Hall Effect
1
Hall Effect
semiconductor sample with high mobility
Michael S. Santana
2DEG, measurements of the resistance per
Partner: Andrew Kulesa
square (sheet resistance), the Hall
Department of Physics, Stony Brook University
coefficient, the carrier sign (electrons or
PHY 445/515
time between collisions), and the mean free
Submitted: May 6, 2015
path (mean distance traveled between
Abstract:
collisions) are made possible [1]. In this
Hall Effect was first observed in 1879 by the graduate student, Edwin Hall, from Johns Hopkins University. Hall Effect is noted when an electrical current in a semiconductor, with a two-dimensional electron gas, flows perpendicularly to an applied magnetic field resulting in a transverse voltage that is both perpendicular to the current and to the applied magnetic field. This voltage, otherwise known as the Hall Voltage, is measured with an appropriate four-probe arrangement. The primary significance of
holes), the carrier relaxation time (mean
experiment, we performed DC Hall Effect measurements at 300 K and 77 K (sample at room temperature, and sample cooled by Liquid Nitrogen) on the GaAs/GaAlAs heterostructures. We measured the Hall coefficient of GaAs to be RH = (2339.956 ± 128.065) m2/C at 300 K, and RH = (2649.006 ± 145.577) m2/C at 77 K. Similarly, we measured the sheet resistance of our GaAs/GaAlAs heterostructure to be (2164.814 ± 465.329) Ω/sq at 300 K… Introduction: [Carriers in E-field: Drude Model; mobility]
this experiment is in the determination of the carrier concentration (density), along
The Drude Model provides a simplified
with resistivity, which allows for the
description of electrical conduction in an
calculation of carrier mobility. Similarly,
electric field. Consider a macroscopic solid
when the Hall Effect is performed at
conductor with a sea of free electrons, each
relatively low temperatures on
traveling with an individual intrinsic velocity
semiconductors containing a two-
inside the conductor. Classical mechanics
dimensional electron gas (2DEG), i.e.
postulates that these intrinsic velocities are
2
thermal in nature, however, quantum
velocity) vd. The current density is given in
mechanics says that, due to the nature of
terms of the terminal velocity and the
the Pauli Exclusion Principle, the intrinsic
carrier concentration, represented by the
velocity of an electron in an electron gas
Drude relation:
within a conductor (metal) or semiconductor is close to that of an
𝑗 = 𝑒𝑛𝑣𝑑 =
electron in an electron gas at absolute zero temperature. In other words, the intrinsic
𝜌=
𝐸 𝜌
𝑚∗ 𝑒 2 𝑛𝜏
velocity of an electron in the electron gas is
Where ρ is the resistivity, m* is the
close to the Fermi velocity. In zero electric
modified mass of the electron as it
field, an electron’s trajectory is random
transverses the periodic potentials of a
allowing for the average of all electron
lattice, and τ is the carrier relaxation time
velocity vectors to yield an average of zero.
(mean time between collisions). In the GaAs
Thus, in a net zero electric field, the center
layer, the electronic effective mass is m* =
of mass of the electron gas is at rest.
0.067m. When studying conductivity
However, if we are to apply an electric field
(inverse of resistivity), it is interesting to
inside the conductor, the electron gas will
talk about the quantity of mobility. Mobility
start drifting in a direction that opposes the
is defined to be the magnitude of drift
direction of the applied field. In a constant
velocity per unit of electric field [1]. From
uniform electric field, we expect the
the above two relations, we attain a
velocity of the electron gas to increase
relation for mobility such that:
indefinitely with an acceleration proportional to the applied field strength, a = eE/m. This phenomena is not observed
𝜇=
𝑒𝜏 𝑚∗
[Two-dimensional conductors]
due to changes in electron trajectories between collisions, which are inelastic and
In two-dimensional conductors, electrons or
remove excess momentum and energy from
holes (electrons in our case) are confined in
the electron gas. Thus, the electrons attain
a quantum well such that the carrier is free
a constant finite velocity (drift or terminal
to move in two dimensions, but is quantized
3
in the third. We define sheet resistance to
The carrier relaxation time coupled with the
be R per sq.:
intrinsic velocity of a carrier (Fermi velocity 𝑚∗ 𝑅∎ = 2 𝑒 𝑛𝜏
or vF) yields the mean free path of the carrier, or, in other words, the mean
And j = E/R∎, a two-dimensional form of
distance a carrier travels between any two
Ohm’s Law. Establishing a relation between
arbitrary collisions. In an effort to compare
the previous formula and the “electrician’s”
microscopic parameters with their
Ohm’s Law gives V/I = (L/w) R∎. Since V/I
theoretical estimates, we will further
represents longitudinal resistance, and
compare our mean free path with a variety
(L/w) represents the number of squares in a
of meaningful length scales.
rectangular sample (between a pair of
Experimental Details:
leads), then: Displayed in the figure below are the 𝑅 = 𝛾𝑅∎
GaAs/GaAlAs heterostructures, with layers
Where γ, (L/w), need not be an integer
grown on a GaAs substrate. The top
number of squares. Since the Hall Effect
capping layer is essential to the top doping
formulas in two-dimensions disregard
layer, and is similarly important in regards
sample thickness, we establish that the Hall
to the protection of the structure as a
Voltage is given by:
whole. Layer three contains dopant atoms
𝑉𝐻 = 𝐼𝑅𝐻 𝐵𝑧
which are included in the GaAs lattice structure. Layer two is undoped, and serves
From the above relation, we can obtain the
as a spacer between the conducting
Hall coefficient in the case of a single
inversion and layer three (source of
carrier, and can likewise solve the following
dopants). This is an example of modulation
relation for the carrier density n:
doping, such that the dopants are
𝑅𝐻 =
1 𝑒𝑛
With RH and R∎, we can further solve for the mobility and the carrier relaxation time.
separated from the carriers in an effort to minimize carrier scattering and maximize or enhance carrier mobility.
4
The previous photograph allows for the calculation of the actual sizes and number of squares between a pair of voltage leads. Using the 100 micron scale, we can calculate γ, (L/w), where L = (552 ± 5) microns and w = (48 ± 5) microns, such that γ = 11.50 ± 0.40. Fig. 1. The sample’s layered structure (not to scale). Taken from [2].
Results:
In the above energy diagram for the
VH vs. B-field (Correction) at 300 K
conduction band in the n-type modulation
0 -0.0002 0
filled in triangle is the inversion layer, which
-0.0004
is representative of the EF state in the conduction band between layers one and
VH (V)
doped GaAs/GaAlAs heterostructure, the
0.15
-0.001
-0.0016 -0.0018
0.2
R-6 & B-4
-0.0008
-0.0014
well).
0.1
-0.0006
-0.0012
two (triangularly-shaped quantum potential
0.05
R-7 & B-3 y = -0.01x + 0.0001 R² = 0.9991
R-8 & B-2
y = -0.0106x + 0.0001 R² = 0.9982 y = -0.0106x + 0.0001 R² = 0.9982
B-field (T)
Fig. 3. The above plot shows the Hall Voltage’s dependency on the applied B-field at 300 K.
VH vs. B-field (Correction) at 77 K
pads (contacts) are lighter in color, and thin gold wires connect to the pads. Taken from [2].
VH (V)
Fig. 2. A microscopic image of our sample. The metal
0 -0.0002 0 -0.0004 -0.0006 -0.0008 -0.001 -0.0012 -0.0014 -0.0016 -0.0018 -0.002
0.05
0.1
0.15
0.2 R-6 & B-4 R-7 & B-3
y = -0.0116x + 0.0001 R² = 0.9981 y = -0.0098x + 0.0002 R² = 0.9974 y = -0.0124x + 0.0001 R² = 0.9987
B-field (T)
R-8 & B-2
5 Fig. 4. The above plot shows the Hall Voltage’s
Using the following relation:
dependency on the applied B-field at 77 K.
𝑉𝐻 = 𝐼𝑅𝐻 𝐵𝑧 Each plot was fitted with a linear regression. R-6 & B-4 (represented as red lead to pad 6 and the black lead to pad 4) are a pair of geometrically opposite fingers, as are R-7 & B-3 and R-8 & B-2. Each respective equation and it’s “goodness of fit,” R2, are included on the plots directly. When plotting VH as a function of applied Bfield, we know that the voltage measured is represented by:
We plotted Hall Voltage at a variety of applied magnetic fields, as shown above for 300 K and 77 K. When the applied magnetic field approached zero, we switched the leads, and thus the direction of the magnetic field, to measure the negative Bfield regime. The slopes of the above plots allow for the determination of the Hall coefficient and the variety of other microscopic parameters at 300 K and 77 K
𝑉𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 = 𝑎𝑉𝐻 + 𝑏𝑉𝑀𝑅
respectively, as discussed in the
Accounting for non-perfect geometry, the
introduction. It should be noted that the
voltage measured for VH is part Hall Voltage
plots of VH vs. B-field at both temperatures
and part voltage resulting from
have been corrected by using the procedure
Magnetoresistance. We can correct for
mentioned above.
MR’s unwanted contribution to the measured Hall Voltage by: 2𝑉𝐻 (𝐵) = 𝑉𝑚𝑒𝑎𝑠𝑢𝑟𝑒 (+𝐵) − 𝑉𝑚𝑒𝑎𝑠𝑢𝑟𝑒 (−𝐵) Similarly, when actively trying to measure Magnetoresistance across a pair of adjacent fingers, we will correct for unwanted Hall Effects by:
By extracting the slopes of each plot at both temperatures, we calculated the Hall coefficient, and coupled with γ and the fact that our mobile carriers are electrons (right hand rule), we are able to compare our microscopic parameters with some other meaningful length scales relating to our sample:
2𝑉𝑀𝑅 (𝐵) = 𝑉𝑚𝑒𝑎𝑠𝑢𝑟𝑒 (+𝐵) + 𝑉𝑚𝑒𝑎𝑠𝑢𝑟𝑒 (−𝐵)
6 300K RH 2339.95585 n 2.66736E+15 R per sq. 2164.81428 µ 1.080903739 t 4.11332E-13 EF 1.52851E-21 vF 223918.9476 mfp 9.21051E-08 λdB 4.85343E-08
77K 2649.006623 2.35617E+15 2164.81428 1.22366461 4.65659E-13 1.35018E-21 210452.0468 9.79989E-08 5.164E-08
Units m2/C m-2 Ω/sq. m2/(Vs) s J m/s m m
computed Fermi velocities into the cyclotron orbit equation yields: 𝑟𝑐 = 6532.85 𝑛𝑚 at 300 K 𝑟𝑐 = 6139.96 𝑛𝑚 at 77 K We know that an electron could not complete a cyclotron orbit at either 300 K
In light of the findings presented in the
or 77 K since c = 1,009,986 nm (c = πr2) in a
above table, at 300 K, we observed the Hall
150mT field. It is likewise important to note
coefficient to be (2339.956 ± 128.065)
that at room temperature (300 K), intrinsic
m2/C, carrier density to be (2.667 ± 0.146) x
semiconductors are typically non-
1015 m-2, the sheet resistivity to be
degenerate. If a semiconductor is to
(2164.814 ± 465.329) Ω/sq., mobility to be
become degenerate, the semiconductor
(1.081 ± 0.239) m2/(Vs), and a mean
begins to act more like a conductor due to
relaxation time of (4.113 ± 0.912) x 10-13 s.
the many electrons in the conduction band.
Similarly, at 77 K, we observed the Hall coefficient to be (2649.007 ± 145.577) m2/C, carrier density to be (2.356 ± 0.129) x 1015 m-2, mobility to be (1.224 ± 0.271) m2/(Vs), and a mean relaxation time of (4.657 ± 1.033) x 10-13 s.
Thus, it is possible that our sample was only degenerate in the inversion layer [2]. Sources of Error: [Geometrical Errors] Of the many sources of geometrical error in our experiment, most notably are the
Using the effective mass of electron mobile carriers derived from band theory, m* = 0.067m, the cyclotron orbit is estimated by: 𝑟𝑐 =
𝛾𝑚∗ 𝑣𝐹 𝑒𝐵
length and width measurements whose ratio was γ and the offset between pairs of fingers used in the measurement of VH. This offset was seen in our plots of VH vs. B-field (non-zero y-intercept), but was avoided by
Substituting the value for the highest B-field
only using the value of the slope in our
we used, 150mT, as well as the previously
calculation of the Hall coefficient.
7
[Thermal emf] Thermal emf is created by a voltage gradient resulting from a temperature gradient. For this reason, thermal emf exists even at zero current as it is currentindependent. Thermal emf can be eliminated or compensated by a variety of methods including but not limited to: waiting until thermal emf disappears as the sample temperature is equalized with its surrounding temperature, or reversing the current and averaging the absolute values (V+ is taken with current in one direction and V- is taken with current in the other direction, then V = (abs(V+) + abs(V-))/2). Either method eliminates thermal offsets. [Sample Position] The sample must be placed perpendicularly in the magnetic field for minimal systematic error on RH. By turning the probe, we witnessed the Hall Voltage change until we reached the maximum Hall Voltage, at which point our sample was relatively perpendicular to the applied field. However, it is unlikely that is our sample was perfectly aligned, so we estimate the mathematical effect of tilting to be cos(θmax), where θmax = 5o and cos(5o) = 0.996. We can correct for
sample positioning by multiplying our magnetic field values by 0.996. The error in sample positioning is negligible compared to the inherent instrumental error of B from the Gaussmeter (0.1 mT). Conclusion: In this laboratory report, we successfully observed the Hall Effect and measured many of the interesting microscopic parameters associated with this phenomenon. From plots of VH vs. B-field, coupled with γ, we calculated the Hall coefficient, the carrier density, sheet resistivity, mobility, and the mean relaxation time at both temperatures. The existence of only one type of carrier in our sample is concluded due to the lack of resistivity change in the B-field. At lower currents we saw some resistivity change in a varying B-field (unknown), however, in a high current regime, magnetoresistance was approximately zero as expected with little or no resistivity change. Furthermore, resulting from the Lorentz force, we use the right hand rule to discover that the type of carriers present in our sample are in fact electrons. We also conclude that an electron at either temperature (300 K or 77 K) could not complete a cyclotron orbit in
8
our maximum applied B-field. It is interesting to note that, in a pure sample, mobility is dependent on lattice scattering
References: [1] Michael Gurvitch, Hall Effect (23 September 2011).
by longitudinal acoustic phonons. This suggests a temperature dependence of μ ∝
[2] D.C. Elton and J. Chia-Yi, Hall Effect
T-3/2 [2]. Thus, it is reasonable that at
measurements of the carrier density and
lower temperatures (77 K vs. 300 K),
mobility of a 3D electron gas in a
mobility of the charged carriers is greater.
GaAs/AlGaAs heterostructure (4 May 2012).
On a side note, using the Fermi velocity and
[3]https://zumbuhllab.unibas.ch/pdf/teachi
mean relaxation time, we are able to
ng/SS06-
calculate the average distance between free
MesoDots/SS06_MesoDotsDMZ_chapter3v
carriers in our sample. The 2-D free-
2.pdf.
electron formula for vF is included in the appendix. Acknowledgements: We would like to thank Professor Aronson and both of the teacher’s assistants for their help throughout this laboratory experiment.
9
Appendix:
And the relative length of these electrons is
[Derivation of 2-D Free Electron Formula]
the Fermi wavelength, or the deBroglie’s wavelength, which is given by [3]:
In a degenerate Fermi-gas, kT
Hall Effect
semiconductor sample with high mobility
Michael S. Santana
2DEG, measurements of the resistance per
Partner: Andrew Kulesa
square (sheet resistance), the Hall
Department of Physics, Stony Brook University
coefficient, the carrier sign (electrons or
PHY 445/515
time between collisions), and the mean free
Submitted: May 6, 2015
path (mean distance traveled between
Abstract:
collisions) are made possible [1]. In this
Hall Effect was first observed in 1879 by the graduate student, Edwin Hall, from Johns Hopkins University. Hall Effect is noted when an electrical current in a semiconductor, with a two-dimensional electron gas, flows perpendicularly to an applied magnetic field resulting in a transverse voltage that is both perpendicular to the current and to the applied magnetic field. This voltage, otherwise known as the Hall Voltage, is measured with an appropriate four-probe arrangement. The primary significance of
holes), the carrier relaxation time (mean
experiment, we performed DC Hall Effect measurements at 300 K and 77 K (sample at room temperature, and sample cooled by Liquid Nitrogen) on the GaAs/GaAlAs heterostructures. We measured the Hall coefficient of GaAs to be RH = (2339.956 ± 128.065) m2/C at 300 K, and RH = (2649.006 ± 145.577) m2/C at 77 K. Similarly, we measured the sheet resistance of our GaAs/GaAlAs heterostructure to be (2164.814 ± 465.329) Ω/sq at 300 K… Introduction: [Carriers in E-field: Drude Model; mobility]
this experiment is in the determination of the carrier concentration (density), along
The Drude Model provides a simplified
with resistivity, which allows for the
description of electrical conduction in an
calculation of carrier mobility. Similarly,
electric field. Consider a macroscopic solid
when the Hall Effect is performed at
conductor with a sea of free electrons, each
relatively low temperatures on
traveling with an individual intrinsic velocity
semiconductors containing a two-
inside the conductor. Classical mechanics
dimensional electron gas (2DEG), i.e.
postulates that these intrinsic velocities are
2
thermal in nature, however, quantum
velocity) vd. The current density is given in
mechanics says that, due to the nature of
terms of the terminal velocity and the
the Pauli Exclusion Principle, the intrinsic
carrier concentration, represented by the
velocity of an electron in an electron gas
Drude relation:
within a conductor (metal) or semiconductor is close to that of an
𝑗 = 𝑒𝑛𝑣𝑑 =
electron in an electron gas at absolute zero temperature. In other words, the intrinsic
𝜌=
𝐸 𝜌
𝑚∗ 𝑒 2 𝑛𝜏
velocity of an electron in the electron gas is
Where ρ is the resistivity, m* is the
close to the Fermi velocity. In zero electric
modified mass of the electron as it
field, an electron’s trajectory is random
transverses the periodic potentials of a
allowing for the average of all electron
lattice, and τ is the carrier relaxation time
velocity vectors to yield an average of zero.
(mean time between collisions). In the GaAs
Thus, in a net zero electric field, the center
layer, the electronic effective mass is m* =
of mass of the electron gas is at rest.
0.067m. When studying conductivity
However, if we are to apply an electric field
(inverse of resistivity), it is interesting to
inside the conductor, the electron gas will
talk about the quantity of mobility. Mobility
start drifting in a direction that opposes the
is defined to be the magnitude of drift
direction of the applied field. In a constant
velocity per unit of electric field [1]. From
uniform electric field, we expect the
the above two relations, we attain a
velocity of the electron gas to increase
relation for mobility such that:
indefinitely with an acceleration proportional to the applied field strength, a = eE/m. This phenomena is not observed
𝜇=
𝑒𝜏 𝑚∗
[Two-dimensional conductors]
due to changes in electron trajectories between collisions, which are inelastic and
In two-dimensional conductors, electrons or
remove excess momentum and energy from
holes (electrons in our case) are confined in
the electron gas. Thus, the electrons attain
a quantum well such that the carrier is free
a constant finite velocity (drift or terminal
to move in two dimensions, but is quantized
3
in the third. We define sheet resistance to
The carrier relaxation time coupled with the
be R per sq.:
intrinsic velocity of a carrier (Fermi velocity 𝑚∗ 𝑅∎ = 2 𝑒 𝑛𝜏
or vF) yields the mean free path of the carrier, or, in other words, the mean
And j = E/R∎, a two-dimensional form of
distance a carrier travels between any two
Ohm’s Law. Establishing a relation between
arbitrary collisions. In an effort to compare
the previous formula and the “electrician’s”
microscopic parameters with their
Ohm’s Law gives V/I = (L/w) R∎. Since V/I
theoretical estimates, we will further
represents longitudinal resistance, and
compare our mean free path with a variety
(L/w) represents the number of squares in a
of meaningful length scales.
rectangular sample (between a pair of
Experimental Details:
leads), then: Displayed in the figure below are the 𝑅 = 𝛾𝑅∎
GaAs/GaAlAs heterostructures, with layers
Where γ, (L/w), need not be an integer
grown on a GaAs substrate. The top
number of squares. Since the Hall Effect
capping layer is essential to the top doping
formulas in two-dimensions disregard
layer, and is similarly important in regards
sample thickness, we establish that the Hall
to the protection of the structure as a
Voltage is given by:
whole. Layer three contains dopant atoms
𝑉𝐻 = 𝐼𝑅𝐻 𝐵𝑧
which are included in the GaAs lattice structure. Layer two is undoped, and serves
From the above relation, we can obtain the
as a spacer between the conducting
Hall coefficient in the case of a single
inversion and layer three (source of
carrier, and can likewise solve the following
dopants). This is an example of modulation
relation for the carrier density n:
doping, such that the dopants are
𝑅𝐻 =
1 𝑒𝑛
With RH and R∎, we can further solve for the mobility and the carrier relaxation time.
separated from the carriers in an effort to minimize carrier scattering and maximize or enhance carrier mobility.
4
The previous photograph allows for the calculation of the actual sizes and number of squares between a pair of voltage leads. Using the 100 micron scale, we can calculate γ, (L/w), where L = (552 ± 5) microns and w = (48 ± 5) microns, such that γ = 11.50 ± 0.40. Fig. 1. The sample’s layered structure (not to scale). Taken from [2].
Results:
In the above energy diagram for the
VH vs. B-field (Correction) at 300 K
conduction band in the n-type modulation
0 -0.0002 0
filled in triangle is the inversion layer, which
-0.0004
is representative of the EF state in the conduction band between layers one and
VH (V)
doped GaAs/GaAlAs heterostructure, the
0.15
-0.001
-0.0016 -0.0018
0.2
R-6 & B-4
-0.0008
-0.0014
well).
0.1
-0.0006
-0.0012
two (triangularly-shaped quantum potential
0.05
R-7 & B-3 y = -0.01x + 0.0001 R² = 0.9991
R-8 & B-2
y = -0.0106x + 0.0001 R² = 0.9982 y = -0.0106x + 0.0001 R² = 0.9982
B-field (T)
Fig. 3. The above plot shows the Hall Voltage’s dependency on the applied B-field at 300 K.
VH vs. B-field (Correction) at 77 K
pads (contacts) are lighter in color, and thin gold wires connect to the pads. Taken from [2].
VH (V)
Fig. 2. A microscopic image of our sample. The metal
0 -0.0002 0 -0.0004 -0.0006 -0.0008 -0.001 -0.0012 -0.0014 -0.0016 -0.0018 -0.002
0.05
0.1
0.15
0.2 R-6 & B-4 R-7 & B-3
y = -0.0116x + 0.0001 R² = 0.9981 y = -0.0098x + 0.0002 R² = 0.9974 y = -0.0124x + 0.0001 R² = 0.9987
B-field (T)
R-8 & B-2
5 Fig. 4. The above plot shows the Hall Voltage’s
Using the following relation:
dependency on the applied B-field at 77 K.
𝑉𝐻 = 𝐼𝑅𝐻 𝐵𝑧 Each plot was fitted with a linear regression. R-6 & B-4 (represented as red lead to pad 6 and the black lead to pad 4) are a pair of geometrically opposite fingers, as are R-7 & B-3 and R-8 & B-2. Each respective equation and it’s “goodness of fit,” R2, are included on the plots directly. When plotting VH as a function of applied Bfield, we know that the voltage measured is represented by:
We plotted Hall Voltage at a variety of applied magnetic fields, as shown above for 300 K and 77 K. When the applied magnetic field approached zero, we switched the leads, and thus the direction of the magnetic field, to measure the negative Bfield regime. The slopes of the above plots allow for the determination of the Hall coefficient and the variety of other microscopic parameters at 300 K and 77 K
𝑉𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 = 𝑎𝑉𝐻 + 𝑏𝑉𝑀𝑅
respectively, as discussed in the
Accounting for non-perfect geometry, the
introduction. It should be noted that the
voltage measured for VH is part Hall Voltage
plots of VH vs. B-field at both temperatures
and part voltage resulting from
have been corrected by using the procedure
Magnetoresistance. We can correct for
mentioned above.
MR’s unwanted contribution to the measured Hall Voltage by: 2𝑉𝐻 (𝐵) = 𝑉𝑚𝑒𝑎𝑠𝑢𝑟𝑒 (+𝐵) − 𝑉𝑚𝑒𝑎𝑠𝑢𝑟𝑒 (−𝐵) Similarly, when actively trying to measure Magnetoresistance across a pair of adjacent fingers, we will correct for unwanted Hall Effects by:
By extracting the slopes of each plot at both temperatures, we calculated the Hall coefficient, and coupled with γ and the fact that our mobile carriers are electrons (right hand rule), we are able to compare our microscopic parameters with some other meaningful length scales relating to our sample:
2𝑉𝑀𝑅 (𝐵) = 𝑉𝑚𝑒𝑎𝑠𝑢𝑟𝑒 (+𝐵) + 𝑉𝑚𝑒𝑎𝑠𝑢𝑟𝑒 (−𝐵)
6 300K RH 2339.95585 n 2.66736E+15 R per sq. 2164.81428 µ 1.080903739 t 4.11332E-13 EF 1.52851E-21 vF 223918.9476 mfp 9.21051E-08 λdB 4.85343E-08
77K 2649.006623 2.35617E+15 2164.81428 1.22366461 4.65659E-13 1.35018E-21 210452.0468 9.79989E-08 5.164E-08
Units m2/C m-2 Ω/sq. m2/(Vs) s J m/s m m
computed Fermi velocities into the cyclotron orbit equation yields: 𝑟𝑐 = 6532.85 𝑛𝑚 at 300 K 𝑟𝑐 = 6139.96 𝑛𝑚 at 77 K We know that an electron could not complete a cyclotron orbit at either 300 K
In light of the findings presented in the
or 77 K since c = 1,009,986 nm (c = πr2) in a
above table, at 300 K, we observed the Hall
150mT field. It is likewise important to note
coefficient to be (2339.956 ± 128.065)
that at room temperature (300 K), intrinsic
m2/C, carrier density to be (2.667 ± 0.146) x
semiconductors are typically non-
1015 m-2, the sheet resistivity to be
degenerate. If a semiconductor is to
(2164.814 ± 465.329) Ω/sq., mobility to be
become degenerate, the semiconductor
(1.081 ± 0.239) m2/(Vs), and a mean
begins to act more like a conductor due to
relaxation time of (4.113 ± 0.912) x 10-13 s.
the many electrons in the conduction band.
Similarly, at 77 K, we observed the Hall coefficient to be (2649.007 ± 145.577) m2/C, carrier density to be (2.356 ± 0.129) x 1015 m-2, mobility to be (1.224 ± 0.271) m2/(Vs), and a mean relaxation time of (4.657 ± 1.033) x 10-13 s.
Thus, it is possible that our sample was only degenerate in the inversion layer [2]. Sources of Error: [Geometrical Errors] Of the many sources of geometrical error in our experiment, most notably are the
Using the effective mass of electron mobile carriers derived from band theory, m* = 0.067m, the cyclotron orbit is estimated by: 𝑟𝑐 =
𝛾𝑚∗ 𝑣𝐹 𝑒𝐵
length and width measurements whose ratio was γ and the offset between pairs of fingers used in the measurement of VH. This offset was seen in our plots of VH vs. B-field (non-zero y-intercept), but was avoided by
Substituting the value for the highest B-field
only using the value of the slope in our
we used, 150mT, as well as the previously
calculation of the Hall coefficient.
7
[Thermal emf] Thermal emf is created by a voltage gradient resulting from a temperature gradient. For this reason, thermal emf exists even at zero current as it is currentindependent. Thermal emf can be eliminated or compensated by a variety of methods including but not limited to: waiting until thermal emf disappears as the sample temperature is equalized with its surrounding temperature, or reversing the current and averaging the absolute values (V+ is taken with current in one direction and V- is taken with current in the other direction, then V = (abs(V+) + abs(V-))/2). Either method eliminates thermal offsets. [Sample Position] The sample must be placed perpendicularly in the magnetic field for minimal systematic error on RH. By turning the probe, we witnessed the Hall Voltage change until we reached the maximum Hall Voltage, at which point our sample was relatively perpendicular to the applied field. However, it is unlikely that is our sample was perfectly aligned, so we estimate the mathematical effect of tilting to be cos(θmax), where θmax = 5o and cos(5o) = 0.996. We can correct for
sample positioning by multiplying our magnetic field values by 0.996. The error in sample positioning is negligible compared to the inherent instrumental error of B from the Gaussmeter (0.1 mT). Conclusion: In this laboratory report, we successfully observed the Hall Effect and measured many of the interesting microscopic parameters associated with this phenomenon. From plots of VH vs. B-field, coupled with γ, we calculated the Hall coefficient, the carrier density, sheet resistivity, mobility, and the mean relaxation time at both temperatures. The existence of only one type of carrier in our sample is concluded due to the lack of resistivity change in the B-field. At lower currents we saw some resistivity change in a varying B-field (unknown), however, in a high current regime, magnetoresistance was approximately zero as expected with little or no resistivity change. Furthermore, resulting from the Lorentz force, we use the right hand rule to discover that the type of carriers present in our sample are in fact electrons. We also conclude that an electron at either temperature (300 K or 77 K) could not complete a cyclotron orbit in
8
our maximum applied B-field. It is interesting to note that, in a pure sample, mobility is dependent on lattice scattering
References: [1] Michael Gurvitch, Hall Effect (23 September 2011).
by longitudinal acoustic phonons. This suggests a temperature dependence of μ ∝
[2] D.C. Elton and J. Chia-Yi, Hall Effect
T-3/2 [2]. Thus, it is reasonable that at
measurements of the carrier density and
lower temperatures (77 K vs. 300 K),
mobility of a 3D electron gas in a
mobility of the charged carriers is greater.
GaAs/AlGaAs heterostructure (4 May 2012).
On a side note, using the Fermi velocity and
[3]https://zumbuhllab.unibas.ch/pdf/teachi
mean relaxation time, we are able to
ng/SS06-
calculate the average distance between free
MesoDots/SS06_MesoDotsDMZ_chapter3v
carriers in our sample. The 2-D free-
2.pdf.
electron formula for vF is included in the appendix. Acknowledgements: We would like to thank Professor Aronson and both of the teacher’s assistants for their help throughout this laboratory experiment.
9
Appendix:
And the relative length of these electrons is
[Derivation of 2-D Free Electron Formula]
the Fermi wavelength, or the deBroglie’s wavelength, which is given by [3]:
In a degenerate Fermi-gas, kT
