UNIVERSITY OF CALIFORNIA, SANTA BARBARA

Department of Physics

Magneto-Trans~ort in the 2-Dimensional Electron Gas and

The ~ u a n t u m ~ a l l Effect

Condensed matter physics is both scientifically rich and technologically important, but there are few examples where the extreme precision of some measured parameter is the center of attraction. The Quantum Hall effect[l] is a strong counter example of this generalization, and the focus of these experiments and laboratory project. The physics of the Quantum Hall effect is scientifically important but has also become the basis of a resistance standard[2]. Paradoxically, it is the impurities and inhomogeneities, the ugly side of condensed matter physics, that allow the quantized Hall effect to express itself.

OBJECTIVES

I . Measure the room temperature sheet resistance of the 2-dimensional electron gas using the Hall bar and van der Pauw geometry's.

2. Measure the sheet resistance as a function of temperature from 300 K to 4.2 K.

3. Measure the Hall resistance and longitudinal resistance as a function of magnetic field at - 4.2 K.

Extract a measure of the 2-dimensional electron density from the low magnetic field, Hall effect.

Use the quantum oscillations in the Hall or longitudinal resistivity to determine the 2- dimensional electron density.

Measure quantized values of the Hall resistance and compare with theory.

Determine the electron mobility as a function of temperature from 4.2 K to 300 K.

4. Measure the temperature dependence of the longitudinal magneto-resistance from 4.2 K to -30K.

Determine the effective mass of electrons in the 2-dimensional electron gas.

I . K. von Klitzing, Rev. Mod. Phys., 58,519 (1986).

2 . B.N. Taylor, Physics Today, August 1989,23.

Determine the temperature dependence of the electron scattering rate in this 2- dimensional electron gas

5. Measure the magneto-resistance with the center contact of the "poor man's" Corbino geometry sample.

BACKGROUND

Introduction:

Electrical conductivity, or lack of it, is a defining property of insulators and metals. Generally speaking, metals exhibit a temperature dependent conductivity that increases as the temperature is lowered while insulators exhibit a conductivity that becomes vanishingly small at sufficiently low temperatures. This broad classification has, of course, many exceptions and caveats. Pure semiconductors behave like insulators but have the property that they can be made to behave like metals by introducing sufficient number of impurities.

The system we deal with here is a 2-dimensional electron gas which can be produced at the interface between two semiconducton, GaAs and AI $3 7As. The electrons are introduced into the semiconductor heterostructure by "doping the AI ,Ga7As, near the interface, with Si. This introduces an electron for every Si atom. But it is energetically favorable for the electrons to reside in the GaAs so they leave their Si donors and move into the GaAs. However they still feel the coulomb attraction to the positively charged Si donor on the AI ,Ga7As side of the interface. and assume a ground state in which they are bound to the interface by the coulomb attraction but can "skate" freely on the GaAs side of the interface. See Fig. I .

Fig. I Energy versus distance through the GaAs / AlGaAs interface. The donated electrons are attracted to the Si donors but reside on the GaAs side of the interface.

Si

The behavior of the conductivity of any system in the presence of a DC magnetic field. magneto-transport. is a powerful tool for characterizing electrical conduction. Often, magneto- transport can be modeled "classically" by assuming the electron motion can be described by "free" particles. Here, the only parameters that reflect the hidden quantum nature of the electron motion is the replacement of the mass by a tensor "effective mass" and the g-factor by an effective g-factor. The quantum Hall effect is an element of magneto-transport, where the quantum mechanical nature of the electron states expresses itself in a very forceful way. Here.

GaAs

we must go beyond the classical model and deal directly with the quantized motion of electrons in a magnetic field and its consequence for magneto-transport.

In the following we develop the features of magneto-transport and semiconductor physics needed to carry out the experiments and interpret their meaning. We will start with the free- electron model of electrical conductivity, with and without a magnetic field, ignoring quantum mechanics but introducing an effective mass. The quantized motion of an electron in a magnetic field and the Pauli exclusion principle will conspire to produce the quantum Hall effect, but only in the presence of imperfections, disorder and potential fluctuations.

"Classical" magneto-transport.

The simplest model of electrical transport, the Dmde model[3], assumes a gas of electrically charged particles that move under the influence of external forces from electric and magnetic fields. We admit at 6z = <v>.6t the outset that the charge may be positive or negative, the mass and the g-factor. which describes the spin splitting in a magnetic field may be described by tensors and that the charged particle may be scattered by imperfections in the lattice caused by impurities or defects. or by thermally induced disorder caused by the Fig. 2. The current passing through

motion of the ion cores at finite temperature. The an area, A.

overall material is neutral. Whatever the sign of the charged carriers, that can move in the system, it is neutralized by an equal amount of fixed charge of the opposite sign.

Conducrivip. In the absence of any external force the average current passing through surface " A of Fig. 2 . . will be zero. However, if there is a net average velocity. <v>, directed normal to the surface. a net charge, 6 q = e . (A . 6 z). n , will pass through the area, A, in a time

St. Here e is the charge on the carrier. n is the volume density of mobile charges and 62 is [he displacement produced by moving at a velocity <v> for a time St. The current I = %I& is given by I = (A.<v>)-e-n and the current density J = I/A = n.e.<v>.

To find the average velocity <v> normal to the area, A, we assume that the influence of an external electric field can be found from the following equation of motion for the average velocity:

3 . N. W. Ashcroft and N. D. Mermin, Solid State Physics, W.B. Saunders Co., Philadelphia. 1976, pp. 2- 15.

1 where m* is the effective mass for the carriers, assumed isotropic here, E is the applied electric field and r is a phenomenological elastic scattering time that describes how quickly the electron gas comes to a steady state when an DC electric field is turned on or off.

You can confirm that the average velocity <v> behaves as follows when a DC electric field E is turned off.

<v(t)> = E (er 1 m*),exp(-t 17) . (7)

r is a useful parameterization of the effect of random scattering of the charged carriers by defects, be they static, introduced by impurities or crystal defects, or dynamic, introduced by the thermal motion of the ion cores.

The factor (er I m*) = p is called the mobility and relates the average drift velocity to the applied DC electric field. The larger T, the longer the time between collisions, the longer it takes for the electron gas to reach steady state, but the higher the average drift velocity <v>. It is important to understand that the average drift velocity <v> is just that. The charged carriers are rushing "to and fro" due to thermal motion or due to combined effects of the Pauli exclusion principal and the uncertainty principal. But this internal motion has no net velocity or current. The simple model constructed here only describes a net average velocity.

The average current density then can be found as

and the conductivity, given by a = J / E ,

The conductivity is controlled by the carrier density, the scattering time and the effective mass. If the experiment is successful, beyond observing the quantum Hall effect, we will determine the carrier density, the scattering time and effective mass for the two dimensional electron gas in our semiconductor heterostructure..

Measurine the conductivitv.

To measure the conductivity of some material we define a sample geometry, pass a current and measure the voltage developed, or apply a voltage and measure the sample current. A piece of material can be fashioned in the shape of a "Hall bar" like that shown in Fig. 3. We have done so for this experiment.

If we pass a current, I, down the "Hall" bar the current density, current per unit area, will be

Fig. 3. Schematic of material fashioned as a "Hall" bar. Current I flows down the length of the bar. Volt meters are attached to the points indicated by VI , VZ. V3, and VJ. The separation between voltage probes is "h", the width of the "Hall" bar is "w" and the thickness is "t".

The electric field required to support this current density is

E = J l o , ( 6 )

and the voltage drop ( V I - V?) or (V3 - V4) is E . I .

If we assume that the material is homogeneous, that is to say the conductivity in the direction of the current flow is uniform across the thickness of the sample, we can deduce the conductivity from the voltage drops and the applied current.

/ \,

/ \ \

i I \

2 - Dimensional ! I Electron Gas (2-DEG)

Fig. 4. Electrical conduction is confined to a sheet of electrons that is only about 10 nm thick. The overall thickness of the material is irrelevant.

Anticipating the fact that the sample we are investigating is in fact a two-dimensional electron gas, we can define a 2-D version of the conductivity.

Here we have simply multiplied by the thickness "t" to remove the thickness from the problem.

The reason for doing this is the fact that the material we will cany out our experiments on is in fact not homogeneous across its thickness. In fact, the current flows in a very thin layer only about 10 nm thick. (See Figs. I and 4.) The transport is quite accurately described as two dimensional in the sense that the electron motion perpendicular to this very thin layer is "quantized" and at low temperatures and at low frequencies the only degrees of freedom that can be excited are the those described by the carriers moving in the plane of the sample. It is important to recognize this fact and ignore the thickness of the sample, which is irrelevant, and define a 2-D conductivity.

From equations (7) and (X), we can see that. whereas a," has the dimensions of mhos 1 meter (MKS units), the 2-D conductivity 020 has the units of mhos I square. Square? The factor (1 1

- w) is the number of "squares" of material between the voltage measuring contacts on the "Hall"

bar. To develop some intuition consider that if we w,ere to take a square of material and simply double its width and double its length but keep its thickness constant the conductance or resistance would not change and indeed we would still be measuring the resistance or conductance of a square. 2-D conductivity defined as mhos / square is a intrinsic variable describing the 2-dimensional electron gas.

Fig. 5. Two measurements of the "resistance" of an arbitrary shaped sample, with uniform thickness and isotropic conductivity in the plane, can be used to determine the conductivity of the material.

While the "Hall" bar is a clean measurement of the conductivity of the material, L.J. van der Pauw showed in 1958 [4] that the conductivity can be determined in an arbitrary shaped sample by making two measurements of the resistance as shown in Fig. 5.

For a slab of three dimensional material of thickness "t", he obtained for the resistivity. which is the reciprocal of the conductivity for a two dimensionally isotropic material.

P3D = o ~ D - ' 9

where RAB,m and R6c.n~ are defined in Fig. 5.

Fig. 6. The function f used to determine the resistivity or conductivity as a function of the resistance ratio Ras.cd RBC.DA

-

1 . L.J. van der Pauw. Phillips Research Reports, 13, 1 (1958).

The function f is shown in Fig 6. Note that the determination of the conductivity or resistivity is relatively insensitive to the ratio R A B . C ~ RBC.DA.

As we discussed for the "Hall" bar geometry, we recast equation (9) in the 2-D limit by dividing by "t" and obtain for the 2-D resistivity expressed as ohms 1 square and conductivity expressed as mhos / square.

The Hall effect.

With the application of a DC magnetic field the conductivity takes on new dimensions. A charged particle moving in a magnetic field experiences a Lorentz force given by e e x 6 , where k! is the electron velocity and 6the magnetic field. It is not surprising then that current flow in a material placed in a magnetic field will produce a force on the carriers perpendicular to their motion. We can use the Dmde model and the equations of motion used earlier to develop a simple phenomenology of the effect of a magnetic field. In the end we will find that the geometry of the sample also plays an important role.

We consider a z-directed magnetic field B and use equations of motion for the x and y directed motion of the electrons in our sample. We will assume at the outset that the electric field is impressed along the x-direction only. But, ultimately the current flow or lack of it in certain directions may induce internal electric fields that we must deal with. Equations describing the x and y directed average drift velocity are

a < - + , > < v > rn* +m** = e E , + e < v y > B , a n d

a t ( 1 1)

We are concerned here with DC transport, so we neglect the time derivatives in the first term of both equations and recover the average drift velocity in the x and y directions.

e r e B < v x > = -. E x + 7 . r < vy > and

m* rn

e B < v x > = + -.r < v >.

rn* Y (14)

Note that eB/m* is the cyclotron frequency, a, the rate at which a charged particle circles the magnetic field. Solving for the average drift velocities we obtain.

(e7 1 m*) Ex < v x i = and

( 1 + ( w , ? ) ~ )

(e7 1 m*) E,r < v > = - ( o 7 ) . Y C

( I + (ac?)')

The current density in the x and y directions is

(ne25/m*) E, J, = and

( I + (oCq2)

As we anticipated, in the presence of a magnetic field, an x-directed electric field will cause current to flow in both the x and y-directions. The conductivity is expressed as a tensor with components

- (ne25/rn*) bxX = on - and ( 1 + ( 0 ~ 7 ) ~ )

The relation between current density and electric field is given by

J , = ox, Ex + oxy Ey (21)

Jy = oyx Ex + ow E y . (22)

The "Hn[l" bar. If we apply these results to our "Hall" bar in Fig. 3., we note that there is no current flow in the y-direction. Jy = 0. This requires that

E, = - OYX -. Ex and OYY

1 electric field, .f, = o ..f, (21,22) is clearly not well suited to describe the Hall bar geometry. The experimental constraints are the current flows in the x and y-direction but the form of "ohms" law expressed by (21.22) suggests that we constrain the voltages or electric fields.

L By simply inverting (2 1.22) we can write .f = p . .? or

where

- -oxy Pxy - (32)

(onxoyy -0xyoyx) '

For the Hall bar geometry we simply note that J , is zero and recover (23-26). The Hall bar C

geometry directly measures the elements of the resistivity tensor, p

The Corbino geomer?. To directly measure the elements of the conductivity tensor we need a geometry that controls the electric field. The Corbino geometry does just that. (Fig. 7.)

Under these conditions there is no "Hall" voltage, Em = 0. The current density in the radial direction is

J , = I / (2m) and (33)

The resistance is

In(b/a) 1 R, = .-

2x ox,

Using (19) we find that the resistance of the Corbino geometry grows rapidly with increasing magnetic field.

Fig. 7. The Corbino geometry consists of concentric circular ohmic contacts. The current flows into the center and radially out to the edge. There can be no "Hall" voltage for t t

would exist in the azimuthal direction.

which should be compared with the longitudinal resistance of the Hall bar which is expected lo show no magneto-resistance. (Eq. (25))

The reason for the resistance is clear from (22). There is an ever increasing fraction of the current in the azimuthal direction

The current spirals out from the inside contact (or spirals in from the outside contact) and [he "effective length" of the current path increases.

For this project we have a "poor man's" Corbino geometry defined by a "blob of Indium solder in the middle of a square. The outside contact is simply four "blobs" of Indium on [he corners. Although this "poor man's" Corbino geometry is not amenable to quantitative analyw. i t does provide the correct topological constraints and the contrast between the behavior of the resistance of the "Corbino" and Hall geometry's in a magnetic field is striking.

Quantum Magneto-Transport

To understand the behavior of electrical transport in quantizing magnetic fields we first explore the quantum mechanical states of a charged particle in a magnetic field using classical mechanics as a guide. The classical equations of motion for an electron in a static magnetic field along the z-direction and static electric field along the y-direction are

(Pay attention to Ey, it will eventually play the role of our Hall field.)

A solution to (38 and 39) is

The carrier executes simple harmonic motion at the cyclotron frequency, o, = e . B / m* , while drifting with a velocity Ey /B in the x-direction. See Fig. 8.

Recognizing the harmonic motion about the magnetic field we leap to the quantum mechanical solution of the harmonic oscillator for our charged particle moving in a magnetic field. The eigenvalues will consist of a ladder of equally spaced levels given by

These levels are called Landau 1 E 1 -

levels and there are many eigenstates Y associated with each Landau level En. This degeneracy of states associated with each Landau level. recognizes the fact that we can specify the orbit center in a somewhat arbitrary way provided that it is inside the area of the sample. Fig. 8. An electron in crossed electric and magnetic

fields moves with a steady velocity normal to both However, there are only a finite while it executes simple harmonic motion at the

number of orthogonal eigenstates cyclotron frequency. associated with each Landau level. The number of states is N = A . B elh. where A is the area, B the field and h/e the quantum of

magnetic flux. Then the 2-dimensional density of allowed states, states that can be occupied by the carriers is

To develop some intuition, imagine that the area of the sample is covered with harmonic oscillator ground states which represent the circulating motion in a magnetic field. The ground state energy is % q@ . This energy represents zero point fluctuation of a particle which is localized in momentum and position by the magnetic field. We can estimate the physical size of the wave function by equating the kinetic energy and magnetic energy to the zero point energy.

The area of the ground state is estimated to be

The number of states allowed per unit area per Landau level is

Take this hand waving argument with more than a "grain of salt". For a more rigorous discussion of the problem read C. Kittel, "Quantum Theory of Solids", pp. 217ff [ S ] .

Associated with each Landau level we have two spin states, spin up and spin down. The eigenvalues then are expressed as

where m, is the spin quantum number +I- Yz, g is the g-factor and p~ is the Bohr magneton.

Figure 9 schematically shows the Landaulspin levels of a 2-dimensional electron gas in a quantizing magnetic field. What do we mean by quantizing? In the figure we have indicated some broadening of the density of allowed states that reflects the effect of potential fluctuations in a real sample. Negative trapped charge. positive trapped charge, stress, disorder will have the effect of locally raising or lowering, locally, the energy of our ladder of Landaulspin levels. Fig. 9. represents the entire sample and therefore the density of allowed states in each Landaulspin level will suffer some spreading due to these inhomogeneities. The system will be quantized

5 . C. Kittel, "Quantum Theory of Solids", John Wiley and Sons, New York, 1966.p~. 217-220.

15

\\\\\\\\\\\\\\\, L=& TP8~ L

\X~~WWS~NS~W~ Density of States Y p I ! Fig. 10. A cross section through the Hall bar,

cutting across the current flow showing - potential fluctuations causing the Landaulspin Density of Allowed States levels to move up and down as a function of

the y-position. At the edge the potential must Fig. 9. Density of allowed states per rise dramatically for the carriers are confined to area per energy interval in a quantizing the sample. magnetic field. The integrated number of states per area per level is n,.

only if the splitting between levels exceeds this level broadening or spreading. Since the Landau level splitting, urn, may exceed the spin splitting, g p ~ B , i t is possible that the system may be quantized vis-h-vis the cyclotron motion, the Landau levels, but not so for the spin levels.

The "real thing ": Inhomogeneities and potential fluctuations. The potential fluctuations that broaden the Landaulspin may be visualized in Fig. 10. Here we have schematically taken a cross section through the Hall bar of Fig. 3.. nomal to the current flow. At each position "y" an orbiting electron in a given Landaulspin level will experience the local potential fluctuations represented by the random rising and falling potential. In fact the distribution of the potential gives rise to the level broadening depicted in the left of Fig. 9 and Fig. 10. At the edge of the sample the potential must rise dramatically to represent the fact that the carriers are confined to the sample.

Lets imagine that we start to fill our sample with carriers. See Fig. I I . The first few carriers will find themselves stuck at the lowest potential minima and unable to carry any current. If we add a few more they must occupy orbits at higher potential, on the sides of the valleys. But these orbits experience an effective electric field given by grad (V(r)) and the guiding center for the orbit will follow equal energy contours. moving with a velocity given by (40). grad (V(r))/B. They will still be trapped, so to speak, in the valleys. When the carrier density has reached a level that covers more than M of the area the character of the trapped orbits changes from orbits circling the edge of a "lake" in a valley to orbits that circle the edge of a "mountain" rising out of the lake. The orbit will circulate in the opposite sense but will still be trapped. However at this point a class of orbits exists with guiding centers that move from the top to bottom contact by moving along the edge. The Landaulspin level has enough electrons to support an "edge channel" along the Hall bar.

I - - Density of Y

States

Fig. I I . As we fill the first Landadspin level with carriers the sample first looks like disconnected puddles " A then disconnected mountain tops "B". But in "B" with more than Yz of the level filled current can be canied along the edge.

As soon as we occupy more than % of a given Landaulspin level, an edge channel opens up, that can carry a current from one contact to the next along that side of the sample.

It is important to realize that at and around % filled the edges of the "lakes" will be tenuously connected to the edges of the "mountains" and current will be able to pass through the bulk of the hall bar. This marks a transition region. If the Fermi level is above this point the current can only pass along the edge.

The only question that remains is what is the Hall voltage produced by this "edge" current. Recall that the velocity of a charged particle in an electric and magnetic field is normal to both and given by equation (40).

Then the current density along the Hall bar, in the x-direction, will be a function of y (See Fig. 1 2 and given by

Density of Y States

Fig. 12. The fluctuating potential in the y-direction, in concert with the magnetic field, causes the orbit centers to move in the x-direction, along the Hall bar. But the velocity and current density will alternate assuming a magnitude and direction proportional to the derivative of the potential or effective electric field.

Recall that the carrier density in a Landaukpin level is no = BI(Ne) and the current densttv becomes

Here, e 2 / h is the quantum of conductance. (Yes, everything seems to be quantized thew days.)

The total current along the Hall bar is

If there is no voltage difference. (Vnght-Vlef,) = 0.. the net current is zero. The currents flowing "every which way" in the presence of the random interface potential and normal B field average to zero. It should be clear that those states at the edge that have energy above the V5 filled Fermi energy will constitute an "edge current" but the magnitude of the current on one side will be equal to the current on the other side maintaining zero net current.

If we pass a current through the sample, we see from (51) that a Hall voltage develops and the Hall resistance will be quantized

The net current, I, is represented by a greater "edge current" on one side of the Hall bar than the other.

If we continue filling our Hall bar with electrons such that "n" spinlLandau levels are occupied, that is to say, such that the Fermi level is placed half way between two levels, the nCh and ( n + ~ ) ' ~ Landaulspin levels, the net edge current, (right -left) will be n times greater.

and the Hall resistance is

Fig. 13. shows the quantum and classical Hall resistance as a function of electron density at fixed magnetic field. The transition regions between Hall plateaus have been broadened to simulate the effect of having a finite range of densities, around the '/2 filled points, for which the current can flow inside the Hall bar, that is to say where the current is not exclusively canied by the edge states. The relatively narrow plateaus occur when the Fermi energy, fixed by the electron density, is situated between two spin split states in the same Landau level. Since these may be relatively close together they exhibit only a weak effect and are rapidly extinguished as the temperature is raised. The broader plateaus correspond to the case where the Fermi energy is situated between two Landau levels. See Fig. 9.

The minima in the longitudinal resistance are striking and counter to one's intuition. They occur where the Hall resistance has plateaued indicating that the Fermi level is half way between Landaulspin levels and there should be no transport. Indeed, one would expect a,, = a, + 0.

The inversion of the conductivity tensor gives p X x = OYY and mathematically

("xx"yy - ~ x y O y x )

implies that in the Hall bar p,, -t 0. as well. However this mathematical argument does little to provide insight as to why the apparent resistance should drop in the Hall plateau. Reference to the edge state model provides a clue. In the Hall plateau region, transport can only occur via edges.

But the current carried at the edge involves carriers moving in only one direction and separated by the carriers moving the other direction by the width of the Hall bar. They are on the other side. Resistance caused by scattering in the backward direction is prohibited. Indeed at the lowest temperatures these minima are very small over a broad range of electron densities or magnetic fields.

m . - H 15000

2 Q u a n t u m RH . .

Electron Density, n 1 n 0

Fig. 13. A sketch of the Hall resistance as a function of electron density in fixed magnetic field. The density is normalized to the Landauispin level density no = B/(h/e). The narrow plateaus are produced by the spin splitting and the broader plateaus by the Landau level splitting. Also shown, is the expected dependence of the longitudinal resistivity (arbitrary units) exhibiting minima in the region of strong broad plateaus in the Hall resistance.

Please note: =, In the experiment, we will keep the density jixed and vary the magnetic field. But with imagination, the connection between Fig. 13 and the experimental results can be made.

What are the conditions needed to quantize the Hall effect? By inspecting Figures 9-12 we can develop the following intuition.

The separation of the Landau levels must exceed any broadening of the Landau levels caused by potential fluctuations. It is important to be able to place the Fermi energy such that i t is sufficiently below high lying states that they can not contribute to the transport. It must be sufficiently above any lower states so they contribute only edge state transport. The temperature must be sufficiently low that the Fermi level is well defined. If we are at a temperature T the Fermi level will be "blurred" by kT. If kT is comparable to the separation between Landaulspin levels then quantization will not be important and we will recover the "classical" result. More about thermal effects below.

Fermi Energy

P

Density of States Y

Fig. 14. At finite temperature holes will be found in the lower Landauispin levels and electrons found in the excited Landauispin levels. This will tend to reduce the minima in the longitudinal resistance. Since the Fermi level is known to be half way between Landaulspin levels we can use the temperature dependence and the Fermi function to extract a measure of the splitting. We must be conscious of the broadening for this will rend to reduce the effective splitting by reducing the thermal energy required to create holes and electrons that can contribute to the transport.

Temperature Dependence and the Electron Mass. Since the quantized Hall effect is sensitive to temperature. We need a well defined Fermi energy. Perhaps we can use the temperature dependence to determine the spacing of the Landaulspin levels and then the g-factor and mass m*. The feature that is easiest to measure and focus on is the minima in p,, . We expect that as we raise the temperature the minima will disappear and we should recover the classical value. Fig. 14 can be used to visualize the basic physics.

At the minimum in p,, the Fermi level lies half way between levels. At slightly elevated temperatures the number of excited electrons and holes, and the p,, minimum, will be proportional to

If we plot ln(p,,) versus 1fT we can extract ( w , - g p B ~ - F ~ ) . If we plot

(rp, - g p B ~ - ~ ~ ) v e r s u s B, the magnetic field, we should extract a combination of the

effective mass and g-factor.

Performing the same analysis on the spin splitting in a given Landau level can determine the g-factor alone. With this in hand, in principal, we can determine the effective mass, m*, the g- factor and FE, the line broadening. ( In the 2-0 system used in this experiment, the g- factor is quite small and we will not be able to measure the spin splim'ng. Then we can ignore it and extract a value form* and 6E. )

PROCEDURE

In this lab project you will be using a probe that has two samples with electrical leads. The samples are made from an Aluminum Gallium Arsenide semiconductor heterost~cture which has been "doped with Silicon to introduce free electrons or carriers that can carry a current. Remarkably, the caniers are confined to a very thin sheet, = 10 nm, so that the electrical conduction is "metallic" and rigorously 2-dimensional. These two samples are going to be used for a variety of magneto-transport experiments. Fig. 15 is a schematic which shows the way that these two samples of AlGaAs are wired to the probe. The bottom sample has the shape of a Hall Bar etched into it so that measurements of the voltage can be accurately measured both parallel and perpendicular to the direction of the current flow. The top sample will enable you to measure the resistivity with a van der Pauw approach, and also the effect of a magnetic field on the conductance measured through a centered contact, a contact completely surrounded by the 2- dimensional electron gas (2-DEG). The probe is connected to a BNC Patch Board via the cables that are attached to the top of it. The circuit diagram for the Patch Board with respect to the probe is represented by Fig. 16 and a complete explanation of how it works follows.

Using the Patch Board

The Patch Board is connected to the sample. You should look at the circuit diagram of the patch board and the wiring schematic for the sample before you do any of the experiments. The following are instructions for using the Patch Board.

The center pin on each BNC connector is connected to one point on one of the samples with the exception of "C" which is connected to two points, one on each sample. The switch above each BNC connector shorts the center pin to the shield of that connector and to ground. When the switch is in the up position the center pin and its shield are shorted together.

IMPORTANT: REMEMBER TO ALWAYS SHORT ALL OF THE CONNECTORS TO GROUND BY FLIPPING THE TOGGLE SWITCH TO THE UP POSITlON WHEN CHANGING THE CONNECTIONS. Doing this will ensure that you do not end up dumping charge into the sampie, adversely affecting your measurements. (The 2-dimensional electron gas is very fragile. If you inadvertently dump excess charge on -the sample by disconnecting and connecting the sample at low temperatures, the thin sheet of free carriers will become very inhomogeneous and the experiment will not be very satisfying. The homogeneous two dimensional gas can be recovered by warming to room temperature. The best approach is to connect all the instruments you wish to the appropriate connector with the shorting switches closed and then open the shorting switches. One shorting switch will remain connected.)

It will be necessary to pass current through the sample. The best way to do this is to connect the BNC cable from the current source to the female BNC connector on the board which is ro be the positive current terminal. Short the connector which is to be the negative terminal. For example, if you wanted current to pass through the sample from points "K" to "C", then you would connect the BNC cable to "K" and short "C" to ground. Do not short "K" or any other connectors that you do not want current to pass through.

KEY FOR THE WIRING SCHEMATIC

0 = COPPER LEADS ON THE END OF M E PROBE

= POINT WHERE GOLD WIRES ARE SOLDERED TO THE AffiaAs SAMPLE

I= GOLD WIRES WHICH CONNECT THE AlGaAs SAMPLE TO THE COPPER LEADS

A,B ... = CORRESPOND TO THE LETTERS ON THE PATCH BOARD

# = HALL BAR SCRIBED INTO THE GaAs SAMPLE; ELECTRONS CANNOT FLOW ACROSS THIS BOUNDARY

Fig. 15. Schematic which shows the wiring o f the AffiaAs samples to the probe.

CIRCUIT DIAGRAM FOR THE PATCH BOARD

HALL GEOMETRY

SWITCH DOWN SW rrCH UP

Fig. 16. Circuit diagram for the BNC patch board.

In order to measure the potential between two points on the sample, you will need two BNC cables, each connected to one of the female connectors which represent those points on the sample. It will be necessary to use BNC to banana plug adapters. The side of the banana plug with the tab on it is connected to the shield.

For every experiment in this lab you should put = 30pA of current through the samples and use the Keithley 227 Current Source.

Measure the resistance with a hand held multimeter at room temperature between all the points on the samples and compare them with the resistances in Appendix A. This is very important because shorts or bad connections will prevent you from making any further measurements. (If you come across any bad connections or shorts. please consult your instructor or Mr. Harding before continuing.)

Hall bar and van der Pauw measurement of resistance. Measure the sheet resistance of the samples using two methods, Hall bar and van der Pauw. The first method simply uses the fact that the electrical resistance of a 2-D system decreases linearly with length and increases linearly with width.

Pass current through the sample with the Hall geometry and measure the longitudinal voltage

(1) Using the dimensions of the Hall Bar, determine the sheet resistivity, ohmslsquare, of the sample. (See Fig. 17 for the sample dimensions.)

Now, use the van der Pauw approach to measure the sheet resistivity.

(2) Using the sample which has the Corbinolvan der Pauw geometry, determine the specific resistivity, p z ~ of the sample.

(3) Compare this result with that obtained using the Hall bar. Speculate on the sources of measured differences.

The Temperature Controller

Resis t iv i~ versus Temperature. For this part of the experiment you will need to become familiar with the T.R.I. Research T-2000 Cryo-Controller. This instrument is connected to two thermometers on the probe which allow you to monitor the temperature inside the sample space. The temperature controller can also be connected to an additional thermometer in the cryostat which enables you to monitor the temperature inside the liquid helium space.(Fig. 18) Finally, there is a heater connected to the probe which is controlled by the temperature controller allowing you to regulate the temperature in the sample space. For this part of the experiment it will onlv be necessary to monitor the temperature. The following will help you operate the temperature controller

Plug in the temperature controller.

PHYSICAL DIMENSIONS OF THE TWO SAMPLES

HALL GEOMETRY

1-.368 -+

Fig. 17. Physical dimensions of the AlGaAs samples. All values are in centmeters.

T .396

F

-

G

D CORBINO/VAN DER PAUW GEOMETRY

CRYO-STAT CROSS-SECTION

El

m

INSULATION JACKET

LIQUID HELIUM SPACE

MAGNET

SAMPLE SPACE

Fig. 18. Cross-sectional view of the cryostat.

Turn on the temperature controller, the switch is in the back.

The temperature controller has two usable thermometer outputs, one for a carbon-glass, and one for a platinum resistor. These outputs are #2 and #4 respectively.

The platinum resistor works well at high temperatures (room temperature to 30K) and the carbon-glass works well at low temperatures (30K-2K). The resistance of the materials which these thermometers are made from changes with temperature. Resistance verses temperature data. stored in its memory. enables the temperature controller to determine the temperature. See the data sheets in Appendix B for the temperature curves.

In order to select a thermometer; push <SELECT>, then the number for the thermometer; then <ENTER>.

One of the platinum resistance thermometers is in the liquid helium space. It is not connected to the temperature controller. It is only necessary to use this thermometer before transferring liquid helium in order to insure that no liquid nitrogen remains in the liquid helium space.

The other platinum resistance thermometer is in the sample space, its number is 4.

The carbon-glass thermometer is also in the sample space, its number is 2.

Recording R verses T during cooling down. The temperature controller puts out a voltage which represents the temperature that is in the LED display. This voltage changes linearly with temperature and is approximately 10 volts at 400K and 7.5 volts at room temperature. This will enable you to plot the resistance verses the temperature on the chart recorder as you cool the sample down from room temperature to 4.2K.

The voltage output on the temperature controller is on the back and is labeled option. Use this to drive the x-axis.

The cool down process takes about an hour. Leaving the chart recorder pen on the paper will cause the pen to bleed. Rather than leaving the pen down during the whole cool down process, simply tap the pen at appropriate intervals. Alternatively record the data by hand.

The thermometers used for this part of the experiment are the carbon-glass and platinum thermometers attached to the sample.

Set the chart recorder up so that i t is reading the longitudinal voltage of the sample with Hall Geometry and the temperature of the sample space.

(4) Record Vlongirudinal verses temperature as you cool down to 4.2K

(5) At the cross over temperature, where you will want to switch from the platinum resistor to the carbon glass resistor, you will note some uncertainty in the actual temperature. It is said.

"With one thermometer you know the temperature, with two you know you don't know it." Comments.

Cooline off the crvostat

Pre-cooling the cvostat. (It is not be necessary to perform this step if the cryostat is already cold (T < 8SK) from the previous day.)

Although you could cool the cryostat from room temperature all the way to 4.2K. the boiling point of liquid helium, it would consume excessive amounts of liquid helium which is fairly expensive. (= $S.OO/Iiter). Instead we will use liquid nitrogen (= $.25/ liter) to pre-cool the cryostat to 77 K, the boiling point of liquid nitrogen. Ideally you want to use just enough liquid nitrogen such that the final temperature inside the cryostat is 77K with no residual liquid-nitrogen remaining inside the cryostat.

WARNING: RESIDUAL WATER IN THE CRYOSTAT WILL FREEZE AND POTENTIALLY RUPTURE THE CRYOSTAT. YOU AND YOUR INSTRUCTOR SHOULD MAKE SURE THAT THE CRYOSTAT IS DRY BEFORE COOL DOWN.

WARNING: RESIDUAL LIQUID NITROGEN IN THE CRYOSTAT, WILL FREEZE ON THE APPARATUS WHEN HELIUM IS TRANSFERRED. THIS FROZEN NITROGEN INSULATES THE SUPERCONDUCTING MAGNET FROM THE HELIUM BATH. HEAT IS GENERATED IN THE MAGNET WHEN IT IS BEING RAMPED UP OR DOWN, THE MAGNET WILL WARM AND BECOME NORMAL. IT WILL BE DIFFICULT OR IMPOSSIBLE TO CHARGE THE MAGNET.

The following details the cool down.

Remove the plugs from ports I and 2 on top of the cryostat. From this point on, refer to Appendix C for a complete schematic the cryostat.

Insert the funnel all the way into port 1

WARNING: NEVER INSERT ANYTHING INTO PORT 2 EXCEPT THE HELIUM VENTING TUBE. THERE IS WIRING DIRECTLY BENEATH THIS PORT WHICH CAN BE DAMAGED IF ANYTHING LONGER THAN THE HELIUM VENTING TUBE IS INSERTED INTO IT.

Make sure that the safety blow off valve is in place before pouring in the liquid Nitrogen.

In order to cool the sample efficiently it is necessary for the probe to be thermally anchored to the sample chamber. To do this simply loosen the quick coupling nut and remove the spacer right above it. Carefully lower the probe until the clamp on the probe rests on this nut. Make sure you tighten the nut after this. This allows the bottom of the probe to touch the wall of the sample chamber which will temporarily be submersed in liquid nitrogen during the pre- cooling.

Carefully pour liquid nitrogen into the funnel and watch the temperature drop. Slow down when you get to 100K. Below l00K the temperature usually drops rapidly as liquid nitrogen is added. Slowing down will prevent excess liquid nitrogen from filling the cryostat after it reaches 77K. Reaching a static temperature of about =80K is sufficient.

If the temperature remains at 77K or gaseous nitrogen can be felt boiling off From port 2, then liquid nitrogen is present. Wait until the temperature starts to rise slightly, (80K). and there is no longer any nitrogen boil off.

It is also possible to check the temperature in the cryostat using the additional platinum resistance thermometer located in the liquid helium space. In order to do this, you will need to turn off the temperature controller. The plug for this thermometer comes out of the connector located next to the magnet current leads on the cryostat. Plug this into the #4 output on the temperature controller. Now when you select #4 you can read the temperature in the helium space in order to help determine if liquid nitrogen is still in the cryostat.

Once you are certain that liquid nitrogen is no longer in the cryostat. you are ready to begin transferring helium. See Mr. Harding or you instructor before you begin the helium transfer.

Remove the funnel from port 1, it may be necessary to use the heat gun to free the swage lock. Remove the swage lock and 0 ring from port 2.

Sample Space Preparation. Before beginning the helium transfer it is necessary to pump out the sample space and back fill it with helium gas. You should familiarize yourself with the valves on the gaseous helium tank and the cryostat so that you don't over pressurize the sample space after evacuating it, or let air into the sample space.

Connect the gas transfer tube to the sample space inlet valve.

Turn on the mechanical pump and lower the sample space pressure to about 300 millitorr.

Close off the valve connected to the pump and let helium gas into the sample space until it reaches atmospheric pressure.

Repeat this process a couple of times to ensure that there is mostly helium gas in the sample space and leave it at something in excess of 1 tom.

(6) Explain why is i t important to fill the sample space with helium.

Helium Measurement "73umping". Thumping is a way to measure the amount of helium in a dewar. The following describes how to thump.

Slowly insert the thumper tube into the helium storage dewar until it hits the bottom. THE SLOWER THE TUBE IS INSERTED THE BE'ITER. This allows the boil off to pre-cool the thumper tube, conserving valuable helium.

Place your thumb over the top of the thumper tube. You should feel vibrations. Slowly raise the tube from the bottom of the storage dewar. When the vibrations you feel first begin to double in frequency, the bottom of the tube is at the surface of the helium.

Compare the height of the liquid helium surface with the specifications on the storage dewar to determine the amount of helium in the storage dewar.

Helium Transfer. Before transferring helium read the following procedure carefully. It is a tricky process. which needs constant monitoring, leaving little time to refer back to this procedure.

Insert the 3-phase plug into the wall socket. Tum on the helium level indicator and switch i t to the continuos mode.

Initially the transfer will be made possible by the pressure of the boil off inside the stor;lge dewar. A helium transfer is best, if it is done very slowly. The way to gauge the speed of the transfer is to observe the rate of boil off from the blow-off port (port 2 on this cryo\taO. There will be a white plume of cold helium gas coming out of port 2. This plume should about 12- 18 inches high.

Gently insert the helium transfer line all the way into port I of the cryostat and tighten the swage lock.

Close the blow off valves on the storage dewar.

Slowly insert the other end of the transfer line into storage dewar until you begin to feel a slight boil off. Tighten swage lock.

When the transfer subsides, lower the transfer line a little bit more in order to continue the transfer.

Repeat this procedure until the transfer line hits the bottom of the storage dewar, maintaining a constant but slow transfer rate.

After this part of the transfer subsides attach the gaseous helium tank to the storage dewar blow off valve with the adapter and open this valve. Slowly open the gaseous helium tank valve until a gentle transfer is reached. You will need to monitor and adjust the pressure in order to maintain a gentle transfer.

Switch the temperature controller to the carbon-glass thermometer when the tempenturc reaches 40K. Stop the transfer when the helium level indicator reaches 90%. Release ihc

pressure in the storage dewar. Remove the transfer line from the storage dewar and then from port I .

Return all the valves on the storage dewar to their proper positions for storage.

Put one of the plugs into port I and put the helium venting tube into port 2. with the end dangling all the way to the floor. This will insure that air is not cryo pumped into the liquid helium space.

Loosen the quick coupling n u t just enough so that you can raise the probe up to replace the spacer without letting air into the sample space. The spacer insures that the sample is in the center of the solenoid where the strongest and most consistent magnetic field is located.

Switch the helium level indicator to the sample mode.

(7) Explain why switching to the sample mode reduces boil off.

Sample space temperature. After completing the helium transfer, the temperature in the sample space may not be 4.2K even though the sample space is submerged in liquid helium. In order to insure that the sample space is at the correct temperature, about 4.2 + 0.6 K, it is necessary to have the sample space tilled with helium and at the proper pressure. This ideal pressure is lower than atmospheric pressure, however, if the pressure is too low there will not be enough gas to conduct heat away from the sample. You will have to experiment with this to find the proper pressure. Make sure that you do not let air into the sample space. Moisture in the air will freeze in the sample space and on the sample and probe.

Magneto-resistance and the Ouantum Hall Effect

Magnet power supply. In this experiment an 8 Tesla super conducting magnet will be used. Current is provided by the Intermagnetics power supply located on the rack. The magnet is equipped with a persistent mode switch. The persistent mode switch is device that enables the magnet to continuously circulate current after the power supply is turned off. This switch (Fig. 19) consists of a piece of super conducting wire that shorts the two magnet power leads together. This wire passes through the center of a coiled heating element. When current is supplied to the heater the wire heats and becomes non-super conducting, or normal. In this state current passes through the magnet coils rather than the wire. When the heater current is turned off the wire becomes super conducting, shorting the magnet power leads, switching the magnet into the persistent mode. Likewise, if the heater is not turned on prior to ramping the magnet current power supply, no current will pass through the magnet. The key thing to realize is that when the persistent mode switch power is on, the switch itself is open and vice versa.

WARNING: NEVER PUT THE MAGNET INTO THE PERSISTENT MODE AND THEN RAMP DOWN THE CURRENT. This leaves the magnet with up to 75 Amps of current flowing through it. If the persistent mode switch is then closed while the magnet is in

PERSISTENT MODE SWITCH DIAGRAM

SUPER

NON-SUPER CONDUCTING WIRE WlRE I

CONDUCTING

MAGNET POWER SUPPLY

MODE SWITCH POWER SUPPLY

SUPER CONDUCTING MAGNET

Fig. 19. Diagram of the persistent mode switch.

this state, 75 Amps of current will instantly be sent to the power supply. This could damage both the magnet and the power supply. -

The following describes how to connect the persistent mode switch power supply and the magnet power supply.

Connect the QTL Constant Current Supply power to the persistent mode switch leads on the cryostat.

Set the current to 45 mA and turn it on, opening the persistent mode switch.

The magnet has a maximum current rating of 80 Amps. To insure that this limit is not exceeded, the magnet power supply should be set to 41.6% of full scale, (180 Amps), which is 75 Amps.

The power supply has a built in sensor that delivers a voltage proportional to the current supplied, 41.9 mV = 75 Amps. Use this voltage to drive the x-axis when you measure resistance versus magnetic field.

The magnet current power supply can be ramped up and down at various rates. These rates represent the time to full scale, not the time to the maximum set current. A good setting is 5 minutes to full scale. Slower than this can be too time consuming and faster should not be attempted because it can cause the magnet to quench. (Quenching is when the magnet coils are driven from the superconducting to normal state. With the magnet coils now having a resistance (about 50R). there is an immense power dissipation, P=12R. The heat created rapidly boils off the helium.)

When you are done for the day it is important that you turn of the power supplies in the following order and at these settings. The magnet power supply should be turned of first and should be ramped all the way down. Then leave the ramping switch in the hold position. After the magnet power supply is off, then turn off the persistent mode switch power supply.

Drive the x-axis of the recorder with the voltage presented on the front of the power supply that is proportional to the current.

7.5 Tesla = 75 Amps = 41.9 mV. Note that when ramping, all of the current does not flow through the magnet. The time rate of change of current across the superconducting coil produces a small voltage that drives a small current through the persistent mode switch even though it is normal. This will make the data weakly dependent on the ramping direction of the magnet.

(8) Measure the Hall voltage as a function of magnetic field to 7.5 Tesla. You have two choices of contact pairs to measure the Hall voltage. Use both. One displays cleaner steps in the

Hall resistance. See Appendix D for sample data. There is also sample data in the binder located in the lab.

, a (9) Measure the longitudinal resistance versus magnetic field to 7.5 Tesla. !' . ,

I :: (10) Determine the two-dimensional electron density from the low field Hall resistance.

(11) Determine the low temperature electron mobility.

(12) From the quantized Hall resistance, index the Hall plateaus by how many Landau levels are filled.

(13) Make a plot of index versus the inverse magnetic field at the longitudinal resistance minimum. Determine the electron density from this plot and compare with that determined from the low field Hall resistance.

(14) Return to the temperature dependent resistivity. Use the electron density to plot the electron mobility versus temperature. Why does it exhibit such a strong temperature dependence?

Determining the effective mass and Landau level width. Here you will use the temperature controller to raise the temperature of the sample to about 25K. By plotting the longitudinal magneto-resistance verses magnetic field at different temperatures, and measuring the temperature dependence of the p,, minimum at different magnetic fields it is possible to extract a measure of the electron effective mass m' and Landau level width 6E.

The temperature controller is able to regulate the temperature inside the sample space. By properly setting the gain and time constants for the system, it will maintain the set temperature while you take the necessary data. Because the sample space is submerged in liquid helium it is not poss~ble to raise the temperature much higher than 25K. While doing this part of the experiment it is a good idea to continuously pump on the sample space in order to reduce helium gas and thereby reduce the heat transfer between the "hot" sample and liquid helium in the cryostat. The following will explain how to initiate a control loop for the temperature controller.

Establishing a control loop: Press the 6 T N > key then the (MENU> key. Now the desired set point temperature can be entered. Example: <5.0> and then <ENTER>. Next you will need to select the thermometer that will be used to determine the temperature. #2, the carbon glass is appropriate for these temperatures. So. press <2> and then <ENTER>. Next you will need to select the gain. For this experiment select <175> and then <ENTER>. Now you will need to enter the integral time constant. For this experiment select <2.0> and then <ENTER>. Finally you will need to select a derivative time constant. For this experiment select <0.5> and then <ENTER>. Next i t will ask you to select alarm levels. this is not necessary and you can skip these by pressing <ENTERs when each one appears.

To stan a loop press <LOOP> and then <ENTER>

To change the set point while the loop is initiated press <FTN> then <ST PT> <NEW VALUE> and then <ENTER>.

To stop the loop press <LOOP> then <0> and then < E N T E b .

For a full description of the capabilities of the temperature controller consult the T-2000 Cryo Controller manual, sections (7.3-8.3).

On the temperature controller there is a heater voltage limit and a heater current limit. These dials read the percentage of full scale listed below each dial. The voltage should be set to 25% of full scale. The current should be set to 60% of full scale.

Note: The temperature controller may over shoot the set point you select. Don't wony about this it will quickly correct itself.

When you are done you will need to lower the temperature back down to 4.2k. First make sure that you have stopped the control loop. Next, while watching the pressure gauge slofil? bleed gaseous helium into the sample space never letting the pressure rise above 500 militorr. As you bleed helium gas into the sample space it will cool down and condense lowering the pressure again. This is called cryo-pumping.

(15) On the chart recorder plot the longitudinal resistance of the sample with Hall geometry verses the magnetic field at 4.2K and 5K,6K,7K,8K,gK...25K all on one sheet of paper. This may seem like a lot of graphs but i t goes quickly and will give you nice data. It may be helpful i f you switch the pen color for each successive graph.

(16) For the minima which occur at approximately 5T and 2.5T. calculate all Ap, and plot In(Ap,,) verses I n .

(17) Extract an activation energy from at least two different magnetic fields. Plot the

activation energy vs. magnetic field B and extract an effective mass for the electrons in this 2 - dimensional electron gas.

(18) From the intercept extract a measure of the line width for the Landau levels.

(19) Compare the line width to the level width you might estimate from the uncertainty principal 6E = rl(l/7).

Examining The Corbino Geometry. The sample on the probe which we used for the van der Pauw measurement can also be used to demonstrate the effects of the Corbino geometry on the conductivity as a function of increasing magnetic field. By passing current from the center contact, through the sample and out through the four contacts on the perimeter of our sample we can see the resistance become increasingly larger as the magnetic field is increased.

(20) Measure the resistance between the center contact and the four outer contacts as a function of magnetic field. Comment on the difference between the magneto-resistance measured this way and that extracted from the Hall bar.

OVER NIGHT SHUTDOWN PROCEDURE

In order to conserve helium and to keep the cryostat cool until the next day, use the following procedure.

Ramp the magnet current source all the way down and make sure it is at 0 by checking the voltage representation of the current on a multi-meter to make sure that it is ramped down all the way. It should read 0 volts. Then turn it off.

Leave the mechanical pump on and connected to the sample space. Pumping on the sample space will remove most of the transfer gas and slow down heat transfer to the liquid helium space.

Turn off the sample current supply, chart recorder, temperature controller, and helium level indicator. Remove the 3-phase plug from wall socket.

Only after you are sure that the magnet has no current supplied to it, can-the persistent mode switch power supply be turned off.

Close all valves on the gaseous helium tank,

Make sure that the helium venting tube is in port two and a plug is in port 1 on the cryostat.

7 . rl ,[,..A ! !!., , . . , , , ,

Acknowledgment

This laboratory project was engineered, assembled and proven by Amir Aho-Shaeer and Kevin Sparkman, UCSB 1996, (shown below) and Michael Wachner, UCSB 1995, (not shown) during the Spring and Fall of 1995 and Winter of 1996. Its successful completion is an impressive measure of their commitment, motivation and talent.

Appendix A Point To Point Resistance Specifications

For The AlGaAs Samples

TABLE OF ROOM TEMPERATURE RESISTANCES

A A XXX B XXX C XXX D XXX E XXX F XXX G XXX H XXX J XXX K XXX

XXX 5.0 7.9 XXX XXX 2.9 XXX XXX XXX XXX X X X XXX XXX XXX XXX XXX X X X XXX XXX XXX XXX XXX X X X XXX XXX XXX XXX

E 19.2 8.5 3.6 1.6 XXX XXX XXX XXX XXX XXX

F 19.6 8.7 3.7 2.0 3.2 XXX XXX XXX XXX XXX

. -. - - - - -

XXX XXX XXX XXX XXX XXX

J 15.7 13.5 13.6 16.5 17.1 17.3 17.9 15.8 XXX XXX

K 9.2 7.5 7.7 10.6 11.2 11.3 12.0 9.9 6.8 XXX

This table represents room temperature resistances between points on the sample with no magnetic field present.

Appendix I3

Resistance Verses Temperature Curves For the Thermometers

C R Y 0 CARBON STANDARD CURVE NO. 3

TEMPERATURE RESISTANCE TEMPERATURE RESISTANCE TEMPERATURE PSSISTANCS

I 1 I I I I I 1

CRY0 RESISTOR S/N STPT

POLYNOMIAL REGRESSION PROGRAM

POLYNOMIAL RANGE I S FROM ORDER 4 TO ORDER 11

POLYNOMIAL REGRESSION OF ORDER 4

REGRESSION COEFFICIENTS

A0 = - 7 . 7 0 2 7 1 2 7 9 A1 = 1 1 . 1 3 6 4 5 8 4 2 A2 = - 5 . 4 1 4 3 0 0 8 4

A3 = 1 . 3 1 9 6 8 5 5 2 A4 = - 1 . 1 2 4 2 1 1 4 1

TABLE OF RESIDUALS

T INPUT R INPUT R GENERATED RESIDUAL DELTA T

C R Y 0 R E S I S T O R S / N S T P T

POLYNOMIAL R E G R E S S I O N PROGRAM

POLYNOMIAL RANGE I S FROM ORDER 4 T O ORDER 11

POLYNOMIAL R E G R E S S I O N O F ORDER 4

R E G R E S S I O N C O E F F I C I E N T S

T A B L E O F R E S I D U A L S

T I N P U T R I N P U T R GENERATED R E S I D U A L DELTA T

Appendix C Schematics Of The Cryostat

These schematics were created by Rose Stuart-Curran using Auto-Cad Release 12.

SCHEMATIC: TOP OF CRYOSTAT

Probe Connect ions

Thecmo-Couple

Samp le Space

Mechanical

CRYOSTAT CONNECTIONS

Thermometer

SCHEMATIC OF CRYOSTAT INSERT

Top of Cryostat

Radiation Shield

Magnet Support Rod

Copper Magnet

Leads

Persistent Mode Switch

Super Conductin Magnet

I

SCHEMATIC OF CRYOSTAT INSERT i

o f Cryostot

Magnet support rod

Radiation Shields

I

SCHEMATIC OF CRYOSTAT INSERT ( BOTTOM )

Rad ia t ion Shie ld

Hel ium Transfer Line Guide

Super Conductna

J

-Magnet Leads

Super Conduct ing Magnet

CROSS-SECTION OF THE PROBE INSIDE THE MAGNET

Appendix D

Sample Data

Data ForTemoerature Dependence OfThe Longitudinal Voltaoe V* Verses Magnetic Field.

CURVE # COLOR

Blue Green Black Blue Green Black Blue Green Black Blue Green Black Blue Green Black Blue Green Black Blue Green Black