Solid state physics part 2 Flashcards
1
Q
Why is it easier to work with conductivities rather than resistivities?
A
We can add them in parallel
2
Q
What is the more general 3D version of Ohm’s law?
A
The current density is equal to the conductivity multiplied by the applied electric field
3
Q
What is the equation for resistivity (rho)?
A
The resistance multiplied by the area divided by the length
4
Q
What is the current density?
A
The current per unit area (current divided by area)
5
Q
How are conductivity and resistivity related with no magnetic field?
A
They are the inverse of each other
6
Q
If the system is cubic and in zero magnetic field, what does the conductivity matrix become?
A
A diagonal matrix with the same value along the diagonal
7
Q
What are the 5 Drude assumptions for the free gas of electrons?
A
Thermal equilibrium is reached through collisions. Electrons scatter only through collisions with ion cores. Between collisions there is no interactions between each other or with ion cores. Collisions are instantaneous and result in electron’s velocity changing. Probability of electron colliding per unit time is 1 over tau
8
Q
What is tau in terms of time?
A
Mean free time, so the time between collisions (also known as an inverse scattering rate)
9
Q
What direction do electron move?
A
Opposite to that of a conventional current
10
Q
What is the Hall effect?
A
The production of a potential difference across an electrical conductor when a magnetic field is applied in a direction perpendicular to that of the flow of current. (moving electrons (a current) in a conductor are pushed to the side of the conductor by a magnetic field)
11
Q
What are holes?
A
Positive charge carriers (default is with electrons so its the absence of an electron)
12
Q
How is the Hall coefficient defined for positive carriers?
A
1 over p (volume density) times e (electric charge)
13
Q
How is the Hall coefficient defined for negative carriers?
A
minus 1 over n (charge carrier density) times e (electric charge)
14
Q
What does the Hall effect determine?
A
The charge of the carriers
15
Q
When is the Hall coefficient positive?
A
The number of positive charges is more than the negative charges
16
Q
What is the Hall resistance?
A
The ratio of the transverse voltage developed across a current-carrying conductor, due to the Hall effect, to the current itself. (current = surface area of block multiplied by current density)
17
Q
What is the Hall resistivity for electrons?
A
The Hall coefficient multiplied by the magnetic field
18
Q
Since Bloch waves are constructed to ‘know’ about the periodicity of the atoms, what does this lead to?
A
No scattering from the periodic arrangement of atoms and a perfect crystal has infinite conductivity
19
Q
In the semiclassical picture, why do we need electron wavepackets?
A
Bloch waves are delocalised across the whole crystal, so wavepackets are relatively localised in comparison
20
Q
In the semiclassical picture, what size are the electron wavepackets in comparison to the lattice spacing and applied fields?
A
The size of the wavepacket is large compared to the lattice spacing and the applied fields vary slowly in comparison to the scale of the wavepacket
21
Q
Why do the electron wavepackets have a k-vector centred on a particular value of k and with nearby vectors in some small finite range added in?
A
To make it a spatially localised state (due to the Heisenberg uncertainty principle). The more localised, the larger range of k must be included
22
Q
What type of velocity do the electron wavepackets have that we consider?
A
Group velocity
23
Q
F=qE direction is defined by what type of charge?
A
A positive test charge
24
Q
What is the crystal momentum of an electron?
A
h bar times k
25
Is crystal momentum the same as true momentum?
No
26
Why is there no current if a band is full with or without an electric field? (not semiconductors)
The energy of one k state is always matched with its equal and opposite state so they are balanced (on the parabola in reduced zone scheme)
27
What happens if there a full band with an applied electric field?
Still no current. All of the electrons march along in unison but equal and opposite state means no net current
28
In a partially filled band with no electric field, is there a net current and why?
No because (as with a full band) all states are symmetrically occupied in energy
29
What happens to a Fermi surface (metal with a partially filled band) if an electric field is applied going to the right and is a current carried?
Fermi surface gets shifted to the left so the states are now asymmetrically occupied and a current can be carried
30
What two things need to be balanced for a steady state to be reached?
Rate of change in momentum due to the electric field and the rate of change in momentum due to scattering
31
What would happen to the Fermi surface if there was no scattering?
It would be continuously moving at a constant rate
32
What is the relaxation time approximation?
On average, electrons are scattered after a time (tau), which is not a function of the wavevector (k)
33
What is equivalent to a certain number of occupied electrons levels carrying a current?
The complementary unoccupied levels being filled with holes each carrying charge +e
34
What do electron-like and hole-like bands look like?
Electron-like bands are a normal parabola and hole-like bands are flipped (negative) parabolas
35
Is the effective mass directly or inversely proportional to the band curvature and what does this mean for the band curvature for heavy and light masses?
Inversely, so the lighter it is, the more curvature it has in its bands than if it were heavier
36
What is the electron effective mass?
h bar squared over the second derivative of the energy with respect to the wavevector
37
How is the electron effective mass related to the hole effective mass?
The hole effective mass is negative the electron effective mass
38
What is the effective mass tensor and how can it usually be simplified?
It looks like a matrix and it is usually diagonalised. For a cubic crystal, the value is the same along the diagonal.
39
Do different bands have different masses?
Yes
40
Does the crystal momentum increase or decrease if we have a missing electron at a certain value of k?
Decrease
41
How is the electron wavevector related to the hole wavevector?
They are negatives of each other
42
How is the electrostatic potential energy (not conventional PE) related between holes and electrons?
They are the negative of each other
43
What does 'electrons sink, holes float' mean?
Electrons prefer to be in lower energy states, therefore, the hole will prefer to sit at the top of the band
44
When an electric field is applied to the right, electrons move to the left, which direction do holes move?
Also the left (same as electrons)
45
What does it mean for electron and hole group velocity since they move in the same direction?
The velocities are the same
46
Since electrons are not scattered by the periodic potential (from Bloch's theorem), what are the sources of scattering from?
Perturbations of the crystal from perfect periodicity (imperfections)
47
Describe the two classes of scattering
Elastic - momentum can change but energy doesn't. Inelastic - energy is gained/lost by the electron
48
What are examples of the two types of scattering?
Elastic: atomic vacancies, impurity atoms. Inelastic: absorption/emission of a phonon
49
What is Matthiessen's rule regarding scattering?
If we can assume the scattering processes are independent, we can take the scattering rates (1 over the mean free time) as probabilities and add them to get one over the total mean free time
50
According to Matthiessen's rule, which scattering process will be the dominant contribution to the total mean free time?
The scattering process which scatters most frequently and therefore has the smallest mean free time (mean free time = inverse scattering rate)
51
What is the mean free path?
The mean distance between collisions, which is the Fermi velocity (defined by the Fermi energy) multiplied by the inverse scattering rate (mean free time)
52
For a typical clean metal, how big is the mean free path in comparison to the unit cell size and what does this mean?
It is a lot bigger, so electrons can travel for many lattice spacings before being scattered (this shows that Bloch wavefunctions are not scattered from a perfect crystal)
53
Is the mean free time (inverse scattering rate) temperature independent or temperature dependent?
Temperature dependent
54
For a metal, does the resistivity increase or decrease as temperature decreases?
Decreases
55
Is the Fermi velocity temperature independent or temperature dependent?
Independent
56
For metals and semiconductors, what type of scattering dominates at high and low temperatures?
At higher temperatures, phonon (lattice vibrations) scattering dominates, whilst at lower temperatures, impurity scattering dominates (phonons 'freeze-out' at low temps)
57
For metals, what defines the residual resistivity, rho nought (the resistivity when the temperature tends to zero)?
Impurities and defects in the crystal
58
For metals, how are the cleanliness of the crystal (amount of impurities), disorder and residual resistivity (rho nought) related?
A 'dirtier' (more imperfect) crystal has more disorder and a higher residual resistivity than a 'cleaner' (less imperfect) crystal
59
Does a smaller RRR (residual resistivity ratio) mean more or less impurities in a metal and so a larger or smaller residual resistivity?
More and larger
60
In a semiconductor, what is the number of carriers sensitive to?
Temperature, chemical doping, light, electric fields
61
What can we control in a semiconductor?
The number of mobile carriers and so also the conductivity
62
What are the two types of semiconductor band structures?
Direct and indirect band gaps
63
What is a direct band-gap?
The lowest energy state in the unoccupied bands is at the same k-vector as the highest energy state of the occupied bands
64
What is an indirect band-gap?
The lowest energy state in the unoccupied bands is at a different k-vector as the highest energy state of the occupied bands
65
What are the unoccupied bands called?
Conduction bands
66
What are occupied bands called?
Valence bands
67
What is the band gap and does it change?
The energy difference between the bottom of the conduction band and the top of the valence band. No it doesn't change
68
How are the energies of the bottom of the conduction band and the top of the valence labelled?
Both with an epsilon and then subscript c for conduction and subscript v for valence
69
What is the difference between insulators and semiconductors and what is similar?
Insulators have a larger band gap but they both have full bands
70
What is an intrinsic semiconductor?
The number of carriers is dominated by electrons thermally excited from the valence band to the conduction band
71
How is the intrinsic carrier density labelled?
n subscript i
72
For semiconductors, there is an energy gap before the conduction band, so instead of assuming it starts at zero in our calculations, what do we do instead?
Add the critical energy from the bottom of the conduction band so it is a finite offset of this energy
73
What is m subscript c and m subscript v regarding semiconductors?
The effective mass of the bottom of the conduction band and top of the valence band respectively
74
What is the non-degeneracy approximation regarding semiconductors?
The Fermi-Dirac distribution is approximately equal to just the exponential term in the normal Fermi Dirac but the exponent is negative
75
What does 1 minus the Fermi-Dirac distribution refer to?
The holes occupancy in the valence band (or the electrons not occupying the states in the valence band)
76
Where is the chemical potential assumed to be regarding semiconductors?
At the centre of the energy gap
77
By convention, is the Fermi level the same or different from the chemical potential?
The same for semiconductors and different for metals (technically speaking, they are different)
78
What is the carrier density, n, in the conduction band equation?
1 over the volume multiplied by the integral between the conduction energy and infinity of the conduction density of states multiplied by the Fermi-Dirac distribution d(energy)
79
What is the hole carrier density, p, in the valence band equation?
1 over the volume multiplied by the integral between minus infinity and the valence energy of the valence density of states multiplied by 1 minus the Fermi-Dirac distribution d(energy)
80
What is N subscript c and N subscript v referring? (about semiconductors)
The effective density of states in the conduction band and valence band respectively
81
For an intrinsic semiconductor, how is the number density of electrons in the conduction band, the number density of holes in the valence band and the intrinsic carrier density related?
They are all equal
82
What is the mass action law for intrinsic semiconductors?
np (electron number density in CB x hole number density in VB) = ni squared (intrinsic carrier density squared)
83
What is an extrinsic semiconductor?
The carrier density is dominated by electrons/holes originating from chemical impurity atoms introduced into the material by doping the system, so n is not equal to p
84
What are the two types of dopant impurities?
Donors and acceptors
85
What are donors in terms of extrinsic semiconductor dopant impurities?
Impurity atoms with more electrons in their outer shell that the host semiconductor, so when ionised, the atoms donate electrons into the conduction band, leaving a positively charged ion
86
What are acceptors in terms of extrinsic semiconductor dopant impurities?
Impurity atoms with fewer electrons in their outer shell than the host semiconductor, so when ionised, the atoms donate holes into the valence band, leaving a negatively charged ions (accepting electrons, giving out holes)
87
Semiconductors that are extrinsically doped using donors are known as what and what does this mean?
n-type, so electrons dominate the carrier type
88
Semiconductors that are extrinsically doped using acceptors are known as what and what does this mean?
p-type, so holes dominate the carrier type
89
What are the impurity bands and what are the two types?
They are the bands formed in the band gap of the extrinsic semiconductor and they are the donor and acceptor levels
90
Where are the donor and acceptor levels relative to the conduction and valence bands and the chemical potential?
Donor level is typically just below the conduction band and above the chemical potential and acceptor level is just above the valence band and below the chemical potential (however they don't have to be here and you can have acceptor levels above donor levels and vice versa)
91
Using the impurity levels and the band gap, why is it easier to thermally ionise the dopants and create mobile carriers?
The energy differences between the conduction band with the donor level and the acceptor level with the valence band can be much smaller than the band gap, so ionisation can happen at lower temperatures (like room temp)
92
What are elements that are acceptors and donors when inserted into silicon? (have to use silicon as a reference because what is an acceptor for one element can be a donor for another)
Donor: phosphorous and acceptor: boron
93
What does shallow and deep mean for impurity levels?
If a donor level is really close to the conduction band it is shallow and if its far away from the conduction band, it is deep. The opposite is true for acceptor levels, so it is in regard to the valence band instead
94
How is the number density for acceptor and donor dopants labelled?
Capital N for both with subscript a and d respectively
95
The semiconductor is charge neutral overall, what equation shows this? (using the number densities of electrons, holes, ionised donor dopants and ionised acceptor dopants)
The electron density plus the ionised acceptor density (negative charge) equals the hole density plus the ionised donor density (positive charge)
96
What do the superscripts on the number density of the dopant types mean?
The charge, so plus, minus or zero. The zero indicates that some dopants aren't ionised
97
If we assume full ionisation or complete ionisation about donor or acceptor dopants, what is their total number density equal to?
The total donor number density is the same as the ionised number density (eg there is zero density of neutrally charged donor dopants) and the same for acceptor dopants
98
What does it mean that in extrinsic semiconductors, the chemical potential approximates the centre of mass of the electrons/holes?
If there are lots of electrons in the conduction band (lots of donor dopants), the chemical potential is closer to the conduction band energy and if there are lots of holes in the valence band (lots of acceptor dopants), the chemical potential is closer to the valence band energy
99
At low temperatures (like room temp), how does the number of intrinsic electrons/holes compare to the dopants?
They are very small, dopants dominate
100
At high temperatures, how does the number of intrinsic carriers change and compare to dopant densities and what does this do to the chemical potential if it was previously dominated by dopants at lower temperatures?
They exponentially increase and dominate and the chemical potential moves towards the middle of the band gap (as with an intrinsic semiconductor)
101
Why is there little temperature dependence during low to middle temperatures for extrinsic semiconductors?
It doesn't take a high temperature to ionised dopant impurities but it takes a high temperature for there to be a significant amount of intrinsic carriers
102
What happens at very low temperatures in extrinsic semiconductors?
There is a 'freeze-out' of extrinsic carriers because dopants need some temperature to ionise into the conduction or valence bands. It exponentially increases at a certain temperature
103
What is compensation in semiconductors?
A semiconductor with both acceptors and donors (adding more of the minority dopant) so the value of (n + p) decreases
104
Does the equation of the density of electrons multiplied by the density of holes equals the intrinsic carrier density squared hold for extrinsic semiconductors at thermal equilibrium?
Yes
105
What is the drift velocity?
It is the average velocity of the charge carriers in the drift current, which is the current due to an applied electric field
106
What is the equation for the drift velocity for electrons?
minus the electron's charge multiplied by the electric field multiplied by the mean free time divided by the effective mass
107
What is the measure of mobility of charge carriers?
A measure of how quickly electrons or holes respond to an applied electric field
108
What is the equation for mobility (mu) for charge carriers?
The absolute value of the drift velocity divided by the absolute value of the electric field (so it's always positive)
109
How do you calculated the longitudinal conductivity (sigma subscript xx) including the mobility of charge carriers (mu)?
The sum over all electron bands of the density of electrons times electron's charge times the electrons'mobility plus the sum over all the holes bands of the density of holes times the electrons charge times the holes' mobility
110
Is the mobility of charge carriers dependent on time or temperature?
Both
111
For semiconductors at low, intermediate and high temperatures, the conductivity is dominated by the temperature dependence of what? (3 answers for each temp)
At low temps, it is dominated by the freeze out of the extrinsic carrier densities, at intermediate temperatures, it is dominated by the tempe dependence of the mobility of electrons and holes and at high temp, it is dominated by the temp dependence of the intrinsic carrier densities
112
What is lithography?
Fabricating semiconductor devices that have dopants in certain areas
113
What is the process of lithography in brief?
There is a wafer that you put resist (photoresist) on and put a mask on top of it to cover parts of it and then shine UV light over it, which will take away or harden any exposed resist
114
What are the three things that can be done to the wafer after the lithography process?
Etch, deposit or implant
115
What is the difference between positive and negative tone resist in semiconductor device fabrication?
When positive, the exposed parts of the polymer become soluble (the resist breaks down) and when negative, the exposed parts becomes insoluble (the resist hardens)
116
When there is a junction between two regions of a semiconductor which are extrinsically doped to be p-type and n-type, what is this called?
p-n junction
117
When the p-n junction is formed, what needs to be true for the chemical potentials on either side of the interface?
They need to line up with each other
118
For p-n junctions, how do they achieve making the chemical potentials match on either side and what does this form?
Charge carriers moved between the layers, creating a built-in potential
119
Is the band gap energy constant throughout a p-n junction?
Yes
120
What is the rigid band-bending in the p-n junctions?
The band bending around the chemical junction which connects the conduction and valence bands of the p and n-types of the regions of the semiconductor
121
At the chemical junction for a p-n junction, where is the chemical potential?
In the middle
122
What is the equations for the electron's charge times the built-in potential for the p-n junction (this is a type of energy)?
The difference between the energy of the conduction bands of the p and n types or the difference between the energy of the valence bands of the p and n types
123
What is the depletion region in a p-n junction and what is it also known as?
The area between the conduction and valence bands where the rigid band bending starts and ends
124
What is the depletion approximation for the p-n junctions?
It assumes there is a space-charge region of width d subscript n on the n-type side and d subscript p on the p-type side. Also, that there are no free charges in the space charge zone
125
For p-n junctions, why is there net positive charge in the space charge region on the n-type side and a net negative charge in the space charge region on the p-type side?
The conduction(valence) band energies move away from the chemical potential when approaching the junction from the n(p)-type side. This leaves only the positive(negative) ionised donor (acceptor) atoms
126
Since the p-n junction system is charge neutral, the positive charge in the space charge region on the n-type side is equal to what and what is the net charge on the zones far from the band bending?
The negative charge in the space-charge region on the p-type side and it is neutral
127
Across the p-n junction, what is the direction of the electric field?
Right to left (positive to negative charge)
128
What direction is the voltage applied across a p-n junction in forward bias?
The positive terminal is connected to the p-type region and the negative terminal is connected to the n-type region
129
In forward bias, is the new built in potential larger and smaller than the equilibrium value (ie when there's no applied voltage) so what is the new built in potential?
Smaller and electron charge multiplied by the difference between the equilibrium built in voltage and the applied voltage
130
In forward bias, is the band bending and depletion zone width increased or decreased?
Decreased
131
What direction is the voltage applied across a p-n junction in reverse bias?
The positive terminal is connected to the n-type region and the negative terminal is connected to the p-type region
132
In reverse bias, is the new built in potential larger and smaller than the equilibrium value (ie when there's no applied voltage) so what is the new built in potential?
Larger and electron charge multiplied by the sum of the equilibrium built in voltage and the applied voltage
133
Why is it harder to get a current in reverse bias across a p-n junction?
Because its harder for electrons to 'climb' the hill as the back bending is steeper
134
In reverse bias, is the band bending and depletion zone width increased or decreased?
Increased
135
What is current rectification at a p-n junction (and this means it's acting as a diode)?
The electrons on the n-type side have to overcome a potential barrier to move to the left, which increases for reverse bias and decreases for forward bias
136
What are the two types of currents for both electrons and holes that occur even if there is no applied voltage at a p-n junction?
Diffusion currents and drift currents
137
What are diffusion currents at p-n junctions?
They are due to concentration gradients near the junction, so the carriers want to diffuse from high concentration to low concentration, so electrons on the n-type side want to go the p-type side and and holes on the p-type side want to go to the n-type side
138
What are drift currents at p-n junctions?
They are due to potential gradients near the junction, so the electrons on the p-type side want to go down the hill and holes on the n-type side want to go up the hill
139
What directions are the diffusion currents for electrons and holes (currents due to concentration) at p-n junctions and why?
The direction for both is right from p to n. This is clear for holes but for electrons, the diffusion current makes the electrons go left from n to p but because electrons are negative the current direction is opposite
140
What directions are the drift currents for electrons and holes (currents due to potential) at p-n junctions and why?
The direction for both is left from n to p. This is clear for holes, but electrons move from the p to the n type side for drift currents and because electrons are negative the direction of the current is opposite
141
Are the diffusion currents and drift currents for p-n junctions positive or negative and why?
Diffusion are positive and drift are negative because positive is defined as left to right
142
At equilibrium where there is no applied voltage, there is no net current, so what can we say about the drift and diffusion currents?
The hole drift current is equal to the negative of the hole diffusion current and the same can be said for the electron equivalents
143
What is the diffusion current limited by at p-n junctions?
The potential energy barrier
144
When the applied field changes, does the drift or diffusion current change by much?
Drift current doesn't change, whilst diffusion current does change
145
What are minority carriers in p-n junctions?
electrons in the p-type region are minority carriers and holes in the n-type region are minority carriers
146
What do generation currents do in p-n junctions and how are they formed?
Create minority carriers (electrons in p type and holes in n type) and they are thermally excited
147
What is recombination in p-n junctions?
Electron in conduction band combining with a hole in valence band and a photon will be released
148
Where does recombination mostly happen in p-n junctions?
Band bending region
149
How is the generation and drift current related for electrons and holes separately?
The drift current is minus the generation current
150
What do we assume about the recombination rate and what does this mean about the generation rate of minority currents?
That it is small and this means the generation rate doesn't depend on the applied voltage (and the generation current is minus the drift current, which doesn't depend on applied voltage)
151
What are the four components of the total current in p-n junctions?
Diffusion and drift currents from both holes and electrons
152
What does the current against applied voltage graph look like for p-n junctions?
It is an exponential, it starts in the negative current, negative voltage section as a straight line, then crosses the origin and exponential increases into positive current, positive voltage region
153
What is the resistance when there is a positive applied voltage (forward bias) for p-n junctions?
Low resistance
154
What is the small reverse bias leakage current when the voltage is negative (reverse bias) for p-n junctions?
There is a small current from minority carriers
155
What happens at large enough reverse bias for p-n junctions?
There can be a junction breakdown, where a large negative current happens
156
What do light emitting diodes (LEDs) do that we assume doesn't occur in most cases?
Recombination, so light is emitted from the space-charge region in forward bias due to recombination
157
What do photodiodes (and p-n solar cells) do and what are they the opposite process of?
They are operated in reverse bias and an incoming photon creates an electron-hole pair in the space-charge zone (opposite to recombination)to give a current and this is the opposite of LEDs
158
What process does the LASER diode work with and how does this work?
Stimulated emission, which is when a photon (with energy equal to the difference in energy levels) approaches an electron in an excited state and the electron is stimulated to fall down a level, releasing a photon with the same energy and phase as the first photon (so now two photons)
159
What is the work function of a metal?
It is the minimum energy (e times phi) required to move an electron from deep within the bulk to a point just outside the surface
160
What is the vacuum level related to the metal work function?
It is the work function amount above the fermi energy
161
What is the vacuum level in relation to the work function energy and the electron affinity energy (separately) for n-type semiconductors?
It is the work function energy above the chemical potential or it is the electron affinity energy (e times chi) above the bottom of the conduction band
162
Why is it easier to use electron affinity energy rather than work function energy?
Work function changes as the chemical potential changes, whereas the electron affinity energy stays the same
163
What is a Schottky junction?
A metal-semiconductor junction. The metal has a large work function and the semiconductor is n-type with both a work function and electron affinity energy labelled
164
What happens in a Schottky junction when it is connected?
The Fermi energy in the metal lines up with the chemical potential in the semiconductor and the vacuum level must also be continuous
165
What is the Schottky barrier height?
It is the energy barrier (e times phi subscript Sb), which is the energy difference between the work function energy of the metal and the electron affinity energy
166
What is the Schottky-Mott relationship equation?
The Schottky barrier height (phi subscript Sb) is equal to the difference between the metal work function (phi subscript m) and the electron affinity (chi)
167
What are the terminals of a 3 terminal device?
Source, gate and drain, all with different voltages
168
What is a basic three-terminal transistor?
The junction field-effect transistor (JFET)
169
What are the two types of JFET that are formed and which is the only one that we will consider?
n-p-n and p-n-p but we will only consider p-n-p systems
170
For the JFET three terminal device(p-n-p), what is the space-charge zone dominated by?
The depletion width in the n-layer because the p layer depletion width is narrow ( I think because the p regions are highly doped)
171
What does it mean with semiconductors if there is n superscript + or p superscript +?
It is a highly doped n-type (or p-type) material
172
In JFETs, what type of bias is applied to the p-n junctions and what does that mean in terms of which voltage is bigger?
Reverse bias and that means the gate voltage is smaller than the source voltage (source voltage zero I think, so gate voltage negative)
173
In JFETs and MOSFETs, is the current flowing vertically larger or smaller than the channel current (source-drain current) and why?
A lot smaller because the junctions are reverse biased
174
In JFETs, when the gate voltage increases (becomes more negative), the space-charge zones increase, which leads to what regarding the channel cross-sectional area and the resistance?
Cross-sectional area decreased and resistance increased
175
In JFETs, which voltage needs to be varied to control the current flowing in the channel?
The gate voltage
176
What is a metal-oxide-semiconductor field-effect transistor (MOSFET) device?
It is also a field-effect transistor but it has an insulating oxide layer between the metallic gate electrode and the semiconducting channel
177
What are the two basic types of MOSFETs and what do they mean?
Normally on and normally off. When the gate voltage is zero, normally on types the source-drain channel conducts and a finite gate voltage is needed to switch it off, whereas, in normally off types, the source-drain channel does not conduct and a finite gate voltage is needed to switch it on
178
In MOSFETs, what is the insulating oxide layer?
A dielectric, with the plates of the capacitor being the top gate electrode and the channel
179
For JFETs, what happens when the difference between the gate and source voltage is large?
The channel resistance is defined by the 'pinch-off' and the channel current doesn't rely on the drain voltage anymore
180
How is the magnetic field defined and what is the force due to it?
Its effect on a moving charge, and the force is equal to the charge multiplied by the cross product between the velocity and the magnetic field flux density
181
What are the two components of the magnetic field?
The material-independent part, the field strength (H) and the material-dependent part, magnetisation (M)
182
What is the equation that combines both parts of the magnetic field?
The magnetic field flux density (B) equals the permeability of free space multiplied by the sum of the field strength (H) and the magnetisation (M)
183
What types of materials have non-zero magnetisation?
Magnetic materials
184
What does the magnetic flux density dot producted with the area equal?
The magnetic flux
185
What is the magnetisation and its units?
Its the magnetic moment per unit volume with units Am^-1
186
How are magnetic moments represented?
With arrows
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Are the magnetisation and magnetic moment an intensive or extensive property of the material?
Magnetisation is intensive, whilst the magnetic moment is extensive
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What is the units and letter representing a magnetic moment?
Am^2 and little m
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What is the units and letter representing the magnetic field/field strength?
Am^-1 and H
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What is the units and letter representing the magnetic flux density?
Tesla or N(Am)^-1 and B
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What is the orbital magnetic dipole moment?
current times area (area of orbit), or minus the Bohr magneton times m subscript l (m subscript l = angular momentum over h bar)
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What is the electron spin magnetic dipole moment?
-2 times the spin quantum number (+ or - half) times the Bohr magneton
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What makes up the magnetisation of a material?
The spin and orbital motion of electrons and nuclei
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Is the nuclear magnetic dipole moment bigger and smaller than the electron magnetic dipole moments?
A lot smaller, so electron contribution dominates the magnetisation
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Is the spin direction in the same or opposite direction to the moment for electrons?
Opposite
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For electrons, is the magnetic dipole moment in the same or opposite direction to the angular momentum vector?
Opposite
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What is the magnetic susceptibility (chi) and its units?
A constant that defines a materials response to an applied magnetic field and it is dimensionless
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What is the equation for magnetic susceptibility?
Magnetisation divided by field strength (M/H)
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What are the two types of field induced magnetism?
Diamagnetism and paramagnetism
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In diamagnetism, is the magnetic susceptibility positive or negative and what does that mean?
Negative, so diamagnets are repelled by magnetic fields
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What happens when a magnetic field is applied to a diamagnet?
An electron current is induced to oppose the magnetic field and the current persists as long as the magnetic field is present
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What are the two types of paramagnetism?
Langevin and Pauli (or itinerant electron magnetism)
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Is the magnetic susceptibility positive or negative for Langevin and Pauli paramagnetism?
Positive
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What happens when a magnetic field is applied to a paramagnet?
They become attracted by the field and the internally induced magnetic field lines up with the externally applied magnetic field
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What type of atoms/ions is the Langevin paramagnetism applicable for?
Free atoms or ions
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Explain the 2 level system in the Langevin paramagnetic system
When an external magnetic field is applied, there becomes a two level energy system, with the top level with a downwards moment of minus the Bohr magneton, whereas the bottom level has an upwards moment of the Bohr magneton
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How is the potential energy related to the moment?
The potential is equal to minus the dot product between the moment and the flux density (B)
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How is the magnetic susceptibility related to temperature in a Langevin paramagnet system?
They are approximately inversely proportional to each other (Curie's law)
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In what case do we consider Pauli paramagnetism (or itinerant electron magnetism)?
For electrons in bands (delocalised electrons)
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For Pauli paramagnetism , when there is no magnetic field applied, what does the density of states look like for the electrons and why?
It looks like a parabola on its side and its split for electrons with spin magnetic dipole moments parallel and antiparallel to the z axis
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For Pauli paramagnetism at no magnetic field, the density of states increases (or decreases for down spin) to what power of the energy?
Half
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For Pauli paramagnetism, what happens to the density of states when a magnetic field is applied in the z direction and why?
The part of the DOS for up spin gets shifted downwards in energy because they align with the magnetic field and the part for down spin gets equally shifted upwards in energy
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For Pauli paramagnetism, when there is an applied magnetic field in the z direction, is there an increase or decrease of occupied up and down spin states and why?
Increase in up spin states and decrease in down spin states, this is because there is more of the DOS of the up spin state below the Fermi energy and less of the down spin states below the Fermi energy
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Is Pauli paramagnetism dependent or independent of temperature?
Independent
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What are the three types of spontaneous long range magnetic order that we are considering?
Ferromagnetism, antiferromagnetism and ferrimagnetism
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What does it mean if a material has spontaneous long range magnetic order?
It occurs even without an applied magnetic field
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What do the magnetic moments look like for antiferromagnetism?
Neighbouring moments are anti-parallel even in the absence of an applied magnetic field
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Is there a net magnetisation for antiferromagnetic materials?
No
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When does order appear for antiferromagnetic materials?
When the temperature is below the Neel temperature
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What do magnetic moments look like for ferrimagnetism?
Neighbouring moments are anti-parallel but with different magnitudes
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What is the magnetisation for ferrimagnets?
It is non-zero for when the temperature is below the Curie temperature
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If the temperature is above the Neel temp for antiferromagnetism and Curie temp for ferrimagnetism, how do the materials act?
They act like diamagnets or paramagnets, which are much weaker
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What does it mean that antiferromagnets and ferromagnetism are correlated states of matter?
By knowing the direction of one moment, you know the direction of the other moments in the material with certainty
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What do magnetic moments look like for ferromagnetism?
Neighbouring moments are parallel even in the absence of a magnetic field
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What is the magnetisation for ferromagnets?
It is non-zero for temperatures less than the Curie temperature
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What is the name for the magnetisation against magnetic field strength graph?
A hysteresis loop
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What are the 3 important points in a hysteresis loop and the letters that represent them?
Remanence (M subscript r), saturation magnetisation (M subscript s) and coercive field (H subscript c)
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Where do the saturation magnetisation parts occur on a hysteresis loop and what do these look like?
At the lowest (negative M and negative H) and highest point (positive M and positive H) and these are just horizontal straight parts
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What is happening with the magnetic moments at the saturation magnetisation regions in a hysteresis loop?
For positive saturation magnetisation, all the moments are pointing upwards lined up with the magnetic field and for negative saturation magnetisation, all the moments are lined up pointing down and also in line with the magnetic field which is pointed down
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What is happening at the positive and negative remanence points in a hysteresis loop?
It is when the field strength is zero (H=0)
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What is happening at the positive and negative coercive field points in a hysteresis loop?
The magnetisation is zero where the moments are alternating up and down
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What direction do you go around a hysteresis loop?
Anticlockwise
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Why is there two lines in the middle of a hysteresis loop?
Work is being done to go around the loop
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Are the positive values of remanence, saturation magnetisation and coercive field symmetrical the same as the absolute value of the negative values of these points or different?
The same
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What type of behaviour do ferromagnetic materials transition to at the Curie temperature?
Paramagnetic
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Why do ferromagnets change to paramagnets at the Curie temperature and beyond?
Higher temperatures cause more entropy (disorder), which leads to destruction of long range order
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What is the exchange coupling energy in ferromagnets?
Approximately gluing the moments together
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What is the estimate for the exchange coupling energy between the moments?
The Boltzmann constant multiplied by the Curie temperature (quite a high energy)
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What is the strength of ferromagnetic magnetisation in comparison to that of paramagnets or diamagnets?
A lot lot stronger
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What is magnetostatic energy?
The energy of producing stray fields (outside the magnetic material)
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How can you minimise the amount of magnetostatic energy for a magnet?
Breaking up into domains with each section having a moment pointed in a clockwise (or anticlockwise) direction so that there are no stray fields
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Is there a net magnetic moment in the macroscopic sample if there is enough magnetic domains that the moments don't hit the sides of the sample?
No, there is no net moment in the macroscopic sample (there is moments locally in each domain)
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What is the approximate stray field energy (magnetic field)?
The volume integral over all space of the magnetic flux density (B) squared
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What is Stoner's model?
Around the Fermi energy, a large density of states favour ferromagnetism so there is a spontaneous splitting of the spin-up and spin-down bands
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When do you use the Stoner model?
For itinerant ferromagnetism to model a solid with high DOS near the Fermi energy
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Is there a net magnetisation for the Stoner model and why?
Yes because there are more up spin bands than down spin bands
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What drives the splitting in the Stoner model?
An internal exchange interaction, which splits the energy of states with different spins
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Why does the Stoner model mean ferromagnetism is favoured?
In non-magnetic systems, there is an equal number of spin up and spin down electrons but in ferromagnetism, there needs to be an imbalance
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Because fermions must have antisymmetry (Pauli) (wavefunction is odd), that means the product of the spin and spatial wavefunctions must consist of what types (even or odd for each)?
One odd and one even (doesn't matter which one)
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(Exchange interaction) Do two neighbouring parallel spin wavefunctions (both up or both down) overlap more or lass spatially than an antiparallel pair and why?
Less because of Pauli
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(Exchange interaction) Since there is less spatial overlap for parallel spin electron pairs, what does this mean for coulomb repulsion energy and potential energy?
Less coulomb repulsion as they are further apart, so less potential energy, which (I think) causes the energy split in the Stoner model
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Superconductivity is associated with what type of electrons? (the same as with ferromagnetism)
Correlated electrons
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In superconductivity, what happens when the temperature is less than a critical temperature for the material?
Zero resistivity and perfect diamagnetism so it opposes applied magnetic field
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The perfect diamagnetism of a superconductor is described by what effect?
The Meissner effect
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What is the Meissner effect?
The complete ejection of magnetic field lines from the interior of the superconductor when the temperature is below the critical temperature
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What is the screened magnetic field inside the superconductor under the critical temperature?
B=0
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What screens the superconductor magnetic field under the critical temperature?
Surface supercurrents
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When does the Meissner state for a superconductor break down?
When the applied magnetic field is too large
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What defines a type 1 superconductor?
The superconductor becomes a normal conductor above a critical magnetic field (H subscript c)
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What needs to be remembered about the magnetisation against field strength graphs (M v H) when we talk about the types of superconductors?
It is -M on the y axis, so when it is in its Meissner state, the gradient is a negative slope (like a diamagnet)
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What type are most superconductors?
Type 2
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What defines a type 2 superconductor?
It has two critical fields (lower: H subscript c1 and higher: H subscript c2). It becomes a normal conductor above the upper critical magnetic field and becomes partly normal between the lower critical field and the upper critical field
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What is the state called for a type 2 superconductor between the lower and upper critical magnetic fields?
Vortex state
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What happens in the vortex state in a type 2 superconductor?
Some magnetic flux is entering the superconductor in the form of quantised vortices (vortices carry some quanta of flux)
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Describe the graph of minus magnetisation (-M) against field strength (H) for a type 1 superconductor
It has constant gradient of 1 (the part represents the Meissner state) until it reaches the critical magnetic field, where it falls vertically downwards and then there is no magnetisation and it is in its normal state with superconductivity destroyed
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Describe the graph of minus magnetisation (-M) against field strength (H) for a type 2 superconductor
It has constant gradient of 1 in its superconducting state until the lower critical field, where the magnetisation drops smoothly (vortex state) until it reaches zero at the higher critical field and then its in its normal state
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What theory describes the microscopic mechanism of superconductivity and what is the outline of it?
BCS theory, which describes the superconducting current as a superfluid of Cooper pairs of electrons, which occurs through phonon-mediated attraction
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How are Cooper pairs created in the BCS theory?
An electron moves through the lattice (of positive ions), which slightly move towards the electron through attraction (lattice vibration = phonon) but the electron is quicker and moves away, whilst the positive ions are still closer together, creating a slight positive charge that a 2nd electron is attracted to
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What spins do the electrons in Cooper pairs have?
One spin up and one spin down
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What do Cooper pairs at the chemical potential do in BCS theory in terms of k space?
A gap is opened in the density of states (allowed energy states) on either side of the Fermi energy
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How does the gap in the density of states caused by the Cooper pairs cause superconductivity?
All excitations must have some minimum energy, so small excitations like scattering are forbidden (no resistance in system), which leads to superconductivity
