2.4 Quantum confined electrons

Transcription

1 2.4. Quantum confined electrons Quantum confined electrons We will now focus our attention on the electron charge densities in case of one, two and three-dimensional confinement. All the relations will be then used in the self-consistent solution of the Poisson/Schrödinger equation Integration over the space Consider a function f defined in the -space. The sum of f over is I = f() (2.7) (2.7) can be written as f() = L3 (2π) 3 f() (2π)3 L 3, (2.7) where L is the sample size along each direction, and (2π)3 L is the volume occupied by one state in the -space. If the available states are 3 continuous rather than discrete, the quantity (2π)3 L can be seen as an 3 infinitesimal volume ( ) 3. As a consequence, the sum becomes an integral, and (2.7) can be expressed as f() = L3 (2π) 3 f()( ) 3 L3 (2π) 3 f()d. (2.72) In general, we can generalize to a generic dimensionality d and mae the following substitution f() 2 Ld (2π) d f()(d) d (2.73) where we have also considered the spin degeneracy. This relation will result to be very useful in the derivation of the electron density whose general expression at the equilibrium is [] n(r) = i f (E i ) Ψ i (r) 2 (2.74) where the sum is performed over the single energy levels E i,andf is the Fermi-Dirac distribution function.

2 8 Chapter 2. Quantum Simulations and let us call a(x, y, z) thetermthatwehaveneglectedinpassingfrom (2.8) to (2.8): a(x, y, z) ˆT yz ψχ ψ ˆT yz χ; (2.83) if the approximation is valid a(x, y, z) mustbemuchsmallerthane E i (y, z) in any point of the domain, which means that the parameter δ, whichwedefineas a(x, y, z) δ max x,y,z [E E i (y, z)]ψ(x, y, z)χ(y, z) (2.84) must be much smaller than. Since χ obeys the 2D density of states, it can be written as χ = Ae j(yy+zz),where 2 2 y 2m y z 2m z = E E i (2.85) As a numerical example, we have considered the case of a silicon MOSFET with channel length equal to 5 nm and oxide thicness equal to 2.5 nm investigated in [2] considering the low lying subband. In Fig. 2. we plot delta as a function of the gate voltage for a continuous doping profile (solid line) and for an atomistic doping profile (dashed line) : in both cases δ is smaller than 3. The eigenfunction Ψ can be then expressed as Quantum confinement in one direction Ψ i,y, z (x, y, z) = ejyy e jzz Ly L z ψ i (x, y, z) (2.86) where L y and L z are the system size in the y and z direction, respectively, while ψ i (x, y, z) isthei th eigenfunction. Using (2.74), we obtain n(x, y, z) = i σ i ψ i (x, y, z) 2 (2.87) where def σ i = f (E i,y,z) (2.88) L y L z y, z E i,y, z = ϵ i y 2m y z 2m z (2.89)

3 2.4. Quantum confined electrons 9 and ϵ i is the i-th energy level in the x-direction Maing the following variable transformation we obtain while (2.89) becomes y = y ; z = z (2.9) 2my 2mz d 2 = d y d z =2 m y m z d yd z (2.9) E i, y, z = ϵ i y z = ϵ i (2.92) (2.9), and using cylindrical coordi- By means of eqs. (2.73),and nates [3] we can evaluate σ i 2 L yl z 4π 2 2 m y m z y, z σ i = = 2π my m z dθ 2π dθ d (2.93) π 2 d f (E i, y, z ) 2 m y m z d f (E i, π y, z ) (2.94) From (2.92), we can replace d in terms of de i, y, z, and obtain and finally σ i = ϵ i my m z σ i = π 2 de i, y, z =2 2 d (2.95) my m z π 2 f (E i, y, z )de i, y, z (2.96) [ ( ϵi E F ln +exp- )] (2.97) where K B is the Boltzmann constant, T is the temperature and E F is the Fermi level. The electron concentration can be then expressed as n(x, y, z) = [ ( )] my ψ i (x, y, z) 2 m z ϵi (y, z) E F π 2 ln +exp i (2.98)

4 2 Chapter 2. Quantum Simulations Two-dimensional quantum confinement The considerations in the previous section, similarly apply whenquantum confinement is predominant in the y z plane. The two-dimensional Schrödinger equation reads, [ 2 2 y m y y z m z ] χ i z = E i,x χ i. (2.99) while the three-dimensional eigenfunctions can be expressed as Ψ x,i = ejxx Lx χ i (y, z) (2.) where χ i (y, z) aretheeigenfunctionsassociatedtotheenergylevelϵ i (x) and the total energy reads E i,x The electron concentration is = ϵ i (x)+ 2 2 x 2m x (2.) n(x, y, z) = i χ i (x, y, z) 2 α i (2.2) where α i = x f (E i,x ) L x (2.3) Using (2.73), α i =2 L + x 2π f (E i,x ) d x = 2 + f (E i,x ) d x (2.4) L x π L x Expressing x as 2mx (E ϵ i ) x = and deriving with respect to E we obtain (2.5) d x = 2mx 2 E ϵ i de (2.6)

5 2.4. Quantum confined electrons 2 Substituting (2.6) in (2.4), we obtain α i = 2m x π (E ϵi ) ϵ i +e E E F de (2.7) and multiplying and dividing by,weobtainthefinalexpression for the electron density n(y, z) = π ( 2mx 2 ) 2 i χ i (x, y, z) 2 F 2 where F 2 is the Fermi integral of order 2 :. ( ) EF ϵ i (x) (2.8) F (y) = x 2 dx (2.9) 2 +ex y Three-dimensional quantum confinement For what concerns three-dimensional quantum confinement, (2.74) reduces to n(r) = Ψ i (r) 2 (2.) i +e E i E F where Ψ i is the orbital associated to the i-th eigenvalue, with energy E i. However for a system with very few electrons, we cannot define afermi level, thence it is improper to fix the Fermi energy. Amoreproperapproachisthenrepresentedbyfixingthenumberof electrons in the system and using in (2.74) the Gibbs statistic. However the Gibbs statistics in the grand canonical ensemble is rather complicated, so it is worth following a simpler approach represented by considering the ground-state of the confined region and filling the energy levels starting from the lowest. The electron density, can then be expressed as n(r) =2 m Ψ i (r) 2 +(N 2m) Ψ m+ (r) 2, (2.) i= where m = N 2 is the number of fully occupied single electron levels.

6 22 Chapter 2. Quantum Simulations 2.5 3D Poisson/Schrödinger equation As introduced at the beginning of this chapter, as dimensions becomes smaller and smaller, quantum confinement effects becomes relevant. A proper simulation approach requires thence solving the Schrödinger equation together with the Poisson equation, in a self-consistent iterative scheme. However, at nanoscale dimensions, not only quantum confinement affects devicebehavior, but alsoother effects whosenature is purely electrostatic, limit the performance of the device. We refer forexample to the charge sharing effects that lower the threshold voltage inmos- FET devices, or to the random distribution of dopants that can affect the conductivity in the channel. In order to tae into account this wide variety of effects, a threedimensional simulation approach has to be followed. In particular the Poisson equation in the three-dimensional simulation domain reads [ε(r) φ(r)] = q [ p(r) n(r)+n + D (r) N A (r)+ρ fix(r) ], (2.2) where φ is the electrostatic potential, ε is the dielectric constant, p and n are the hole and electron densities, respectively, N + D is the concentration of ionized donors, N A is the concentration of ionized acceptors, and ρ fix is the fixed charge concentration. In all the simulations that we have performed (which will be shown in the following chapters), we have considered quantum confinement of the electrons, while a semiclassical expression has been assumed for the other densities. For what concern the exchange term V exc,wehaveconsideredthe expression in [4] q2 [ V exc = 3π 2 4π 2 n(r) ] 3, (2.3) ε ε r which differ by the exchange potential found by Slater by a factor of 2 3. We have found that the above expression, valid for a 3D system, has an impact on the results only inside the confined regions, while it turns out that exchange corrections to the potential are negligible in the other regions of the analyzed structures. From a numerical point of view, the coupled Poisson and Schroedinger equations are solved by means of a Newton-Raphson method with apredictor-correctorscheme[4],thatwehaveverifiedtobefasterand