9 Density of states for some nanostructures and Two dimensional electron gas
Prof. Subhasis Ghosh
3.3.1 DOS in 3D
3.3.2 DOS in 2D
3.3.3 DOS in 1D
3.3.4 DOS in 0D
3.4 Two Dimensional Electron Gas
CALCULATION OF DENSITY OF STATES (DOS): Quantum Wells, Wires and Dots
Figure . Electron state is defined by a point in k-space.
Note that the 2 arises from the constraints of periodic boundary conditions as proposed to the more general where n=0, 1, 2, 3… The volume of a given mode is then= . The number of modes (N) in the sphere is,
Say the particle in an electron and we consider spin (up and down), then we multiply N by 2.
Figure . Density of states in 3 dimension (Eq.3.48)
3.3.2 DOS in Two Dimensions (well)
Here we have 1D that is quantized. Let’s us assume it is the z-direction. The total energy of this system is a sum of the energy along the quantized direction plus the energy along the other 2 free directions. It is expressed as
The area of a given mode is then kx,ky with the total number of modes (N) in the area being
Again if the particle is an electron and we consider spin, multiply by 2 to get
Figure Density of states in 2 dimension. Shaded area presents occupied states.
3.3.3 DOS in One Dimensions (Wire)
Consider now the situation where there are two dimensions confined and only 1 degree of freedom (say the x-direction). The total energy of the system can be written as
This is the energy density for a given n, m value, the expression taking into account all m, n combination is
Figure Density of states in 1 dimension. Shaded area presents occupied states.
3.3.4 Zero dimensions (Quantum Dot)
Here since all three dimensions are confined. The density of states is basically a series of delta functions. The total energy of the system is
where m, n, o are integers and
The density of states is
3.3.5 More density of states
Density of states in the conduction band
For this we need to know the probability that an electron will occupy a given stats of energy E. The Probability, P(E), is referred as the Fermi Dirac distribution. In addition we need to know the density of states ( ′). The density of states has units of number of unit volume per unit energy. Therefore ′ is the number of states per unit volume. The number of occupied states at a given energy per unit volume is therefore
the total concentration of electrons in the conduction band is therefore the integral over all available energies
where Ecis the energy where conduction band starts. For the case of three dimensional material
Taking account into conduction band begins, the density of states can be written as
the total concentration of electrons in the conduction band is given as nr=
The integral is called the Fermi integral or Fermi Dirac integral.
Consider the case where,E-EF ≫KT and the Fermi Dirac distribution function becomes
This is the expression for the effective density of states of the conduction band.
Where Ev is the energy where valance band starts. The total concentration of holes in the valance band is the integral over all energies.
Summary
Fermi level of an intrinsic semiconductor
If the bulk semiconductor is intrinsic, there has been no doping of the material and hence no extra electrons or holes anywhere. in this situation
One can therefore see that at T=0 the Fermi energy of an intrinsic semiconductor is at the halfway point between the top of the valance band and the bottom of the conduction band.
Density of states in the conduction band
We start with the Fermi Dirac distribution for electrons and also the density of states
Consider only one of the subband. In this case the density of states simplifies to
Now recall from the previous section that the number of states at a given energy per unit volume
the total concentration of electrons in this first subband is the integral over all available energies. Rather than use ntot as before let’s just stick to nc from the start
Since the band really begins at en as opposed to Ec like in the bulk the integral change from
Density of states in the valance band
As with the conduction band case we need the probability of occupying a given state in the valance band. This denoted ℎ( ) and is evaluated from
first. The total concentration of holes in this first subband is the integral over all energies. we get
Fermi level position :2D
Divide by 2 to go back to only 1 spin orientation since in an optical transition spin slips are generally forbidden
The expression applies to either conduction band or valance band. Applying the following equivalence
where pj is the desired joint density of the states. Now from the conservation of momentum, transition in k are vertical such that the initial k value in the valance band is the same k value as in the conduction band (ka=kb=k) where ka is the k value in the valence band and kb is the value in the conduction band. The energy of the initial state in the valance band is
Likewise the energy of the final state in the conduction band is
The energy of the transition is
2D Well
Area in k-space
Where the area occupied by a given mode or state is .Here we assume that represents the confined direction
Together, the number of modes in the area is
Multiply by 2 to account for spin
Now consider the density
With the energy density given by
Starting with the energy density
Divide by 2 to get rid of the spin since formally speaking, spin flip optical transitions are forbidden
Now applying the following equivalence
one obtains
wherePj(E ) is the desired joint density of states. As before in the 3D case, the conservation of momentum means that transition in k-space are vertical. That is the initial k value in the valance band is the same as the final k value in the conduction band (Ka =Kb =K) where Ka(Kb ) is the valance (conduction) band values.
The energy of the initial state in the valance band is
Likewise the energy of the final state in the conduction band is
This leads to
Or
Such that when replaced into our main expression the desired expression for the joint density of states is
1D wire
Consider the length in k-space
Lk=2k
The length occupied by a given mode or state is where
The number of states in the given length is
Multiply this by 2 to account for spin, we get total number of states as
Consider the density ie number of states per unit length
And the energy density is given by
Or alternately
Starting with the energy density
Divide by 2 to consider only one spin orientation since spin flip transition are generally forbidden
Now apply the following equivalence
wherePj(E) is the desired joint density of states. As before in the 3D and 2D case, the conservation of momentum means that transition in k-space are vertical so that Ka=Kb =K) where Ka(Kb) is the valance (conduction) band values.
The energy of the initial state in the valance band is
Likewise the energy of the final state in the conduction band is
3.4.1 Two-Dimensional Electron Gas
In low dimensional systems, the quantum effects were first observed is two-dimensional electron gas (2DEG). There are two basic systems, (i) Si metal-oxide-semiconductor field-effect transistors (MOSFETs) and (ii) GaAs/AlGaAs heterostructures where 2DEG has been studied extensively. A typical Si device with 2DEG is shown in Fig. 3.4.1. Here, Si surface serves
Figure 3.4.1: Band diagram showing conductance band EC , valence band EV and quasi- Fermi level EF . A 2DEG is formed at the interface between the oxide (SiO2) and p-type silicon substrate as a consequence of the gate voltage Vg .
as a substrate while SiO2 layer on Si behaves as an insulator. 2DEG is induced electrostatically by application a positive gate voltage Vg . The sheet density of 2DEG can be described as
where Vt is the threshold voltage which is the minimum gate-to-source voltage required to create a conducting path between the source and drain terminals in MOSFET.
The other important 2DEG system is based on modulation-doped GaAs-AlGaAs single and double heterostructures. As the bandgap in AlGaAs is higher than that in GaAs, by doping only higher bandgap semiconductor (AlGaAs), it is possible to move the Fermi level inside the forbidden gap.
When AlGaAs is grown on GaAs substrate, a unified level of chemical potential is established, and an inversion layer is formed at the interface, as shown in Fig. 3.4.2..
Figure 3.4.2: Band structure of the interface between n-AlGa As and intrinsic GaAs, (a) before and (b) after the charge transfer.
If metal semiconductor field effect transistor (MESFET) is fabricated on single heterostructures in which 2DEG is created by a modulation doping, then a narrow channel can be squeezed by selective depletion in spatially separated regions. This is the simplest way to create a lateral confinement using split metallic gates as shown in Fig. 3.4.3 The SEM micrograph of a typical device is shown in Fig. 3.4.4.
Figure 3.4.3: A narrow channel created in AlGaAs/GaAs based modulation doped heterosturcture using split gate technique.
Figure 3.4.4: Scanning electron microphotographs of real device (taken from Phys. Rev. Lett. 59, 3011, 1987).
3.4.2 Basic Properties of Low-Dimensional Systems
Let us take z-axis perpendicular to the plane of 2DEG. As mentioned before, in 2DEG the motion of electron is free in x-y plane and quantized along z-axis. Hence, the wave function of the electrons in 2DEG can be decoupled as
where r is the vector in x-y plane of 2DEG. In case of Si-MOSFET or GaAs/AlGaAs single heterostructure, the confining potential can be approximated as triangular one and given by
Using separation of variables the Schro¨dinger equation for the wave function χ(z) is given by
Each level (for different values of n) creates a sub-band for the in-plane motion. Here the effective mass m of electron or hole , determined by the bandstructure of GaAs is much smaller than the mass of a free electron.
Figure 3.4.6: Density of states for a quasi-2D system.
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