Description of the physical sys- tem

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Chapter 1

Description of the physical

sys-tem

1.1 Introduction

In this chapter we will explain the physics underlying the systems that we will simulate with our computer program. We will start by describing what a mesoscopic system is and how it can be made. Some approximation is possible when dealing with mesoscopic systems, thus we will introduce the concept of effective mass and of a single band envelope function.

After such preliminary concepts, we will deal with the problem of the conductance for mesoscopic systtems, and how it is related to the Green’s function which we will be explaining in the following chapter.

1.2 Mesoscopic systems

Essentially a mesoscopic system can be defined as a system with a di-mension between the macroscopic, where the classical laws apply, and the microscopic, the atomic level, realm of quantum mechanics.

A conductor is said to be mesoscopic when its size is intermediate be-tween the macroscopic length scale and the microscopic (atomic) scale, so

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that classical transport models are not applicable and quantum phenomena become relevant.

In order to study mesoscopic conductors we need to find a way to make them. The answer to this problem lies, for example, in the exploitation of heterostructures.

An example can be provided by the GaAs/AlGaAs heterojunction, a structure in which some n-doped AlGaAs atomic layers are grown over layers of intrinsic GaAs.

As we can see in Fig. 1.1, AlGaAs is characterized by a wider gap, with respect to GaAs. Their respective Fermi levels are misaligned, specifically it is higher in the AlGaAs, thus electrons spill over from it, leaving behind positively charged donors, which gives rise to an electrostatic potential that causes the bands to bend as shown.

If we analyze the electron density, we see that it is sharply peaked near the interface between GaAs and AlGaAs, where the Fermi energy is inside the conduction band, thus forming a thin conducting layer usually called two-dimensional electron gas, 2-DEG in short.

The two-dimensional electron gas lies in a region characterized by very high mobility, since it is spatially separated from the donors (sometimes some

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AlGaAs n

GaAs

b)

a)

2−DEG

Fig. 1.1 Conduction and valence band line-up at a junction between an n-doped AlGaAs and an intrinsic GaAs, before and after the charge transfer has taken place.

additional layers of undoped AlGaAs are added to work as buffer, this ensures a better mobility, but reduces the electron density).

By confining the other two dimensions, in example by using the tech-nique of the split gate as shown in Fig. 1.2, it is possible to create quasi-one

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and quasi-zero dimensional systems (quantum wires and quantum dots, re-spectively).

GATE VOLTAGE

Fig. 1.2 The 2-DEG is confined along theyaxis by means of the split gate technique.

One last thing that needs to be mentioned concerns which electrons actu-ally contribute to the current. At low temperatures, when the equilibrium is reached, only electrons whose energy is near the Fermi energy “carry cur-rent”. Indeed, at low temperatures only the states with energy lower than the Fermi energy are occupied, thus in the space of the wave vectors k, all

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the occupied states are inside the circle of radius kF (we are considering only

two dimensions).

This is clearly shown in Fig. 1.3.

E

e

000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111

k

∆

k

F x y

Fig. 1.3Spatial distribution of the occupied states at equilibrium and for low temperatures, with and without the presence of an electric fieldeE.

When we apply an electric field the entire distribution is shifted by a value ∆k, but if such value is really small compared to the Fermi wave vector, ∆k ≪ kF, from a global point of view, only a few states change their occupancy.

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Of course, all the sea of electron has been shifted, so every electron changed state, but only a few state that were occupied are free after the field is applied (and vice versa). We can think that the field moved a few electrons from states near −kF to states near +kF, so only a small part of the electrons

contributed to the current flow: those with energy near to the Fermi energy EF.

1.3 Single band envelope function and effective mass

approximation

Inside an heterostructure it is possible to study the electron dynamics in the conduction band if the following three assumption are verified:

1. Non degenerate conduction band

2. Applied field are only loosely dependent o time (small variation over a long time, so that there are no interband optical transitions)

3. Applied fields are weak (so that Zener tunneling is forbidden) Under such assumptions we can use the singble band envelope function approssima-tion. In such a case electrons are described by an effective Hamiltonian where the only terms that appear are those related to the external fields and the discontinuities of the bands due to the heterojunctions. The re-maning part of the cystalline interaction is included in a renormalization of the electronic mass.

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Denoting with E(k) the dispersion relation, since it is periodic over the reciprocal space, it can be expanded with Fourier over the wave vectors of the lattice: E(k) =X j EjeiRj ·k (1.1) where by Rj we denoted the vectors for the direct lattice. From (1.1) it is

possible to define the differential operator E(−i∇): E(−i∇) =X

j

EjeRj ·∇

(1.2)

The exponentials are the shift operators over the vectors Rj.

By applying this operator to the Bloch eigenfunctions we find

E(−i∇)ψk(r) = X j EjeRj ·∇ ψ_k(r) = X j Ejψ_k(r + Rj)

than, by applying the Bloch theorem we can write

X j Ejψ_k(r + Rj) = X j EjeiRj ·k ψ_k(r) and since X j EjeiRj ·k ψ_k(r) = E(k)ψ_k(r) we finally find that

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Equation (1.3) meaning is that to apply to a Bloch eigenfunction the unper-turbed Hamiltonian of the crystal is equivalent to apply to it the operator E(−i∇).

The Hamiltonian of the system can be written as H = H0+ V (r), where

V (r) is an external potential. Since we are interested only in the dynamic for the conduction band, we can describe the carriers with wavepacket built with the eigenfunctions ψ_k(r):

η(r, t) = X

k

a(k, t)ψ_k(r) (1.4)

By applying the Hamiltonian to the wavepackets we find: Hη(r, t) =X k a_k(t)Hψ_k(r) =X k a_k(t)E(k) + V (r) ψ_k(r) =X k a_k_{(t) [E(−i∇) + V (r)] ψ}_k(r) = [E(−i∇) + V (r)]X k a_k(t)ψ_k(r) = [E(−i∇) + V (r)] η(r, t) (1.5)

Under the assumption of weak fields and low temperature, the electrons in the conduction band will remain near its bottom. For GaAs the minimum is at k = 0, which means that the electrons are bound in a part of the reciprocal lattice where the wave vector is almost null.

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We should also remember that for the Bloch eigenfunctions

ψ_k(r) = e

ik·r

√

Ωuk(r)

with Ω being a normalization volume, the atomic part u_k(r) is weakly affected by variations of the wave vector. Thus we can write

η(r, t)_{≃ F (r, t)u}0(r), F (r, t) = X k a_k(t)e ik·r √ Ω (1.6) and the function F (r, t), too, is weakly affected by variations of the wave vector, since it is the sum of plane waves with wave vectors almost equal to zero.

The function F (r, t) is called envelope function. Now we can write

E(−i∇)F (r, t)u0(r) = X j EjeRj ·∇ F (r, t)u0(r) =X j EjF (r + Rj, t)u0(r + Rj) = u0(r) X j EjeRj ·∇ F (r, t) = u0(r)E(−i∇)F (r, t)

and thus we are finally able to write the effective Schr¨odinger equation for F :

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Since, as previously discussed, electronic transport is restricted to small val-ues of the wave vector, we can approximate the dispersion relation up to second order: E(k) ≃ Ec+ ¯h2 k 2 2m⋆ (1.8)

where Ec is the bottom of the conduction band.

By applying (1.8) to (1.7) we find the effective mass equation − ¯h 2 2m⋆∇ 2 + V (r, t) F (r, t) = i¯h ∂ ∂tF (r, t) (1.9) where we included Ec into V (r, t).

These calculations hold for a bulk semiconductor, in order to apply them to a heterostructure we need to verify the hypothesis that the atomic parts of the wave functions are the same in the whole heterostructure.

Defining a coordinate system where the axis z is normal to the het-erostructure layers, while the axis x and y define a plane parallel to them, for the specific kind of system we will be working with, the confinement along the z axis is so strong that we actually work in the hypotesis of a two-dimensional system.

The axis x will be parallel to the direction of motion, and y will be transverse to it and will be affected to confinement, for example by means of

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a split gate, as we mentioned earlier. In this case Schr¨odinger equation is " −i¯h∇ + eA2 2m⋆ + V (x, y) # ψ(x, y) = Eψ(x, y) (1.10)

and its solution are

ψ(x, y) =X

i

eikix_χ

i(y)

1.4 Conductance

In a mesoscopic system the conductance is not a local property, it de-pends from global properties of the device. As an example, we can think of a ballistic quantum wire, it is supposed to have no resistance, since transport is ballisitc, but it, indeed, has a resistence different from zero.

If we consider the system in Fig. 1.4, we can express the conductance G from point 1 to point 2 by using the Landauer-B¨uttiker formula, which holds for the already mentioned conditions of low temperature and near equilibrium (so that the Fermi level can be considered constant in the entire system):

G = 2e 2 h X n,m Tn,m (1.11)

where Tn,m is the generic element of the transmission matrix of the device

(n are the output modes, while n are the input ones).

This formula can be modified to include the effect of higher temperatures by introducing the thermal smearing function −df/dE, where f is the Fermi

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LEAD1 LEAD2

1 1L 2L 2

CONDUCTOR

Fig. 1.4 A conductor is connected to two large contacts by two leads.

distribution. We can thus calculate the conductance as a function of the temperature Tp using the following formula:

G(EF, Tp) =

Z ∞

0

G(E, 0)−df dE dE

Anyway, the conductance we obtain by using equation (1.11) includes the resistence for the contacts. That is, it calculates the conductance between two planes deep inside the contacts. Thus, we could think of trying to calculate the conductance between the leads (two planes in the leads, those marked as 1L and 2L), but this is not necessary. We assume the contacts to be “reflectionless”.

What we mean by “reflectionless” is that an electron exiting from a narrow conductor into a wide contact has a negligible probability of being

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reflected. It is important to note that going from a wide contact to a narrow conductor is all but a “reflectionless” process.

We still need to explain the meaning of the transmission matrix. Let us consider the term Tn,m: it is the ratio between the probability current for

mode n (due to mode m) at the output and the total probability current for mode m at the input.

Since the probability current of a mode of unitary amplitude is equal to the group speed v of such mode, and since the probability current is a function of the square of the absolute value of the wavefunction, for a mode of amplitude A and group speed v the probability current is equal to |A|2v. Thus, if we define Am the amplitude of the incoming mode m and An

the amplitude of the outgoing mode n, we can write:

Tn,m= |A n| 2 |vn| |Am| 2 |vm| = t′ n,m 2 (1.12) Where we denoted by t′

the matrix whose generic element is

t′n,m= |An| |Am| s |vn| |vm|

1.5 Shot noise

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In a mesoscopic system charges are not independent from one another, since only a finite number of modes is available for conduction. Thus the formula is different from that for macroscopic systems:

SI = 4 e2 h e |V | X n wn(1 − wn) (1.13)

where V is the average value of the applied voltage and wnare the eigenvalues

for the matrix t′H_t′

.

It should be noted that the trace for such matrix is the sum of the square of the absolute value of every single element of t′

, and since the trace is the sum of the eigenvalues:

X n wn = X n eig(t′ t′H_{) = tr(t}′ t′H_{) =}X n,m |t′ n,m| 2 =X n,m Tn,m

We can obtain the macroscopic formula for the shot current in case for every wn the expression wn≪ 1 holds true.

We can now see how important is the evaluation of the transmission matrix, since if we know it we both know the conductance and the shot noise for our device.

1.6 Current mapping

This is the method we will employ in our program in order to evaluate the spatial distribution of the current throughout the system.

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The idea is to use a probe as a local scatterer. Because of scattering, the reflection probabilities of the modes will change, thus modifying the trans-mission matrix and, consequentely, the conductance of the whole system.

Since we assume the voltage applied to the device to be constant, a variation of the conductance will cause a variation of the current. The higher the current flowing through the zone affected by the probe, the more the transmission matrix will be affected.

By moving the probe from one point to another, over a lattice covering all the device, we can map the variations of conductance, or rather of current, over all the system.

In the following chapters we will explain how we will use the Green’s functions method to study a system, but we would accomplish nothing if we were not able to relate them to the conductance.