Lecture 20 - Semiconductor Structures

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1 Lecture 0: Structures Kittel Ch 17, p , extra material in the class notes MOS Structure metal Layer Structure Physics 460 F 006 Lect 0 1 Outline What is a semiconductor Structure? Created by Applied Voltages Conducting channels near surfaces Controlled by gate voltages MOSFT Created by material growth Layered semiconductors Grown with atomic layer control by MB Confinement of carriers High mobility devices -d electron gas Lasers Covered briefly in Kittel Ch 17, p , added material in the lecture notes Physics 460 F 006 Lect 0 Structure MOS Structure Layer Structure MOS Structure If the metal gate is biased positive, are repelled and are attracted to Interface Insulator prevents direct contact with the metal metal Physics 460 F 006 Lect 0 3 Physics 460 F 006 Lect 0 4 MOS channel If the metal is biased positive, there can be an electron layer formed at the interface Insulator prevents direct contact with the metal lectrons are bound to interface but free to move along interface across the full extent of the metal region How can this be used? Layer of Physics 460 F 006 Lect 0 5 MOSFT Transistor - I The structure at the right has n regions at the surface of the semiconductor Contacts to the n regions are source and drain If there is no bias on the metal gate, no current can flow from source to drain Why? There are two p-n junctions one is reverse biased whenever a voltage is applied Off state Drain n-type region created by doping gate Source Physics 460 F 006 Lect 0 6 1

2 MOSFT Transistor - II A positive bias on the metal gate creates an n-type channel Current can flow from source to drain through the n region On state MOSFT transistor! Layer of Drain gate Source Physics 460 F 006 Lect 0 7 Layered Structures lectrons and can be confined and controlled by making structures with different materials Different materials with different band gaps fi and/or can be confined Structures can be grown with interfaces that are atomically perfect - sharp interface between the different materials with essentially no defects Physics 460 F 006 Lect 0 8 Layered Structures Can be grown with interfaces that are atomically perfect - a single crystal that changes from one layer to another xample: GaAs/AlAs Single crystal (zinc blende structure) with layers of GaAs and AlAs Grown using MB (Molecular Beam pitaxy) Can tune properties by making an alloy of GaAs and AlAs, called GaAs Physics 460 F 006 Lect 0 9 Layered Structures xample of bands in GaAs-AlAs structures Both and are confined (Other systems can be different) GaAs Physics 460 F 006 Lect 0 10 Uses of Layered Structures Confinement of carriers can be very useful xample - light emitting diodes Confinement of both and increases efficiency n-type Uses of Layered Structures Confinement of light can be very useful xample - light emitting diodes, lasers Confinement of light is due to larger dielectric constant of the low band gap material - total internal reflection light Physics 460 F 006 Lect 0 11 light - confined to direction along layer Used to make the lasers in your CD player! Physics 460 F 006 Lect 0 1

3 Uses of Layered Structures The highest mobility for (or ) in semiconductors are made with layer structures xample pure GaAs layer between layers of doped n-type n-type Pure GaAs n-type Quantum Layered Structures If the size of the regions is very small quantum effects become important. How small? Quantum effects are important when the energy difference between the quantized values of the energies of the is large compared to the temperature and other classical effects In a semiconductor the quantum effects can be large! High mobility for the in GaAs because the impurity dopant atoms are in the Physics 460 F 006 Lect 0 13 Physics 460 F 006 Lect 0 14 lectron in a box Here we consider the same problem that we treated for metals the electron in a box see lecture 1 and Kittel, ch. 6 There are two differences here: 1. The have an effective mass m*. The box can be small! This leads to large quantum effects We will treat the simplest case a box in which each electron is free to move except that it is confined to the box To describe a thin layer, we consider a box with length L in one direction (call this the z direction and define L = L z ) and very large in the other two directions (L x, L y very large) Physics 460 F 006 Lect 0 15 Schrodinger quation Basic equation of Quantum Mechanics [ - ( h/m ) + V( r ) ] Ψ ( r ) = Ψ ( r ) where m = mass of particle V( r ) = potential energy at point r = (d /dx + d /dy + d /dz ) = eigenvalue = energy of quantum state Ψ ( r ) = wavefunction n ( r ) = Ψ ( r ) = probability density From Lecture 1 Physics 460 F 006 Lect 0 16 Schrodinger quation - 1d line Suppose particles can move freely on a line with position x, 0 < x < L 0 L Schrodinger q. In 1d with V = 0 -( h /m ) d /dx Ψ (x) = Ψ (x) Boundary Condition Solution with Ψ (x) = 0 at x = 0,L Ψ (x) = 1/ L -1/ sin(kx), k = m π/l, m = 1,,... (Note similarity to vibration waves) Factor chosen so 0 L dx Ψ (x) = 1 lectrons on a line Solution with Ψ (x) = 0 at x = 0,L Ψ xamples of waves - same picture as for lattice vibrations except that here Ψ (x) is a continuous wave instead of representing atom displacements 0 L (k) = ( h /m ) k From Lecture 1 Physics 460 F 006 Lect 0 17 From Lecture 1 Physics 460 F 006 Lect

4 lectrons on a line For in a box, the energy is just the kinetic energy which is quantized because the waves must fit into the box (k) = ( h /m ) k, k = m π/l, m = 1,,... Quantization for motion in z direction n = ( h /m ) k z, k z = n π/l,,,... Lowest energy solutions with Ψ n (x) = 0 at x = 0,L In Lecture 1 we emphasized the limit that the box is very large Ψ n (x) Here we emphasize the case where the box is very small Approaches continuum as L becomes large n = From Lecture 1 k Physics 460 F 006 Lect 0 19 x Physics 460 F 006 Lect 0 0 Total energies of lectrons Including the motion in the x,y directions gives the total energy for the : (k) = ( h /m* ) (k x + k y + k z ) Quantized two-dimensional bands n (k x, k y ) = ( h /m* ) (n π/l) + ( h /m* ) (k x + k y ),,... = n + ( h /m* ) (k x + k y ) = ( h /m* ) (n π/l) + ( h /m* ) (k x + k y ),,... This is just a set of two-dimensional free electron bands (with m = m*) each shifted by the constant ( h /m* ) (n π/l),,,... Physics 460 F 006 Lect 0 1 n = k x, k y Physics 460 F 006 Lect 0 Quantized two-dimensional bands What does this mean? One can make twodimensional electron gas in a semiconductor! xample - fill bands in order Density of States in two-dimensions Density of states (DOS) for each band is constant xample - fill bands in order n = lectrons can move in dimensions but are in one quantized state in the third dimension n = µ DOS k x, k y Physics 460 F 006 Lect 0 3 Physics 460 F 006 Lect 0 4 4

5 Quantum Layered Structures If wells are very thin one gets quantization of the levels and they are called quantum wells Confined in one direction - free to move in the other two directions Let thickness = L L MOS Structure - Again lectrons form layer Mobile in two dimensions Confined in 3rd dimension If layer is thin enough, can have quantization of levels due to confinement Similar to layer structures discussed next Physics 460 F 006 Lect 0 5 Physics 460 F 006 Lect 0 6 lectrons in two dimensions If the layer is think enough all are in the lowest quantum state in the direction perpendicular to the layer but they are free to move in the other two directions (k) = ( h /m* ) (n π/l) + ( h /m* ) (k x + k y ),,... This can happen in a heterostructure (the density of is controlled by doping) L Hall ffect See lecture 18 here we consider only of density n = #/area The Hall effect is given by = Hall / J B = -(e) lectrons confined to thin layer B e (SI units) J Or a MOS structure (the density of is controlled by the applied voltage) Physics 460 F 006 Lect 0 7 Hall z y x Physics 460 F 006 Lect 0 8 Hall ffect xpected result as the density n is changed The Hall constant = is given by = Hall / J B = V Hall /IB = -(e) where I = J x L y, V Hall = Hall x L y Hall ffect Consider a MOS device in which we expect n to be proportional to the applied voltage n Physics 460 F 006 Lect 0 9 V Physics 460 F 006 Lect

6 (QH) What really happens is Quantized values at the plateaus = (h/e )(1/s), s = integer Now the international standard for resistance! How can we understand this? In a magnetic field, in two dimensions have a very interesting behavior The energies of the states are quantized at values hω c (ω c = qb/m* = cyclotron frequency from before) (Similar to figure 10, Ch 17 in Kittel) Landau levels hω c V Physics 460 F 006 Lect 0 31 DOS Physics 460 F 006 Lect 0 3 Now what do you expect for the Hall effect, given by R H = 1/(nec) As a function of the applied voltage V, the fill the Landau levels When a level is filled the voltage must increase to fill the next level (QH) The Hall constant is constant when the levels are filled legant argument due to Laughlin that it work in a dirty ordinary semiconductor! Quantized values at the plateaus = (h/e )(1/s), s = integer Values are fixed by fundamental constants Now the international standard for resistance! DOS Physics 460 F 006 Lect 0 33 Voltage determined by details of device V Physics 460 F 006 Lect 0 34 Summary What is a semiconductor Structure? Created by Applied Voltages Conducting channels near surfaces Controlled by gate voltages MOSFT Created by material growth Layered semiconductors Grown with atomic layer control by MB Confinement of carriers High mobility devices -d electron gas Lasers Covered briefly in Kittel Ch 17, p , added material in the lecture notes Physics 460 F 006 Lect 0 35 Next time nanostructures Physics 460 F 006 Lect

7 Quantum Dots Structures with () confined in all three directions Now states are completely discrete Artificial Atoms GaAs AlAs Physics 460 F 006 Lect 0 37 Quantization in all directions Now we must quantize the k values in each of the 3 directions = ( h /m* ) [(n x π/l x ) + (n y π/l y ) + (n z π/l z ) ] n x, n y, n x = 1,,... Lowest energy solutions with Ψ n (x,y,z) = 0 at x = 0,L x, y = 0,L y, z = 0,L z has behavior like that below in all three directions Ψ n (x) x n = Physics 460 F 006 Lect 0 38 Confinement energies of lectrons The motion of the is exactly like the electron in a box problems discussed in Kittel, ch. 6 xcept the have an effective mass m* And in this case, the box has length L in one direction (call this the z direction - L = L z ) and very large in the other two directions (L x, L y very large) Quantization in the confined dimension For in a box, the energy is quantized because the waves must fit into the box (Here we assume the box walls are infinitely high - not true but a good starting point) (k z ) = ( h /m* ) k z, k z = n π/l,,,... Key Point: For ALL cases, the energy (k) = ( h /m* ) (k x + k y + k z ) We just have to figure out what k x, k y, k z are! Physics 460 F 006 Lect 0 39 k Physics 460 F 006 Lect 0 40 lectrons in a thin layer To describe a thin layer, we consider a box with length L in one direction (call this the z direction and define L = L z ) and very large in the other two directions (L x, L y very large) Solution Ψ = 3/ L -3/ sin(k x x) sin(k y y) sin(k z z), k x = m π/l, m = 1,,, same for y,z (k) = ( h /m ) (k x + k y + k z ) = ( h /m ) k Approaches continuum as L becomes large k Physics 460 F 006 Lect

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