report quantised conduction tn2992

8/13/2019 Report Quantised Conduction TN2992

1/22

Quantised conductionTN2992

Joeri de Bruijckere, Thomas Horstink, Stevie-Ray Janssen and Martijn Schmeetz

TN2992: Experimenteel eindproject February 14, 2013


2/22

Contents

1 Introduction 1

2 Theory 22.1 Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1.1 Fermi-Dirac statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.1.2 Free electron gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2.1 Classical conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2.2 Quantum conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 2-Dimensional-Electron-Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Ballistic transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.5 Quantised conductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.6 Low Resistance Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Experiment 143.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2 Sample description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3 Measurement results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

i


3/22

1. IntroductionIn summary we have reported the rst measurements of the conductance of single ballistic point

contacts in a two-dimensional electron gas. A novel quantum effect is found: The conductance is quantised in units of e2/ [1] [2]. This quantum effect was discovered by accident at the TU Delftuniversity in 1988. Unexpectedly, plateaus were observed in the conductance. The step gain wasmeasured to be equal to integer multiples of e2/ .

The purpose of this report is to provide the reader with a theoretical basis concerning quantisedconductance and to provide the reader with experimental results of the performed reproduction of theexperiment mentioned above. This report is the result of minor students (listed on the front page)and is part of the TN2992 course at the TU Delft.

The report is structured as follows: The rst part governs descriptions of the theories behind thequantum conductance and some of the elements required for the experiment, including: fermion char-

acteristics, a classical and quantum mechanical approach on conduction, an explanation of 2DEG, therequirement for ballistic transport, an overview of the quantised conductance effect, concluding withsome theory on low resistance measuring. The second part governs the experimental result, includingsections dedicated to the setup, processing of the obtained results and a summary of conclusions of the reproduced experiment.

1


4/22

2. Theory2.1 Fermions

The charge carriers in the sample of our experiment are, like in almost every electrical circuit, electrons.The Standard Model of particle physics treats electrons as elementary particles , which means that theyare among the fundamental building blocks of our universe and cannot be divided into smaller particles.This idea is generally accepted among physicists; as the Standard Model in its entirety is. Still, manytheories have been proposed, advocating the existence of even smaller particles: preons . But so farnone have been found in experiments [3] and since the Standard Model has been so successful, preonmodels gain little interest nowadays.

The elementary particles of the Standard Model have the property of being completely identical toparticles of the same kind. You could, for example, not distinguish between two electrons; their DNA

is exactly the same. This holds consequences for describing systems that have multiple elementaryparticles of the same kind. It makes that the wave function of the system cannot simply be written asa product of multiple one-particle wave functions, because you could not tell which particle belongsto which wave function. For such a system of multiple identical particles (for simplicity, we considertwo: one in state a and one in state b), a wave function has to be constructed as:

(r 1, r 2) = a (r 1)b(r 2) b(r 1)a (r 2) (2.1)

where the vectors r 1 and r 2 represent position coordinates of the particles. The normalisation constantis left out, for its little importance in this explanation. It can be seen that there are two possible wavefunctions for this system of identical particles: one goes with a plus sign, in which case we call itsparticles bosons and the other goes with a minus sign, for which we call its particles fermions .Now, we suppose that the two identical particles occupy the same state, say a . For bosons, the twoproducts in Equation (2.1) get simply added. But, for fermions something peculiar happens:

(r 1, r 2) = a (r 1)a (r 2) a (r 1)a (r 2) = 0 .

The wave function reduces to zero! Therefore, the probability of nding two identical fermions in thesame state, is zero. This remarkable conclusion is a fundamental principle in quantum mechanics andis known as the Pauli exclusion principle . It holds only for identical fermions. As we mentioned, thewave function for a system of identical bosons just adds the two products of wave functions, so theseparticles are not forbidden to occupy the same state and do not obey the exclusion principle.

It happens to be that electrons are fermions, so they should also obey the Pauli exclusion principle.In atoms, this is what prevents the electrons from decaying all into their ground states, which woulddecrease the diversity in all our atoms and would make a very dull universe. By the exclusion principle,the electrons in our sample should all be in different states, having different combinations of spin andmomenta. The energies are therefore varying from electron to electron. A particular branch of physics,called Fermi-Dirac statistics , describes these energies of single particles, for a system of multipleidentical fermions. It can be used to determine the probability for fermions to be in a state with acertain energy.

2.1.1 Fermi-Dirac statistics

In a system that contains numerous electrons, like our sample, a lot of different possible states areoccupied by the electrons. For large numbers of particles, we can use statistical mechanics to describethe system. Fermi-Dirac statistics are concerned with the energy states within a system of single

2


5/22

particles, that obey the Pauli exclusion principle. It describes how the electrons (or more generally:the identical fermions) occupy the different energy states. Fermi-Dirac statistics is only applicableto fermions. Bosons, particles that do not obey the Pauli exclusion principle, are described by asimilar kind of statistics, called Bose-Einstein statistics . But, since we are dealing exclusively withelectrons, we do not need Bose-Einstein statistics.

According to Fermi-Dirac statistics the expected fraction of electrons n with a certain energy E isgiven by the equation:

n(E ) = 1

e(E )/k B T + 1 (2.2)

where is the chemical potential, kB is the Boltzmann constant and T is the temperature. Thisequation is known as the Fermi-Dirac distribution . At zero temperature this distribution has arectangular shape, as in Figure 2.1 (a), which means that all energy states are occupied up to acertain energy: the Fermi energy , indicated as E F in Figure 2.1. The Fermi energy is the energycorresponding to the highest occupied state at absolute zero . When the temperature increases, theshape of the distribution becomes smoother, as in Figure 2.1 (b). The Fermi energy is then no longerthe highest energy in the system. The chemical potential is also called the Fermi level . This is theenergy for which the occupation number is half the total number of particles. Close to absolute zerothe Fermi level is equal to the Fermi energy and the terms get therefore often mixed up. When thetemperature rises however, the Fermi level increases and the terms become less equal.

Figure 2.1: Fermi-Dirac distribution at absolute zero (a) and above absolute zero (b)

Close to absolute zero temperature only the energy states around the Fermi energy are partiallyoccupied. The states at lower energies are completely lled, so no electrical conduction can take placeat these levels. That means that at very low temperatures, only the electrons having an energy closeto the Fermi energy can be conducting. This is an important aspect in the understanding of quantisedconductance and we will use this fact later on.

2.1.2 Free electron gas

The electrons in our sample can be treated by approximation as particles in a box, where they arenot subjected to any forces. This idealised model is known as the free electron gas . Because of the heterostructure, the electrons in our sample are conned in one direction, so we actually have atwo-dimensional free electron gas. If we suppose that the electrons are conned in the z-direction andthe dimensions of the sample are of lengths lx and ly , the potential V (x, y ) for this system is:

V (x, y) = 0 if 0 < x < l x and 0 < y < l y otherwise (2.3)

Solving the Schr odinger equation for this system leads to wave functions in the following form:

n x n y = sinnx lx

x sinnyly

y (2.4)

3


6/22

where nx = 1 , 2, 3,... and ny = 1 , 2, 3,... are positive integers, each combination representing a sta-tionary state. That leaves us with a set of discrete stationary states. For simplicity the normalisationconstant is omitted. Now, we introduce a vector that characterises the direction and magnitude of anelectrons momentum: the wave vector , k (kx , ky). Its magnitude is related to the wavelength byk 2/ and to the momentums magnitude by k p/ .

One way of representing the stationary states of this system is by drawing them in a k-space asintersection points of the lines that satisfy kx = ( /l x ), (2/l x ),... and ky = ( /l y), (2/l y),... Theintersection points representing states of equal energies lie on a circle in this k-space. The outermostcircle, where the energies equal the Fermi energy E F , is known as the Fermi circle , drawn in Figure2.2. For all points on the Fermi circle the magnitude of the wave vector k = kF .

Figure 2.2: Fermi circle in k-space with the possible stationary states indicated as dots.

Actually, Figure 2 is a simplication of a k-space where we cannot draw a smooth circle through thedots (stationary states) with the highest energies. The resolution of the discrete points is too lowand the drawn Fermi circle does not make a lot of sense in this case. In reality, the number of dotsis a large number in the order of Avogadros number. Through such a large number of dots, itis possible to draw a very smooth Fermi circle, like the one in Figure 2.2. The circle represents theline between occupied and unoccupied states at absolute zero. The analogue for a three-dimensionalsystem would be the Fermi surface , for the states at the Fermi energy would lie on a sphere.

2.2 Conduction

Electrical conductance is a quantity which denes a materials ability to conduct an electric current,often noted by the Greek letter . It is the reciprocal quantity of the better known quantity resistance(). Conduction is expressed in the SI-unit siemens (S) and S = AV . Conductance has in the past

been described with classical mechanics and has correctly described certain effects like the Hall effect .But it also had its shortcomings, which lead to redescribing the phenomenon of electric conductionusing quantum mechanics.

We will use conduction in a metallic solid to describe the conductivity, which is essentially the move-ment of electrons in the solid. A metal in solid state arranges the atoms in a lattice where outer-shellelectrons ( valence electrons ) can freely move between the atoms. We say that a free electron gas ispresent in the solid, which enables the electrons to move within the solid. When a potential differ-ence is applied over the solid, the electric eld which is created by the potential difference forces theelectrons to move in the direction of the positive potential, creating a current.

2.2.1 Classical conduction

The rst description of the behaviour of the free electrons in a metal lattice was described by theDrude Model , proposed in 1900 by Paul Drude. Drude assumed the free electrons to act as a gas

4


7/22

within the lattice, so electrons move fast in random directions. With this assumption it is logical thatthe electrons often collide with the positively charged atoms (ions) in the lattice and other electronsmoving through the solid. Keep in mind that these electrons move this way, even if there is no potentialdifference applied. It is part of Drudes assumption that the electrons behave as a gas. The speedthe electrons move with within the lattice is the thermal velocity . In Drudes model the collisionsare considered isotropic so the kinetic energy equals the thermal energy which is solely depended on

temperature T . The thermal energy of an electron can be derived by assuming that it can move freelyin 3 dimensions, as U t = 32 kbT , equating this with the kinetic energy of an electron we get:

12

mev2 = 32

kbT (2.5)

v = 3kbT m e (2.6)Where kb is the Boltzman constant , T is the temperature and m e is the mass of an electron.

But now suppose we do apply that potential difference over the metal. The electrons will be affected bya force pointing to the positive potential due the electric eld induced by the potential difference. This

force accelerates the electrons in the opposite direction of the eld, while their free motion propertiesstay intact. Now every time an electron collides with something, the electric elds resulting forcepushes the electron in the opposite direction of the electric eld. What we observe is a so calledpinball effect where the electrons do move in the direction of the force but still interact with the ionsand other electrons in the form of collisions. The net result of this pinball machine is movement of the electrons (along quite a bumpy road) in the opposite direction of the electric eld with a smallvelocity, called the drift velocity vd.

The electric current can now be described in two ways; using Ohms law, but also using the newlyacquired vd . Remembering that electrical current is nothing more than the amount of charge thatcrosses an area per unit of time, we can say that:

I = V R

= Qt

= neAv d (2.7)

Where n is the amount of electrons, e is the elementary charge, A is the cross-sectional area. Anotheraspect that can be used to describe conduction is the mean free path , , which is the average length of the path an electron can cover before it collides with an ion in the lattice. Assuming the magnitudeof the electrons velocity equals roughly the thermal velocity, we can say that = v where is theaverage time between two collisions. We should note that the probability of an electron having acollision in a time interval dt equals dt [4] and is independent of speed and momentum but does relyon r , the radius of the ions in the lattice.

Drudes model was a nice beginning but it had some fundamental aws, varying from the descriptionof heat conduction, to the actual electric conduction and its dependency on temperature. By the1930s quantum mechanics was used to adjust Drudes model to the free electron model, sometimesreferred to as the Drude-Sommerfeld model .

2.2.2 Quantum conduction

The Drude model left out certain aspects that were not known at the time; most importantly the wave-characteristics that particles have according to quantum mechanics, but also the quantised energiesthat electrons have when roaming freely through the lattice. To describe the behaviour of electronsin a free electron model using quantum mechanics, we say that electrons are in a potential well with

innite barriers; the electrons do not leave the metal and are free to roam inside[5].Now note that the electrons possible energy levels are quantised and well dened. They are the integermultiples of the ground state E 0. Drudes model states that only the valence electrons participated in

5


8/22

conduction. This remains the same in Sommerfelds model, except that these free electrons do not havethe average thermal energy 32 kbT but have several distinctive energy levels. Some of these electronsare at the highest energy level, the fermi energy E F . The electrons with energy E F participate inconduction, because these electrons are the only ones capable of jumping to the next empty energylevel. Suppose the kinetic energy of lattice ions equals kbT . The maximum energy an electron cangain from a collision with such lattice ion is kbT ! Only the electrons within kbT of the Fermi energy

can move to the next state. Since not all the electrons can occupy the high energy levels due to thePauli exclusion principle there are only a few electrons that contribute to the conduction.

So only a few electrons, with a specic energy E F , participate in the electric conduction. Now insteadof using the thermal velocity (a result of the thermal energy ) we use E F . Equating E F to the kineticenergy we gain the Fermi velocity:

vF = 2E F me (2.8)This change is fundamentally different from Drudes assumptions. The electron energy and velocityare not as dependent on T as they were before! This gets rid of a lot of inconsistencies between theDrude model and experimental results regarding electron velocity and temperature.

Another important aspect when viewing electrical conduction with quantum mechanics is that thecollisions between an electron and lattice ion are no longer considered as if two balls bounce off eachother. We see the electron as a travelling wave through the lattice. Now it is logical that if theelectrons wavelength is longer then the space between two ions in the lattice, the electron can movefreely without colliding with the ions. Now, if the lattice contains impurities this will interfere withthe electron thus adjusting the mean free path , . But these impurities rise from the thermal vibrationof the ion, so is also no longer dependent on the radius of the ion but rather on the radius of itsthermal vibrations, which logically is depended of T . This corrects the relation problems between and T which were present in the Drude model.

2.3 2-Dimensional-Electron-Gas

In order to study quantum conductance, a 2-Dimensional-Electron-Gas (2DEG) was used. A 2DEGis a gas of electrons in which the electrons can only move in 2 directions. Movement in the thirddimension is restricted to certain quantised energy levels, which allows motion in that direction to beignored. This 2DEG can then be used to represent a wire of which the conductance can be investigated.[6] [7] [8] [9] [10] [11]

To create a 2DEG, a heterostructure of semiconductors was used. The rst is an n-type AlGaAsand the other is intrinsic GaAs. An n-type semiconductor is a semiconductor that has been doped.Doping is the addition of impurities (e.g. atoms of a different material than the semiconductor ismade of) to the semiconductor crystal structure. There are two types of doped semiconductors:

The p-type, in which atoms with fewer electrons than atoms in the semiconductor materialare added to the crystal structure. This creates holes which can move through the structureand potentially create electric current.

The n-type, in which atoms with more electrons than atoms in the semiconductor materialare added to the crystal structure. The semiconductor then contains excess electrons which canfreely move within the material.

An intrinsic semiconductor on the other hand, is a semiconductor made up of a pure element orcompound. So it contains no impurities whatsoever.

In metals, electrons can change energy levels quite easily, because of the low energy costs. They havevery good conductivity, because there are numerous available energy states at the Fermi level. Insemiconductors, the electrons can only have energy levels in so called energy bands. These energy

6


9/22

bands are separated by so called gaps. Such a gap is a measure for the energy needed to jumpto a higher energy band. The energy bands are being lled from the bottom following the Pauli-exclusion principle, which states that no two quantum-fermions can occupy the same quantum statesimultaneously.

Figure 2.3: The difference between the energy bands of an intrinsic semiconductor and an n-typesemiconductor. The n-type semiconductor has an additional dopand band, containing the excess freeelectrons [6].

The bands are occupied up to the valence band. The band above the valence band is called theconduction band and these bands are separated by a gap. The conduction band only contains veryfew free electrons for semiconductors under normal conditions. The band gap determines how muchenergy it takes for an electron to move from the valence band to the conduction band.

The two different semiconductors have different sized energy bands (see gure 2.3). The n-typesemiconductor also has a dopand band, which contains the excess electrons. These electrons can moveto the conduction band more easily.

The semiconductor n-AlGaAs has a larger difference between the valence energy and the conductionenergy than i-GaAs (see gure 2.4). The Fermi energy of n-AlGaAs is higher than the conductionenergy of i-GaAs. When they are brought together in a heterostructure, the excess electrons of n-AlGaAs can move very easily from their dopand band to the conduction band of i-GaAs, since theyhave a higher energy.

This charge transfer causes the conduction band and valence band to line up. This shifts the valenceand conduction energy levels. All the free electrons from n-AlGaAs get trapped in a well at the junction between n-AlGaAs and i-GaAs, formed by the shift in energy levels. This well is limited by

the new Fermi energy. The energy needed for these electrons needed to escape is too high.These electrons are now trapped in the well and can only move in the plain where n-AlGaAs andi-GaAs touch. This is called a 2-dimensional-electrons-gas.

2.4 Ballistic transport

If we want to explore the true nature of electrical conduction we have to look at the behaviour of electrons without interfering too much with a systems imperfections. As we now know, the mean freepath of an electron in a crystal lattice is often ( always ) very small and it becomes rather difficult toobserve the electron solely on that path. This makes measuring the pure conductivity pretty clumsilybecause it is near impossible to correct the conductivity for all the systems imperfections. Withballistic conduction we describe the transport of electrons in a medium with negligible resistance dueto scattering. Also, it is a very interesting question from an elementary point of view; what exactly

7


10/22

Figure 2.4: A heterostructure of semiconductors with their valence and conduction bands before (a)and after (b) charge transfer. At the junction of the two semiconductors a 2DEG is formed ( E c is theconduction energy, E f the Fermi energy, and E v the valence energy) [7].

drives resistance if there is no scattering? Is it zero like in a superconductor? Is there some elementaryvalue for conduction?

In order to observe ballistic transport we must try to make the electrons mean free path long enoughso that it is longer then the system through which the electron propagates. In this way, the electronsmotion is constant and is not subjected to alternations due to impurities. Keep in mind that normally,especially at temperatures well above the absolute zero, scattering due to impurities dominates themotion behaviour of the electron.

In order to achieve the long mean free path we must alter the factors that shorten , the scatteringeffects. For there are multiple sources for different scattering events, we must rst gure out whichare applicable to the electrons in our 2DEG . Then we can use the Matthiesen rule for adding up thescattering events to get a proper total scattering value. Matthiesens rule says: 1 / = i 1/ i wherei indicates all the different scattering events. Note that this formula does not really represent the

real relationship of the mean free path, because many individual scattering events effect each other.Scattering could change the velocity, which alters the impact of other scattering events. Matthiesensrule is a nice guideline though.

The fact that the electrons are inclined in a two dimensional space alters the scattering possibilities[12]; The most dominant scattering sources are the so called ionised impurities of the AlGaAs/GaAs heterostructure, but also phonon scattering and interface impurities take their part into describingthe mean free path. Many of these scattering sources have different ways to be treated; ionised andinterface impurities decrease if the AlGaAs and GaAs donors are further removed from the 2DEG -but this also decreases the electron density and therefore also the electron mobility [12]. The electronmobility characterises how quickly an electron can move when pulled by an electron eld.

Interface impurities are the deviations from a perfect crystal lattice. These deviations account forscattering as well. This value is usually very small [12], because the crystal structures of AlGaAs andGaAs t quite neatly on each other, from a lattice point of view. Phonon scattering characterisesthe scattering which is induced by the vibrational motion of the lattice. There are different types of

8


11/22

phonon scattering, but for now we will only use the fact that they are all temperature dependent andmore or less negatively affect .

Some of these scattering phenomena can be changed by altering the systems geometry, some canbe changed by using purer materials; for instance, in the AlGaAs certain Ga atoms are replacedwith Al atoms. This is done in a random order which impuries the lattice, thus creates scatteringevents. The lattice vibrations, caused by thermal vibrations for instance, can be reduced by coolingthe system down, resulting in less scattering. Applying techniques to decrease the amount of possiblescattering events in a system increases the mean free path, giving a longer distance over which electricalconductance can take place without scattering. This is exactly what has been done with the 2DEGsamples used for ballistic transport.

Also remember that, by having ballistic transport, with no interaction between the systems mediumand the electron, it is possible to account for quantum mechanic effects of conduction; normally scat-tering interaction with the system by colliding entangles the electrons system with the mediumssystem and this destroys the observable quantum effects!

Now how long should this be in order to observe quantised conductance in our sample? The point contact created with the 2DEG and electron depletion creates a gate with a width, say W and a lengthof L. In order for the electron to propagate between the gates freely without scattering, the meanfree path, should be much larger than L. If this is the case, the contact is known as a Sharvin point contact , which allows for quantised conduction.

2.5 Quantised conductance

In Section 2.1.1 we mentioned that conduction is mainly caused by electrons with the Fermi energy.At low temperatures the electrons with the Fermi energy are the electrons with the highest energies.High energies correspond to short wavelengths and thus do the electrons with the Fermi energy have

the shortest wavelengths. For conduction to take place, the conducting material should be wider thanthis wavelength. Otherwise, you could say that the electrons would not t through the material.This becomes clearer when you imagine the electron to be a wave. Only when the connementis wide enough for a standing wave to t in (see Figure 2.5), electrons will go through. Such aconnement could only take place at nano-scale, for the wavelengths of electrons are in this order of magnitude.

Figure 2.5: Connement of width W with the rst standing wave to t in.

The standing wave in Figure 2.5 represents the lowest energy state to t through a connement of width W , having a wavelength of = 2W . So only when the width becomes comparable to W F / 2where F is the wavelength for electrons with the Fermi energy, conduction becomes possible. Thiswould result in a step when we would measure the conduction as a function of W ; a step from zeroconduction for W < F / 2 to nonzero conduction for W > F / 2.

If we would increase the width even further, we would measure more steps. The conduction increasesstepwise with increasing width. This is due to the fact that for every certain interval of W an extrastationary state becomes able to pass through the connement, which results in a contribution to thetotal conduction.

9


12/22

The stepwise behaviour can be made clear with the Fermi circle, Figure 2.2. The stationary states withthe Fermi energy are discrete points on the circle, which means that there is no continuous transitionbetween the various states on the circle. If the connement is wide enough for the rst electrons topass, then these electrons would be in the stationary state with their momentum parallel directedalong the passage. Obviously the electrons directed along the passage, pass most easily. For electronsthat are angled away from the direction of the passage, a wider connement is required for them to

pass and take part in the conduction. Each subsequent state on the Fermi circle requires a widerconnement and because the states are discrete, the conduction with increasing width will increasein steps rather than linear. It happens to be that each state contributes to the conduction with anamount of e2/ where e represents the elementary charge and the reduced Planck constant. Thequantised conduction for N propagating states can therefore be written as:

G =N

n =1

e2

(2.9)

This formula is conrmed by experiments, which have resulted in patterns like Figure 2.6. Here, theconductance is not plotted as a function of the width, but as a function of gate voltage, which hasbasically the same effect. The gate voltage depletes electrons below it and because of its geometry apassage at nano-scale can be created; a quantum point contact. With decreasing negative gate voltagethe electrons get less depleted and the passage becomes wider.

Figure 2.6: Conductance as a function of gate voltage in a 2DEG at 0.6 K [1].

Quantised conductance only occurs at very low temperatures. The reason for this is that ballistic

transport can take place at these temperatures. By this, the electrons in the propagating modes canpass unaffected through the connement, i.e. with zero resistance. Despite this, the conduction in thesystem does not become innite. This is due to the fact that only a small number of modes is capableof passing through the connement and thus only a small fraction of electrons can be conducting.All the rest is blocked by the connement, which causes the resistance. When the connement getswidened so that an extra mode can pass, all electrons in that mode will become conducting at thesame time, causing a step in the conductance.

At higher temperatures the electrons get affected by impurities, which results in a nonzero resistancefor even the propagating modes. If we again widen the connement for an extra propagating mode,the electrons in that mode not immediately start conducting altogether. Instead, this mode now hasnonzero resistance and its electrons start conducting in a more continuous way.

Moreover, when the temperature increases, the Fermi-Dirac distribution tends to a smooth curve likein Figure 2.1 (b), which means that more states become available for taking part in the conduction.The states that are conducting lie no longer on a thin Fermi circle, but rather on a circle with

10


13/22

a much thicker line. When the connement gets widened the propagating modes are added morecontinuously, because of their arbitrarily distributed energies. All this spoils the discrete behaviourof the conductance. It smears out the steps, giving a more linear correlation between conductanceand width. When the temperature increases, the stepwise behaviour of Equation (2.9) reduces to theclassical description of conductance for a quantum point contact. In two dimensions this is given by[1]:

G = e2

kF W

(2.10)

In this equation the conductance is obviously linearly proportional to the width and the discretesteps have vanished. Before quantised conductance was discovered, this equation was thought to begenerally correct. No discrete behaviour was expected.

2.6 Low Resistance Measurements

The resistance of the device can be measured by either varying the current, measuring the voltage orvarying the voltage and measuring the current using a digital multimeter (DMM). The resistance canthen be calculated using Ohms Law:

V = IR (2.11)

Consider a two-wire resistance measurement setup as in Figure 2.7. In this setup a current sourceand voltmeter are used to measure the desired resistance. The voltage reading however will not onlymeasure the voltage across device V R , but it will also take into account lead and contact resistanceV lead , offset voltages and noise effects.

Figure 2.7: Two-Wire Resistance Measurement Setup; where V M is the measured voltage and V R thevoltage over the test resistance.

Consider rst the leading resistance contribution only, where the measured resistance is given by:

V M I

= R + (2 R lead ) (2.12)

The expected resistance will have a very low magnitude in comparison to the lead resistance. Thetwo-wire resistance measurement setup will thus not suffice for the desired measurement. A four-wireresistance measurement setup can be used to overcome this problem caused by the lead resistances(Figure 2.8). The sense current will be very small due to the high resistance of the voltmeter in

11


14/22

comparison to the very small resistance of the devices. Because the sense current is negligible; V M V R , and the measured resistance can be determined with much greater accuracy:

V M I

= V R

I (2.13)

Figure 2.8: Four-Wire Measurement Setup

The circuit measures a nonzero voltage offset even when there is no current source. This offset iscaused my multiple error sources including thermoelectric EMFs, offsets generated by recticationof radio frequency interference and offsets in the voltmeter input circuit. There are many ways tocounteract these different error sources which are not mentioned in this report. The offset can bemeasured in a dry setting, such that it can be subtracted from the measured value to obtain a moreaccurate result.

The noise effects can be reduced using a Lock-In amplier. The Lock-In amplier basically consists of two systems: a synchronous switch and a amplier/lter, see Figure 2.9. The switch is designed suchthat negative signals are multiplied with -1 such that the output is fully positive. If a reference signalwith the same frequency as the input signal, is applied to the input signal:

V (t) = V 0sin (2f m t + ) (2.14)

The phase of the reference signal can be modied such that both the signals are in phase. If theyare, the result is then passed to the amplier/lter stage, where the signal is modied such that theoutput signal will be a constant DC signal. The result is a strong low signal-to-noise ratio signal. (If

d is 12

then the output signal equals to zero.)

12


15/22

Figure 2.9: Schematic Lock-In Amplier

13


16/22

3. Experiment3.1 Introduction

The phenomenon quantised conduction was rst observed by the Delft-Philips group in 1988. Aspart of the course TN2992 experimenteel eindproject we will try to replicate the phenomenon usingequipment supplied by the Quantum Transport group at the Department of Applied Physics, DelftUniversity.

At 4.2 K we use a high mobility 2DEG of a GaAs-AlGaAs hetereostructure to create ballistic transportand use the electron depletion technique to create a variable gate width. With this set-up we expectto measure quantised conduction with steps of e2/ .

The experiment has been conducted on 12 December 2012 at the Quantum Transport group at theDepartment of Applied Physics. Measurement results suggest one single step of conductance has beenobserved with size 8K 1 which is within range of the expected value e2/ 1. We propose thesamples limited geometry and used doping to account as explanation for the observation of only one step instead of multiple steps.

3.2 Sample description

The sample used in our experiment was actually not designed for measuring quantised conductance.It serves as a chip to create and manipulate quantum dots on. Nevertheless, it has the requiredcharacteristics for making quantised conductance apparent. Just like in the original experiment, our

sample has a heterostructure of GaAs and AlGaAs in which a 2DEG is conned. Furthermore, it hasa gate with electrodes separated by distances (180-280 nm) in the same order of magnitude as theFermi wavelength. A schematic overview of the sample is given in Figure 3.1.

14


17/22

Figure 3.1: Schematic overview of the sample used in this experiment. In the gure both gates (red)and ohmics (green/blue) are indicated.

The gure gives a view perpendicular to the plane in which the 2DEG is conned. To make quantumpoint contacts, negative voltages have to be applied to opposing gates. The electrons beneath them get

depleted and depending on the voltage, the electrons can pass or cannot pass between two electrodes.Because the sample contains multiple gates, there are several possibilities of making quantum pointcontacts. The two most suitable combinations have been used during the experiment, as will bediscussed later on.

The rst is the combination of QPC1 and LS. The shortest distance between these two electrodes is280 nm. To measure the resistance for the point contact created by these gates, ohmics 10 and 9have to be used. The other combination involves the gates T and D, which are separated by a distanceof 180 nm. This point contact can be measured by a combination of the ohmic contacts 10 (or 9)and 3 (or 4).

The ohmic contacts can be connected to a measurement device in order to make resistance measure-

ments of the paths between them. The connections of all gates and ohmics are directed to a singlebus. A schematic overview of the bus is given in Figure 3.2. The connections are indicated by thesame labels as in Figure 3.1. With this connector, the sample can be connected to a measurementdevice and a voltage supply.

Figure 3.2: Schematic overview of the connector of the sample.

15


18/22

3.3 Measurement results

The experimental set-up has now been discussed. The following step is to discuss what measurementshave actually been performed in order to observe quantum conductance. The goal was to use twoohmics and two gates to achieve an electric current for which the resistance could be measured.

Before the actual measurements could be performed, the resistance of the ohmics of the sample wasmeasured. This resistance will be subtracted from the actual measured resistance later on. Themeasurement for the ohmic resistance was performed at 4 K.

A voltage source and a ampere meter were connected in series to the circuit. Next, these were connectedover two of the ohmics, that were used later on for the actual sample measurements. First ohmics O9and O10 were connected. The results of this measurement are displayed in table 3.1.

Ohmics O9 - O10 O9 - O4Input voltage (V) 0.05 0.05Offset voltage (V) 0.051 0.01Output voltage (V) 0.262 0.168S3b voltage source amplication (V/V) 1.00E-03 1.00E-03M1b ampere meter amplication (V/A) 1.00E+08 1.00E+08Actual voltage (V) 0.00005 0.00005Actual current (A) 2.11E-09 1.58E-09Resistance () 23696.68 31645.57

Table 3.1: The measured resistances over the two ohmic pairs used in the sample measurements.

The expected values of the resistance over the ohmics was in the k range. By performing simple cal-culations (see equation 3.1), the settings for the amplication of the ampere meter could be deducted.The resistance could best be measured with a voltage in the order of V. Since the input voltage was

50 mV, the amplication of the voltage source was set on 1 .00 10 3

V/V, to convert the voltage toV.

R = U I

I = U R

= 50 10 6

10 103 = 5 .0 108A (3.1)

The expected value of the resistance was in the order of 10 k, and the output voltage should bearound 1 V, meaning that the amplication of the ampere source could be calculated. It turned outto be 1.00 108 V/A.

Using these settings, the measurement could be performed, and the output voltage and offset voltagewere measured. The offset voltage was subtracted from the output voltage, the values were convertedusing the amplication factors and the actual voltage and current could be calculated. These valuesgave in turn the actual resistance over the ohmics (see table 3.1). As expected, the values were in thek range.

In attempt to reproduce the experimental step-wise conductance plot, a voltage sweep was executedover the gate QPC1-LS. The result is shown in Figure 3.3. Instead of the expected multi-step plot,the result shows only a single step. Rather than a single jump, the plot shows a transition phase, afterwhich a nearly constant value of L = 3.8464 10 5 1. This value has to be corrected for the leadresistance caused by the ohmics leading to eq3.2.

L = Lmeas

+ 1

Rohmic= 3 .8464 10 5 + 1 / 23696.68 = 8 .0664 10 5 1 (3.2)

16


19/22

Figure 3.3: Result of Sweep voltage from -320mV to -200mV (201 data points) over gate QPC1-LS

Figure 3.4: Result of Sweep voltage from -500mV to 0mV (201 data points) over gate T-D

17


20/22

The theoretical conductance gain per step is given by equation 3.3. The obtained result differs onlyby 3% from the theoretical conductance gain, such that it is allowed to conclude that a single stepwas obtained.

G = e2

= 7 .818 10 5 1 (3.3)

It was tried to use a different connection to reproduce the step-wise conductance plot. In this attempta sweep ranging from -500mV to 0mV was used. The result is shown in g3.4. Again only a single stepis observed. The nal conductance including the ohmic resistance correction is eq3.4, which deviates11% from the theoretical step gain.

L = Lmeas + 1

Rohmic= 3 .7773 10 5 + 1 / 31645.57 = 6 .9372 10 5 1 (3.4)

3.4 Conclusion

One step in conduction of e2/ has been observed where several steps were expected. In vain of multiple attempts with different setup congurations, no more steps were unveiled. The conductiondid limit to a plateau values of 8 .0664 10 5 1 and 6 .9372 10 5 1 which are within acceptablerange of the theoretical expected value 7 .818 10 5 1.

An explanation for the single step can be that the width of the gates is not sufficiently large, eventhough the gate sizes in the sample were comparable to previous experiments [1]. It was assumedthat the electron depletion has improved and that the W max is more constant over a longer distance.Moreover, the doping of the GaAs-AlGaAs heterostructure in the used sample is optimised for otherapplications then measuring quantum conductance. In other words, the sample used is not designed

for measuring quantum conductance.The deviation in step height in comparison to the theoretical result is caused by a multitude of uncertainties, including: the uncertainty in ohmic resistance measurements and the uncertainty insample resistance measurements. To minimise the measurement uncertainties, one could use noisereduction methods such as the AC lock-in technique and reduce the offset error.

18


21/22

Acknowledgements

To get a grasp of understanding certain quantum phenomena such as quantum conductance, one needsan understanding of quantum mechanics in general. In particular for bachelor-students from differentfaculties such as aerospace and mechanical engineering this knowledge is not readily present.

This project enabled us to learn a bit about quantum mechanics and in particular quantum conduc-tance. We would like to thank Dr. Ad Verbruggen for his supervision and support during this project.Dr. Pierre Barthelemey for helping us perform the experiments and the necessary preparations. Mo-hammad Shaei for supplying the samples and Pasquale Scarlino for enthusiastically giving us a tourthrough the lab.

The project has been a great learning experience and it was an enjoying project to end our physicsminor with.

19


22/22

References

[1] B. J. van Wees, H. van Houten, C. W. J. Beenakker, J. G. Williamson, L. P. Kouwenhoven,D. van der Marel, and C. T. Foxon. Quantized conductance of point contacts in a two-dimensionalelectron gas. Phys. Rev. Lett. , 60:848850, Feb 1988.

[2] H. Van Houten and C. Beenakker. Quantum point contacts. Physics Today , pages 2227, Jul1996.

[3] D. Lincoln. The inner life of quarks. Scientic American , 307:3643, Nov 2012.[4] Classical and quantum conductivity. http://en.wikipedia.org/wiki/Classical_and_

quantum_conductivity , 2013.

[5] Igor Kuskovsky. Classical and quantum free electron models of electrical conductivity. Departmentof Physics of the Division of Mathematics and Natural Sciences, Queens College of the CityUniversity of New York, 2007.

[6] Stanford University Rsasaki. Chapter 7. two dimensional electron gas system (2deg). http://www.stanford.edu/ ~rsasaki/AP388/slide7 , 2013.

[7] 2deg. http://en.wikipedia.org/wiki/2DEG , 2013.

[8] Semiconductor. http://en.wikipedia.org/wiki/Semiconductor , 2013.

[9] Warwick University D. R. Leadley. Reduced dimensional structures. http://homepages.warwick.ac.uk/ ~phsbm/2deg.htm , 2013.

[10] Rice University Rui-Rui Du. 2deg materials and basic characterization. http://wls.iphy.ac.cn/Chinese/1219/2/rrdu.pdf , 2013.

[11] Department of Physics Branislav K. Nikolic and University of Delaware Astronomy. Het-erojunctions, interfacial band bending, and 2deg formation. http://www.physics.udel.edu/~bnikolic/teaching/phys824/lectures/band_bending_2deg.pdf , 2013.

[12] Dominik Zumbhl. Gaas heterostructures and 2d electron gas. Quantum Coherence Lab at theDepartment of Physics, University of Basel, 2010.

report quantised conduction tn2992

Documents