

transition temperature is coherent with those observed in other experiments.

As stated in “3.1. Electrical Conductivity”, only one of the silicon diodes was used to plot the obtained

values. This calls for the caution of taking the values obtained to be absolute values, due to the

possible presence of a temperature gradient within the copper plate. However, the coherence of the

transition temperature observed for the superconductor can be taken as an indication that the accuracy

of the measurements is within an acceptable range for this experiment.

A “4-pole resistance measurement” was used in order to acquire accurate values of the resistance.

Although a more sensitive method that can be adapted in order to measure low resistances is to

measure the decay current around a closed superconducting loop. [12]

It is quite obvious from the first law of thermodynamics that the change in energy corresponding to a

temperature change in a mass will increase if the mass is increased. Thus, the application of vacuum

becomes practical as the presence of mass in the container will decrease. This will cause the cooling

process to become faster for a constant energy.




16


The deviation of the calculated band gap energy for the germanium and silicon to the tabulated value

was within 3% and 0,6% respectively. Taking into account the fact that the surfaces of the samples

were quite reflective and the fluctuation of the signal from the oscillator at lower wavelengths, the

acquired value is within an acceptable range. Overheating was prevented by the appliance of a

chopper. Another solution could the appliance of a fan that keeps the temperature of the resistor

constant at its optimized drift temperature.

A DC-supply was used in the experiment in order to minimize fluctuations in the photon energy, as

would be the case with an AC-supply that would make the photon energy increase and decrease

periodically. This would obstruct the acquisition of accurate values with an oscillator.


The deviation of the calculated band gap energy for the 𝐼𝑛𝑆𝑏-sample to the tabulated value was within

37%. The band gap energy was determined within a temperature range of 203 < 𝑇 < 300 𝐾 in this

experiment, whereas the tabulated value was determined at 300 𝐾. As stated in ”2.4. The Hall Effect”,

the band gap energy has a linear dependency to the temperature at higher temperatures, thus implying

that the obtained value in this experiment is the mean value over the temperature range.

The charge carrier density’s tendency to follow the proportionality stated in Eq. 20 can be deduced

from Figure 10. Perusing Figure 11 and Figure 9, one can see that the increase in the charge carrier

density and the conductivity occurs simultaneously. This suggests that the thermal energy at

approximately 160 𝐾 (corresponding to 𝑇−1 ≈ 0,006 𝐾−1 in the graphs) becomes high enough to

excite the electrons from the valence band to the conduction band. In the temperature range 90 < 𝑇 <

190 𝐾 (corresponding to 0,006 < 𝑇−1 < 0,011 Figure 11) the charge carrier density is constant

which suggests that the thermal energy is too low to excite the electrons to the conduction band.

According to the theory, the charge carrier density should increase slightly at the lowest temperatures,

due to the electrons being excited from the valence band to the acceptor band. This cannot be verified

in this experiment, as the temperature needs to be decreased further. A slight decrease in the

conductivity can be seen in the temperature range 90 < 𝑇 < 190 𝐾 in Figure 9. This is expected when

one considers the fact that the charge carrier density being constant within the same temperature range,

and the fact that the resistivity increases with increasing temperature.

The deviation between the experimentally obtained proportionality of the carrier mobility to the

temperature and the theoretically expected value was within 17%. The general behavior that is

observed in Figure 12 and Figure 13 is in line with the theoretically expected tendency of the charge

carrier mobility at intermediate temperatures, which is a decrease due to the scattering from thermally

excited lattice vibrations.

It is worth mentioning that the measuring equipment has its imperfections. An example is that the

increase of the temperature was obtained by raising the sample from the holder, thus forcing a

temperature gradient within it. Although this temperature gradient was taken into account by

measuring the ohmic voltage and Hall voltage in both directions, the value of the measured

temperatures depend on the placement of the thermocouple. The application of the vacuum cooling

system similar to the one used to study the electrical conductivity would be preferable, as it would

minimize the temperature gradient, as well as the human error in the collecting of the data.




17

6. Conclusion

Despite its simple approach, the free electron model and the complementarities from the nearly free

electron model successfully predicts the outcomes in all the conducted experiments.




18

Bibliography

[1] Hook, J. R., Hall, H. E. (2010), “Solid State Physics”, 2nd

ed. England: John Wiley & Sons. pp.

79-85



117-132



92-96



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ed. England: John Wiley & Sons. pp. 86



134-135



121-132

[9] Iwasiewicz, A., Hassmyr, L., Edman, L. (2004), “Solid State Physics – Hall Effect”, Umeå

University, Department of physics.

[10] Schmidt, F., Hassmyr, L., Edman, L. (2004), “Solid State Physics – Band Gap in

Semiconductors”, Umeå University, Department of physics.

[11] Dam, B., Huijbregtse, J. M., Klaassen, F. C., van der Geest, R. C. F., Doornbos, G., Rector, J. H.,

Testa A. M., Freisem, S., Martinez, J. C., Stäuble-Pümpin, B., Griessen, R. (1999), ”Origin of high

critical currents in 𝑌𝐵𝑎2𝐶𝑢3𝑂7−𝛿 superconducting thin films”, Vrije Unversiteit, Division of Physics

and Astronomy.



235-236

[13] Nordling, C., Österman, J. (2008), “Physics Handbook – for Science and Engineering”, 8th ed.

Studentlitteratur AB pp. 140

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