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Physics 460 F 2006 Lect 20 1
Lecture 20: Semiconductor Structures
Kittel Ch 17, p 494-503, 507- 511
+ extra material in the class notes
metal Oxide insulator
Semi- conductor
MOS Structure
Semi- conductor Large-gap
Layer Structure
Semi- conductor Large-gap
Semi- conductor Small-gap Physics 460 F 2006 Lect 20 2
Outline
- What is a semiconductor Structure?
- Created by Applied Voltages Conducting channels near surfaces Controlled by gate voltages MOSFET
- Created by material growth Layered semiconductors Grown with atomic layer control by “MBE” Confinement of carriers High mobility devices 2-d electron gas Quantized Hall Effect Lasers
- Covered briefly in Kittel Ch 17, p 494-503, 507- 511 - added material in the lecture notes
Physics 460 F 2006 Lect 20 3
Semiconductor Structure
metal Oxide insulator
Semi- conductor
MOS Structure
Semi- conductor Large-gap
Layer Structure
Semi- conductor Large-gap
Semi- conductor Small-gap
Physics 460 F 2006 Lect 20 4
MOS Structure
- If the metal “gate” is biased positive, holes are repelled and electrons are attracted to Interface
- Insulator prevents direct contact with the metal
Metal (+ bias)
Oxide insulator
Semi- conductor
p-type
valence band maximum
conduction band minimum
μ (^) Layer of electrons at interface
Physics 460 F 2006 Lect 20 5
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
- Electrons are bound to interface but free to move along interface across the full extent of the metal region
- How can this be used?
Metal (+ bias)
Oxide insulator
Semi- conductor
p-type
Layer of electrons at interface
Physics 460 F 2006 Lect 20 6
MOSFET Transistor - I
- The structure at the right has n regions at the surface of the p-type 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”
Metal gate
Oxide insulator
Semi- conductor
p-type
Source
Drain
n-type region created by doping
Physics 460 F 2006 Lect 20 7
MOSFET 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”
- MOSFET transistor!
Metal gate (+ bias)
Oxide insulator
Semi- conductor
p-type
Source
Drain
Layer of electrons at interface
Physics 460 F 2006 Lect 20 8
Semiconductor Layered Structures
- Electrons and holes can be confined and controlled by making structures with different materials
- Different materials with different band gaps fi electrons and/or holes 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 2006 Lect 20 9
Semiconductor Layered Structures
- Can be grown with interfaces that are atomically perfect - a single crystal that changes from one layer to another
- Example: GaAs/AlAs Single crystal (zinc blende structure) with layers of GaAs and AlAs
- Grown using “MBE” (Molecular Beam Epitaxy)
- Can “tune” properties by making an alloy of GaAs and AlAs, called Ga (^) 1-x Alx As
Semi- conductor Large-gap e.g. Ga (^) 1-x Alx As
Semi- conductor Large-gap e.g. Ga (^) 1-x Alx As
Semi- conductor Small-gap e.g. GaAs
Physics 460 F 2006 Lect 20 10
Semiconductor Layered Structures
- Example of bands in GaAs-AlAs structures
- Both electrons and holes are confined
- (Other systems can be different)
Ga (^) 1-x Alx As
GaAs Ga (^) 1-x Alx As
valence band maximum
conduction band minimum
Small-gap
Physics 460 F 2006 Lect 20 11
Uses of Layered Structures
- Confinement of carriers can be very useful
- Example - light emitting diodes
- Confinement of both electrons and holes increases efficiency
valence band maximum
conduction band minimum
holes
electrons
p-type n-type
light
Physics 460 F 2006 Lect 20 12
Uses of Layered Structures
- Confinement of light can be very useful
- Example - light emitting diodes, lasers
- Confinement of light is due to larger dielectric constant of the low band gap material - total internal reflection
holes
electrons
light - confined to direction along layer
Used to make the lasers in your CD player!
Physics 460 F 2006 Lect 20 19
Electrons on a line
- For electrons in a box, the energy is just the kinetic energy which is quantized because the waves must fit into the box
E (k) = ( h^2 /2m ) k 2 , k = m π/L, m = 1,2, ...
E
k
Approaches continuum as L becomes large
From Lecture 12 See Kittle, Ch 6
In Lecture 12 we emphasized the limit that the box is very large
Physics 460 F 2006 Lect 20 20
Quantization for motion in z direction
- E (^) n = ( h^2 /2m ) k (^) z^2 , k (^) z = n π/L, n = 1,2, ...
- Lowest energy solutions with Ψn ( x ) = 0 at x = 0,L
Ψ n ( x )
x
n = 1
n = 2
n = 3 Here we emphasize the case where the box is very small
Physics 460 F 2006 Lect 20 21
Total energies of Electrons
- Including the motion in the x,y directions gives the total energy for the electrons:
E ( k ) = ( h^2 /2m* ) (k (^) x^2 + k (^) y^2 + k (^) z^2 )
= E (^) n + ( h^2 /2m* ) (k (^) x^2 + k (^) y^2 )
= ( h^2 /2m* ) (n π/L)^2 + ( h^2 /2m* ) (k (^) x^2 + k (^) y^2 ) n = 1,2, ...
- This is just a set of two-dimensional free electron bands (with m = m) each shifted by the constant ( h^2 /2m ) (n π/L)^2 , n = 1,2, ...
Physics 460 F 2006 Lect 20 22
Quantized two-dimensional bands
E (^) n (k (^) x, k (^) y) = ( h^2 /2m* ) (n π/L)^2 + ( h^2 /2m* ) (k (^) x^2 + k (^) y^2 ) n = 1,2, ...
E
kx , ky
n = 1
n = 2
n = 3
Physics 460 F 2006 Lect 20 23
Quantized two-dimensional bands
- What does this mean? One can make two- dimensional electron gas in a semiconductor!
- Example - electrons fill bands in order
E
kx , ky
n = 1
n = 2
n = 3
μ
Electrons can move in 2 dimensions but are in one quantized state in the third dimension
Physics 460 F 2006 Lect 20 24
Density of States in two-dimensions
- Density of states (DOS) for each band is constant
- Example - electrons fill bands in order
E
n = 1
n = 2
n = 3
DOS
Physics 460 F 2006 Lect 20 25
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
valence band maximum
conduction band minimum
Small-gap
holes
electrons
L
Physics 460 F 2006 Lect 20 26
MOS Structure - Again
- Electrons 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
Metal (+ bias)
Oxide insulator
Semi- conductor
p-type
valence band maximum
conduction band minimum
μ (^) Layer of electrons at interface
Physics 460 F 2006 Lect 20 27
Electrons in two dimensions
- If the layer is think enough all electrons are in the lowest quantum state in the direction perpendicular to the layer but they are free to move in the other two directions
- E ( k ) = ( h^2 /2m* ) (n π/L)^2 + ( h^2 /2m* ) (k (^) x^2 + k (^) y^2 ) n = 1,2,...
- This can happen in a heterostructure (the density of electrons is controlled by doping)
- Or a MOS structure (the density of electrons is controlled by the applied voltage)
Small-gap
holes
electrons
L
μ Layer of at interfaceelectrons Physics 460 F 2006 Lect 20 28
Hall Effect
- See lecture 18 – here we consider only electrons of density n = #/area
- The Hall effect is given by ρHall = E (^) Hall / J B = -(1/ne) (SI units)
B
E
e J
EHall x
y z
Electrons confined to thin layer
Physics 460 F 2006 Lect 20 29
Hall Effect
- Expected result as the density n is changed
- The Hall constant = is given by ρHall = E (^) Hall / J B = V (^) Hall/IB = -(1/ne) where I = J x Ly , V (^) Hall = E (^) Hall x Ly
ρHall
n
1/n
Physics 460 F 2006 Lect 20 30
Hall Effect
Consider a MOS device in which we expect n to be proportional to the applied voltage
ρHall
V
1/n
μ (^) Layer of electrons at interface
Physics 460 F 2006 Lect 20 37
Semiconductor Quantum Dots
- Structures with electrons (holes) confined in all three directions
- Now states are completely discrete
- “Artificial Atoms”
Semi- conductor Large-gap e.g. AlAs
Semi- conductor Small-gap e.g. GaAs
Physics 460 F 2006 Lect 20 38
Quantization in all directions
- Now we must quantize the k values in each of the 3 directions E = ( h^2 /2m* ) [(n (^) x π/Lx)^2 + (ny π/L (^) y)^2 + (nz π/L (^) z) 2 ] nx , ny , nx = 1,2, ......
- Lowest energy solutions with Ψn ( x,y,z ) = 0 at x = 0,Lx , y = 0,Ly , z = 0,Lz has behavior like that below in all three directions
Ψ n ( x )
x
n = 1
n = 2
n = 3
Physics 460 F 2006 Lect 20 39
Confinement energies of Electrons
- The motion of the electrons is exactly like the “electron in a box” problems discussed in Kittel, ch. 6
- Except the electrons have an effective mass m*
- And in this case, the box has length L in one direction (call this the z direction - L = Lz) and very large in the other two directions (Lx, Ly very large)
- Key Point: For ALL cases, the energy
E ( k ) = ( h^2 /2m* ) (k (^) x^2 + k (^) y^2 + k (^) z^2 )
- We just have to figure out what kx, k (^) y, k (^) z are!
Physics 460 F 2006 Lect 20 40
Quantization in the confined dimension
- For electrons 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) E (k (^) z) = ( h^2 /2m* ) k (^) z^2 , k (^) z = n π/L, n = 1,2, ...
E
k
Physics 460 F 2006 Lect 20 41
Electrons 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 = Lz) and very large in the other two directions (L (^) x, Ly very large)
- Solution Ψ = 23/2^ L-3/2^ sin(k (^) xx) sin(k (^) yy) sin(k (^) zz) , k (^) x = m π/L, m = 1,2, …, same for y,z E (k) = ( h^2 /2m ) (k (^) x^2 E + k (^) y^2 + k (^) z^2 ) = ( h^2 /2m ) k 2
k
Approaches continuum as L becomes large
