Chapter 2

Semiconductor Heterostructures

In this lecture you will learn:

• Energy band diagrams in real space • Semiconductor heterostructures and heterojunctions • Electron affinity and work function • Heterojunctions in equilibrium • Electrons at Heterojunctions • Semiconductor Quantum wells Herbert Kroemer (1920-) Nobel Prize 2000 for the Semiconductor Heterostructure Laser

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Band Diagrams in Real Space - I N-type semiconductor P-type semiconductor Energy Energy Ec Ef  KT n  Nc e

Ef Ev  KT p  Nv e Ec E Ef c

  E k Ef k v Ev

For devices, it is useful to draw the conduction and valence band edges in real space:

Ec Ec Ef

Ef E v Ev x x

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

1 Band Diagrams in Real Space - II Electrostatic potential and electric field: An electrostatic potential (and an electric field) can be present in a crystal:  r and Er  r The total energy of an electron in a crystal is then given not just by the energy band dispersion E n  k  but also includes the potential energy coming from the potential:    En k  En k  er  Therefore, the conduction and valence band edges also become position dependent:   Ec  Ec  er  Ev  Ev  er 

Example: Uniform x-directed electric field Ec E  f Er  E xˆ   x Er  Ex xˆ  r  x  0  Ex x E v Ec x  Ec x  0  eEx x

x N-type semiconductor

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Electron Affinity and Work Function

Electron affinity “” is the energy required to remove an electron from the bottom of the conduction band to outside the crystal, i.e. to the vacuum level

Energy  Vacuum level 0 Potential in Conduction a crystal band

0 x Work function “W ” is the energy required to V remove an electron from the Fermi level to the vacuum level W 

Ec • Work function changes with doping but Ef affinity is a constant for a given material

Ev x

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

2 Semiconductor N-N Heterostructure: Electron Affinity Rule Heterostructure: A semiconductor structure in which more than one semiconductor material is used and the structure contains interfaces or junctions between two different semiconductors

Consider the following heterostructure interface between a wide bandgap and a narrow bandgap semiconductor (both n-type):

1 2

Eg1 Eg2 The electron affinity rule tells how the energy band V edges of the two semiconductors line up at  1 2 a hetero-interface Ec1 Ef 1 Ec2

Ef 2 Eg1 Eg2 E Ev1 v2

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Semiconductor N-N Heterojunction V

 1 2 Electrons Ec1 Something is wrong here: the Fermi level (the chemical E Ec2 f 1 potential) has to be the same everywhere in Ef 2 Eg1 equilibrium (i.e. a flat line) Eg2 Ev 2 Ev1

• Once a junction is made, electrons will flow from the side with higher Fermi level (1) to the side with lower Fermi level (2)

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

3 Semiconductor N-N Heterojunction: Equilibrium

Depletion region Accumulation • Electrons will flow from the V region 2 side with higher Fermi level (1)  to the side with lower Fermi 1 level (2) Ec1 Ec2 Ef 1 Ef 2 • Electron flow away from Eg2 semiconductor (1) will result in a E E region at the interface which is g1 v 2 depleted of electrons (depletion region). Because of positively Ev1 charged donor atoms, the depletion region has net --- 1 +++ 2 positive charge density +++ ------Eg1 +++ Eg2 • Electron flow into --- +++ semiconductor (2) will result in a --- +++ region at the interface which has an accumulation of electrons (accumulation region). The Note: the vacuum level follows the electrostatic accumulation region has net potential: negative charge density V x  V x  0  e  x  x  0 

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Semiconductor N-N Heterojunction: Equilibrium V

 1 2 • Electron flow from Ec1 semiconductor (1) to Ef 1 Ec2 semiconductor (2) continues eVb until the electric field due to the Ef 2 formation of depletion and Eg1 Eg2 accumulation regions becomes so large that the Fermi levels on Ev 2 Ev1 both sides become the same

Depletion • In equilibrium, because of the region eVb Accumulation electric field at the interface, V region 2 there is a potential difference between the two sides – called  1 the built-in voltage Ec1 Ec2 Ef 1 Ef 2 • The built-in voltage is related Eg2 to the difference in the Fermi E E levels before the equilibrium g1 v2 was established:

Ev1 eVb  Ef 1  Ef 2

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4 Semiconductor P-N Heterojunction V

 1 2 Electrons Ec1 Ef 1 Ec2

eVb E Eg1 g2 Ef 2 Ev1 Ev 2 Holes

Once a junction is made:

• Electrons will flow from the side with higher Fermi level (1) to the side with lower Fermi level (2)

• Holes will flow from the side with lower Fermi level (2) to the side with higher Fermi level (1)

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Semiconductor P-N Heterojunction: Equilibrium

eVb Depletion 2 • Electron flow away from region Depletion semiconductor (1) will result in a V region region at the interface which is E depleted of electrons (depletion 1 c2 E E region). Because of positively c1 g2 charged donor atoms, the Ef 2 Ef 1 depletion region has net Ev 2 positive charge density Eg1 • Hole flow away from E v1 semiconductor (2) will result in a region at the interface which is 1 +++ --- 2 depleted of holes (depletion +++ --- region). Because of negatively Eg1 +++ --- Eg2 charged acceptor atoms, the +++ --- depletion region has net +++ --- negative charge density

Note: the vacuum level follows the electrostatic potential: V x  V x  0  e  x  x  0 

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

5 Semiconductor P-N Heterojunction: Equilibrium V

 1 2 Ec1 • Electron flow from E semiconductor (1) to f 1 Ec2 eVb semiconductor (2) and hole flow E from semiconductor (2) to Eg1 g2 semiconductor (1) continues Ef 2 until the electric field due to the E Ev1 v 2 formation of depletion regions becomes so large that the Fermi levels on both sides become the eVb Depletion 2 same region Depletion V region • The built-in voltage is related Ec2 to the difference in the Fermi 1 E E levels before the equilibrium c1 g2 was established: Ef 2 Ef 1 Ev 2 eVb  Ef 1  Ef 2

Eg1

Ev1

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Types of Semiconductor Heterojunctions V 1 2 Ec1

Type-I: Straddling gap Ec2

Eg1 Eg2

Ev2 Ev1 V 1 2 E Ec1 c2

Type-II: Staggered gap Eg2 Ev2 Eg1

Ev1 V 2 Ec2  1 Eg2

Type-III: Broken gap Ev 2 Ec1

Eg1

Ev1

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6 Band Offsets in Heterojunctions V

 1 2 Ec1 E c Ec2

Eg1 Eg2 Ev2 Ev1 Ev

The conduction and valence band offsets are determined as follows:

Ec  2  1

Ev  Eg  Ec  Eg1  Eg2  Ec

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

A PN Heterojunction

Vacuum level q1 Ec1 q2 Ec Ec2  Nd  Ef2 Ef 2  Ec2  KT ln   Nc2  Eg1 Eg2  Na  E Ev2 E  E  KT ln  f1 E v1 f 1   v  Nv1  Ev1

 x

p  N -- ++ po a nno  Nd 1 (p-doped) -- ++ 2 (n-doped) 2 ni1 -- ++ n2 n  npo  p  p  i2 N -- ++ no a Nd

-x xn p  x

 NaNd  qVbi  Ef 2  Ef 1 qVbi  Eg2  Ev  KT ln .  Nc2Nv1 

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

7 A PN Heterojunction in Equilibrium

q1

Ec1 Vacuum level

Eg1 Ec q2

Ec2 Ef Ef Ev1 Eg2

Ev Ev2

-x x x p  n The Depletion Approximation: (x)

+qNd

-xp +  - xn x

-qNa

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

A PN Heterojunction in Equilibrium

Electric Field: E(x) d  x E x   x dx -xp

 qNd xn  x   0  x  x xn x   n  2  qNa x  xp Ex    x p  x  0  1 0 elsewhere   (x) V 2 bi Electrostatic Potential: 2 Nd xn Na xp q q 22 Charge per unit area: 21  -x xn x qNd xn  qNa xp  Q p

1 2 qNd xn  qNa x p 2 Nd Vbi  xp   12  q Na 1Na  2Nd  2 2 1 Nd xn Na xp Vbi  q  q 2 Na Vbi  2 22 21 xn   12  q Nd 1Na  2Nd 

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8 A PN Heterojunction in Reverse Bias

- - ++ 1 (p-doped) -- ++ 2 (n-doped) V<0 -- ++

-W -x x W p p  n n x + - V Quasi Fermi Levels and their Splitting:

q1

Ec1 Vacuum level Eg1 Ec

q2 Ef1 Ev1 -qV Ec2 Ef2 Eg2 Ev Ev2

-xp  xn x

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

A PN Heterojunction in Reverse Bias

-- ++ V<0 1 (p-doped) -- ++ 2 (n-doped) -- ++

-W x W p -xp  n n x + - V

Depletion regions grow in width:

1  2 Nd Vbi V  2 xn V   1 2  q Na 1Na  2Nd  1  2 Na Vbi V  2 x p V   12  q Nd 1Na  2Nd 

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

9 A PN Heterojunction in Forward Bias

- - ++ V>0 1 (p-doped) - - ++ 2 (n-doped) - - ++

-W x W p -xp  n n x + - V

Vacuum q1 Electrons level Ec1 q2 Ec E g1 Ec2 Ef2 qV E Ef1 g2 Ev1 Ev2 Ev Holes

-xp  xn x

Now diffusion exceeds drift!! Minority carrier injection………

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

A PN Heterojunction in Forward Bias

Assumption: Vacuum q1 Electrons level Ec1 q2 Main bottleneck for current Ec E flow are the quasineutral g1 Ec2 regions and not the Ef2 qV E depletion regions Ef1 g2 Ev1 Ev2 Ev Holes

-xp  xn x

Electron concentration on the p-side:

Ef 2(xp )Ec (xp ) KT Ef 1(xp )Ec (xp ) KT Ef 2(xp )Ef 1(xp ) KT n(xp )  Nc1 e  Nc1 e e 2 ni1 qV KT qV KT  e  npo e Na Ev (xp )Ef 1(xp ) KT p(xp )  Nv1 e  Na

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

10 A PN Heterojunction in Forward Bias

Assumption: Vacuum q1 Electrons level Ec1 q2 Main bottleneck for current Ec E flow are the quasineutral g1 Ec2 regions and not the Ef2 qV E depletion regions Ef1 g2 Ev1 Ev2 Ev Holes

-xp  xn x Hole concentration on the n-side:

Ef 2(xn )Ec (xn ) KT n(xn )  Nc2 e  Nd Ev (xn )Ef 1(xn ) KT Ev (xn )Ef 2(xn ) KT Ef 2(xn )Ef 1(xn ) KT p(xn )  Nv2 e  Nv2 e e 2 ni2 qV KT qV KT  e  pno e Nd

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

A PN Heterojunction in Forward Bias Minority carrier concentrations:

p(x)

n(x)

-W -x  x W p p n n x Electrons on the p-side: Excess electrons injected in nx  npo Re x Ge x  the p-side will recombine e1 with the holes

nx J x  q D Diffusion current e e1 x Need to solve: 0 n 1   J x  G x  R x t q x e e e

2  nx nx  npo  D  e1 2 x e1

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

11 A PN Heterojunction in Forward Bias Minority carrier concentrations:

P-side: N-side: 2 2  nx nx  npo  px px  p  D   no e1 2 2 2 x e1 x Lh2 2  nx nx  npo L  D    h2 h2 h2 2 2 x Le1

Le1  De1e1

Boundary conditions: Boundary conditions: qV KT qV KT n(xp )  npo e p(xn )  pno e n(W )  n p po ?? p(Wn )  pno ??

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

A PN Heterojunction in Forward Bias Minority carrier concentrations:

Wp  x  sinh  qV  L     e1   KT  nx  npo  npo e  1  Wp  x  x p Wp  x p    sinh     Le1 

Wn  x  sinh  qV  L    px  p  p  h2  e KT  1 x  x  W no no   n n Wn  xn  sinh     Lh2 

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

12 A PN Heterojunction in Forward Bias Majority carrier concentrations and charge neutrality:

P-side:

One must have: px  px  Na  nx  nx  npo

N-side:

One must have: nx   nx   Nd  px   px   pno

Excess majority carrier density must balance the excess minority carrier density

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

A PN Heterojunction in Forward Bias Minority carrier current:

P-side: ~0 nx J x  qn E x  q D e e1 e1 x Wp  x  cosh  qv n2 D  L     q i1 e1  e1  e KT  1 W  x  x   p p Na Le1 Wp  xp  sinh    N-side:  Le1  ~0 px  J x  qp E x  q D h h2 nh x

Wn  x  cosh  qv n2 D  L     q i2 h2  n2  e KT  1 x  x  W   n n Nd Lh2 Wn  xn  sinh     Ln2 

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

13 A PN Heterojunction in Forward Bias Minority carrier current:

Je(x) Jh(x) -W -x  x W p p n n x

Since there is no obstacle to current flow in the depletion regions, and if we ignore electron-hole recombination in the depletion region, we must have:

JT

Je(x) Jh(x) -W -x  x W p p n n x Total current:

JT  Je x  Jh x Must be constant throughout the device

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

A PN Heterojunction in Forward Bias Total current:

A JT

Je(x) I Jh(x) -W -x  x W p p n n x + - V 2 2  qv  n De1 Wp  xp  n D W  x   J  q i1 coth   i2 h2 coth n n  e KT  1 T N L  L  N L  L     a e1  e1  d h2  h2  

 qv  I  AJ  I e KT  1 T o    

2 2 ni1 De1 Wp  xp  ni2 Dh2 Wn  xn  Io  qA coth   coth  Na Le1  Le1  Nd Lh2  Lh2 

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

14 A PN Heterojunction in Forward Bias Majority carrier current:

JT

Je(x) Jh(x) -W -x  x W P-side: p p n n x

Jh x  JT  Je x

N-side:

Je x  JT  Jh x

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

A PN Heterojunction in Forward Bias

Quasi Fermi Levels:

Vacuum level q1

Ec1 q2 Ec E g1 Ec2 E qV f2 Ef1 Eg2 Ev1 Ev2 Ev

-xp  xn x

 qv  I  AJ  I e KT  1 T o    

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

15 A PN Heterojunction in Reverse Bias

Quasi Fermi Levels:

q1

Ec1 Vacuum level Eg1 Ec

q2 Ef1 Ev1 -qV Ec2 Ef2 Eg2 Ev Ev2

-xp  xn x

 qv  I  AJ  I e KT  1 T o    

Reverse bias current: I  -Io Why?

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

A PN Heterojunction in Forward Bias

Electron-Hole Recombination in the Depletion Region:

Vacuum level q1

Ec1 q2 Ec E g1 Ec2 E qV f2 Ef1 Eg2 Ev1 Ev2 Ev

-xp  xn x 1   J x  G x  R x q x e e e

xn  Je xn  Je  x p  q  Re x  Ge x dx xp

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

16 A PN Heterojunction in Forward Bias

Electron-Hole Recombination in the Depletion Region:

JT

Je(x) Jh(x) -W -x  x W p p n n x 1   J x  G x  R x q x e e e

xn  Je xn  Je  x p  q  Re x  Ge x dx xp

Similarly:

xn Jh  x p  Jh xn  q  Rh x  Gh x dx xp

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

A PN Heterojunction in Forward Bias Electron-Hole Recombination in the Depletion Region:

JT

Je(x) Jh(x) -W -x  x W p p n n x

qv 2 2   xn n De1 Wp  xp  n D W  x   J  q i1 coth   i2 h2 coth n n  e KT  1  q  R x  G x dx T N L  L  N L  L    e e  a e1  e1  d h2  h2   xp

2 Ge x  Re x  np  ni 

2 Ef 2 Ef 1 KT np  ni e

 qv  G x  R x  e KT  1 e e    

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

17 Effective Mass Schrodinger Equation Energy Consider a semiconductor with energy band dispersion:  2     E k  E   k  k . M 1 . k  k c c 2 o e o Ec The Bloch functions are solutions of the equation: Ef  22      V r   r  E k   r   Lattice  c,k c c,k Ev k  2m    e ik.r   r  u  r c,k V c,k  What if one needs to solve the equation: ko

2 2           VLattice r  U r  r  E r  2m 

Some extra potential (perhaps due to some crystal impurity, defect, or external electric field)

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Effective Mass Schrodinger Equation 2 2 Energy           VLattice r  U r  r  E r  2m  One can in most cases write the solution as: Ec Ef  r  r  r c,ko  Ev k Envelope function

Where the envelope function satisfies the “effective mass  Schrodinger equation”: ko ˆ     Ec ko  i  Ur r  E  r

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

18 The Envelope Function Energy ˆ     Ec ko  i  Ur r  E  r Electron  r  r   r wavefunction c,ko  k

Slowly varying envelope function Bloch function

 ko r

  r c,ko

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

The Effective Mass Schrodinger Equation: An Example Energy Consider a conduction energy band with the dispersion: 2 2 2 k k 2 2 2   kx  kox   y  oy   kz  koz Ec k  Ec    2mxx 2myy 2mzz  ko   ˆ    k What is this equation: Ec ko  i  Ur r  E  r

Note that one has to make the following replacements in the energy dispersion relation:

     E k  Eˆ k  i   k  k  i k  k  i k  k  i c c o x ox x y oy y z oz z ˆ  The operator E c  k o  i   is then:  2 2 2 2 2 2 E k  i  E       c o c 2 2 2 2mxx x 2myy y 2mzz z The effective mass Schrodinger equation becomes:

2 2 2 2 2 2             2  2  2  Ec  Ur  r  E  r  2mxx x 2myy y 2mzz z 

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

19 Electrons at Heterojunctions

Ec2 Ec Ec1

Eg1 Eg2 E v1 Ev Ev 2 Question: What happens to the electron that approaches the interface (as shown)? How does it see the band offset? Does it bounce back? Does it go on the under side?

The effective mass equation can be used to answer all the above questions

In semiconductor 1:  r   r   r 1 1 c1,ko ˆ     Ec1ko  i  Ur 1r  E 1 r In semiconductor 2:

 r   r   r 2 2 c2,ko ˆ     Ec2 ko  i  Ur 2r  E 2 r

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Electrons at Heterojunctions; Effect of Band Offsets

Ec2 Ec Ec1  Ur   0 Eg1 Eg2 E v1 Ev Ev 2 Assume for the electron in the conduction band of semiconductor 1: 2 2   k     Notice that the E k  E   1r   1r   c1,k 0 r  c1 c1 2m o conduction band edge e1 energy (i.e. E or E ) 2 c1 c2   appears as a constant    2  E  r  E  r  c1  11  potential in the effective 2me1   mass Schrodinger And for the electron in semiconductor 2: equation 2k 2       r    r    r  Conduction band offset Ec2 k  Ec2  2 2 c2,ko 0 2me2 at the heterojunction 2 therefore appears like a   2         Ec2  2r  E 2 r potential step to the  2me2  electron

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

20 Electrons at Heterojunctions: Boundary Conditions

(1) Continuity of the wavefunction at the boundary:   1r x 0   2r x 0     If one assumes:   r   r  1r   2r  c1,ko c2,ko x 0 x 0

(2) Continuity of the normal component of the probability current at the boundary: In text book quantum mechanics the probability current is defined as:  Jr   * r   r  c.c.   * r   r  r   * r 2im 2im 2im Or in shorter component notation: J r   * r    r  c.c.  2im  Probability current is always continuous across a boundary We need an expression for the probability current in terms of the envelope function

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Electrons at Heterojunctions: Boundary Conditions Probability Current: In a material with energy band dispersion given by: 2 2     1    En k  En  k  ko .M . k  ko  En   k  ko k  ko 2 , 2m The expression for the electron probability current (in terms of the envelope function) is:  *    J r   r  r  c.c.  2im

Continuity of the probability current: The continuity of the normal component of the probability current across a heterojunction gives another boundary condition for the envelope function: 1 1    r     r  1 x 0  2 x 0  mx 1  mx 2

1 mxx  1  r 1  r For: M 1   1 m  1  2  yy  mxx1 x x 0 mxx2 x x 0  1 mzz 

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

21 Electrons at Heterojunctions: Boundary Conditions

Semiconductor 1 Semiconductor 2

x  0 x (1) Continuity of the envelope function at the boundary:   1r  x 0  2r  x 0

(2) Continuity of the normal component of the probability current at the boundary: 1 1    r     r  1 x 0  2 x 0  mx 1  mx 2

If in both the materials the inverse effective mass matrix is diagonal then this boundary condition becomes:

1 mxx  1  r 1  r M 1   1 m  1  2  yy  mxx1 x x 0 mxx2 x x 0  1 mzz 

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

The Effective Mass Theory for Heterojunctions

Ec2 Ec Ec1 x 0 Assume in semiconductor (1): Assume in semiconductor (2):   ko  0 ko  0 2 2 2k 2 2 2 2 2 2k2 2 2   kx  y  kz   kx  y  kz Ec1k  Ec1    Ec2 k  Ec2    2mx1 2my1 2mz1 2mx2 2my 2 2mz2

In semiconductor (1):

ˆ     Ec1ko  i  Ur 1r  E 1 r ˆ    Ec1 i 1r  E 1 r  2 2 2 2 2 2          E  r  E  r  2 2 2 c1 11   2mx1 x 2my1 y 2mz1 z 

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

22 The Effective Mass Theory for Heterojunctions  rt 1r  2 r  Ec2 Ec Ec1 x In semiconductor (1): 0

 2 2 2 2 2 2         E  r  E  r  2 2 2 c1 11   2mx1 x 2my1 y 2mz1 z 

 i k x1x ky y kzz Assume a plane wave solution: 1r  e

2 2 2k 2 2 2  kx1  y  kz A plane wave Plug it in to get: E  Ec1    solution works 2mx1 2my1 2mz1

We expect a reflected wave also so we write the total solution in semiconductor (1) as:

 i k x1x ky y kzz i k x1x ky y kz z 1r  e  r e

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

The Effective Mass Theory for Heterojunctions  rt 1r  2 r  Ec2 Ec Ec1 x In semiconductor (2): 0

 2 2 2 2 2 2         E  r  E  r  2 2 2 c2  2 2   2mx2 x 2my2 y 2mz2 z 

 i k x2x ky y kzz Assume a plane wave solution: 2r  t e

2 2 2k2 2 2 A plane wave  kx2  y  kz Plug it in to get: E  Ec2    solution works 2mx2 2my 2 2mz2 here also

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

23 Boundary Conditions at Heterojunctions  rt 1r  2 r  Ec2 Ec Ec1 x 0 2 2 2k 2 2 2  i k x1x ky y kzz i k x1x ky y kzz  kx1  y  kz 1r  e  r e E  Ec1    2mx1 2my1 2mz1 2 2  i kx 2x ky y kzz 2 2 k 2 2  r  t e  kx2  y  kz 2 E  Ec2    2mx2 2my 2 2mz2 (1) Envelope functions must be continuous at the interface:

1x  0  2 x  0 i k y k z i k y k z i k y k z  e y z  r e y z  t e y z  1 r  t

Note that this boundary condition can only be satisfied if the components of the wavevector parallel to the interface are the same on both sides

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Boundary Conditions at Heterojunctions  rt 1r  2 r  Ec2 Ec Ec1 x 0 2 2 2k 2 2 2  i k x1x ky y kzz i k x1x ky y kzz  kx1  y  kz 1r  e  r e E  Ec1    2mx1 2my1 2mz1 2 2  i kx 2x ky y kzz 2 2 k 2 2  r  t e  kx2  y  kz 2 E  Ec2    2mx2 2my 2 2mz2 Energy conservation: 2 2 2k2 2 2 2 2 2k2 2 2  kx1  y  kz  kx2  y  kz E  Ec1     Ec2    2mx1 2my1 2mz1 2mx2 2my 2 2mz2 2 2 2 2 2 2 2 2 k k  ky  1 1  k  1 1    x2   x1  E       z    c     2mx2 2mx1 2  my 2 my1  2  mz2 mz1  2 2 2 2  kx2  kx1    Veff ky ,kz 2mx2 2mx1 Note that the effective barrier height depends on the band offset as well as the parallel components of the wavevector

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

24 Boundary Conditions at Heterojunctions  rt 1r  2r  Ec2 Ec Ec1 x 0 2 2 2k 2 2 2  i k x1x ky y kzz i k x1x ky y kzz  kx1  y  kz 1r  e  r e E  Ec1    2mx1 2my1 2mz1 2 2  i kx 2x ky y kzz 2 2 k 2 2  r  t e  kx2  y  kz 2 E  Ec2    2mx2 2my 2 2mz2 (2) Probability current must be continuous at the interface:

Conservation of 1  1  1  2 probability current at mx1 x x 0 mx2 x x 0 the interface ik i k y k z i k y k z ik i k y k z  x1  e y z  r e y z  x2 t e y z mx1 mx2 k k  x1 1 r  x2 t mx1 mx2

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Transmission and Reflection at Heterojunctions rt Ec2 Ec Ec1 x 0 We have two equations in two unknowns: k k 1 r  t x1 1 r  x2 t mx1 mx2 The solution is: 2 1 m k m k t  r  x1 x2 x2 x1 1 mx1kx2 mx2kx1 1 mx1kx2 mx2kx1 Where: 2 2 2 2  kx2  kx1   Veff ky ,kz 2mx2 2mx1

Special case: If the RHS in the above equation is negative, then kx2 becomes imaginary and the wavefunction decays exponentially for x>0 (in semiconductor 2). In this case: r  1 and the electron is completely reflected from the hetero-interface

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

25 Semiconductor Quantum Wells

Ec2 Ec2 Ec1 Ec A thin (~1-10 nm) narrow bandgap material AlGaAs GaAs AlGaAs sandwiched between two Ev1 wide bandgap materials Ev 2 Ev 2

Semiconductor quantum wells can be composed of pretty much any semiconductor from the groups II, III, IV, V, and VI of the periodic table TEM micrograph GaAs GaAs InGaAs quantum well (1-10 nm)

GaAs InGaAs

GaAs

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Semiconductor Quantum Well: Conduction Band Solution

Ec2 L Ec2 Ec Ec1 x Assumptions and solutions: 0 2 2 2 2   k   k Ec1k  Ec1  Ec2 k  Ec2  2me 2me

ˆ   ˆ   Ec1 i 1r  E 1 r Ec2 i 2 r  E 2 r 2 2  2 2                Ec1 1r  E 1 r     Ec2  2 r  E 2 r  2me   2me 

Symmetric  x L 2 iky y kzz ik y k z e e cos k x e y z   x 2r  B i k y k z x  L 2 1r  A  x L 2 y  z iky y kz z e e  sinkx x e   x L 2 iky y kzz Anti-symmetric   e e 2r  B x   L 2  x L 2 iky y kzz  e e

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

26 Semiconductor Quantum Well: Conduction Band Solution

Ec2 L Ec2 Ec1 Ec x 0 Energy conservation condition: 2 k 2  k 2 2   2  k 2 E  E   x ||  E   || c1 c2 2 2 2 2me 2me k||  ky  kz 2m    e E  k 2 2 c x  The two unknowns A and B can be found by imposing the continuity of the wavefunction condition and the probability current continuity condition to get the following conditions for the wavevector kx:  2m e E  k2   k L   2 c x  tan x     Wavevector kx cannot   2  kx kx be arbitrary!  Its value must satisfy  2me 2 2 Ec  kx these transcendental   kxL     cot    equations   2  kx kx

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Semiconductor Quantum Well: Conduction Band Solution r Ec2 Ec2

Ec

Ec1 L x Graphical solution: 0  2me 2  Ec  kx  k L   2 Different red curves for Increasing Ec values  tan x       2  k k  x x 2m  e E  k2 k L 2 c x   x     cot      2  kx kx

In the limit Ec  ∞ the values of kx are: 0   3 2 5 kxL 2 kx  p L ( p = 1,2,3…….. 2 2 2

• Values of kx are quantized • Only a finite number of solutions are possible – depending on the value of Ec

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

27 Electrons in Quantum Wells: A 2D Fermi Gas

Ec2 Ec2 E 2 Ec

E1 Ec1 L x 0 Since values of kx are quantized, the energy dispersion can be written as: 2 2 2k 2  kx  || 2 2 2 E  Ec1   k||  ky  kz 2me 2me 2 2  k||  Ec1  Ep  p = 1,2,3…….. 2me 2 2   p  In the limit Ec  ∞ the values of Ep are: Ep    p = 1,2,3…….. 2me  L  • We say that the motion in the x-direction is quantized (the energy associated with that motion can only take a discrete set of values) • The freedom of motion is now available only in the y and z directions (i.e. in directions that are in the plane of the quantum well) • Electrons in the quantum well are essentially a two dimensional Fermi gas!

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Energy Subbands in Quantum Wells

Ec2 Ec2 E 2 Ec

E1 Ec1 L x E 2 2 0   k|| E Ec p,k||  Ec1  Ep  2me p =1,2,3…….. 2 2 2 k||  ky  kz

The energy dispersion for Ec1  E3 electrons in the quantum wells can be plotted as shown Ec1  E2 It consists of energy Ec1  E1 subbands (i.e. subbands of kz Ec1 the conduction band)

 Electrons in each subband k ky || constitute a 2D Fermi gas

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

28 Density of States in Quantum Wells

Ec2 Ec2 E 2 Ec

E1 Ec1 L x 0

Suppose, given a Fermi level position Ef , we need to find the electron density: We can add the electron present in each subband as follows: 2  d k||  n 2 f E p,k E     2 c ||  f p 2

Ec1  E3 If we want to write the above as: Ef Ec1  E2  E  E n   dE gQW E f E  Ef c1 1 E c1 Ec1

 Then the question is what is the density of states gQW(E ) ? k||

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Density of States in Quantum Wells 2 2   k|| Ec p,k||  Ec1  Ep  2me Start from: 2  Ec1  E3 d k Ef ||  Ec1  E2 n   2   2 f Ec p,k||  Ef p 2 Ec1  E1 And convert the k-space integral to energy space: Ec1   me  n    dE   f E  E   2  f k|| p Ec1Ep       m   dE  e   E  E  E f E  E    2  c1 p f Ec1 p     gQW E  This implies: m 3 e 2    m  m g E   e   E  E  E 2 e QW    2  c1 p 2 p       me 2  

Ec1 Ec1  E1 Ec1  E2 Ec1  E3

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

29 Density of States: From Bulk (3D) to QW (2D)

E E E E

Ec1  E3

Ec1  E2

Ec1  E1

Ec1 Ec1

  k g E k g3DE || QWg2DE m m m e 2 e 3 e 2 2 2      

The modification of the density of states by quantum confinement in nanostructures can be used to: i) Control and design custom energy levels for laser and optoelectronic applications ii) Control and design carrier scattering rates, recombination rates, mobilities, for electronic applications iii) Achieve ultra low-power electronic and optoelectronic devices

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Semiconductor Quantum Well: Valence Band Solution Ev1 E E v E v 2 L v 2 x Assumptions and solutions: 0 2 2 2 2  k   k E k  E   Ev1k  Ev1  v 2 v2 2m 2mh v ˆ   ˆ   Ev 2 i 2 r  E 2 r Ev1 i 1r  E 1 r  22   22      E  r  E  r     E  r  E  r  v 2  2 2  v1 11   2mh   2mh   22   22      E  r  E  r     E  r  E  r  v 2  2 2  v1 1 1  2mh   2mh    x L 2 iky y kzz Symmetric  e e 2r  B x  L 2  iky y kzz  x L 2 iky y kzz  coskx x e e e 1r  A  iky y kz z  sinkx x e   x L 2 iky y kzz   e e 2r  B x   L 2 Anti-symmetric  x L 2 iky y kzz  e e

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

30 Semiconductor Quantum Well: Valence Band Solution Ev1 E E v E v 2 L v 2 x Energy conservation condition: 0

2 2 2 2 2 2  kx  k||     k|| E  Ev1   Ev 2  2mh 2me 2m    h E  k 2 2 v x  The two unknowns A and B can be found by imposing the continuity of the wavefunction condition and the probability current conservation condition to get the following conditions for the wavevector kx:  2m h E  k2   k L   2 v x  tan x     Wavevector kx cannot   2  k k be arbitrary!  x x  2mh 2 2 Ev  kx   kxL     cot      2  kx kx

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Semiconductor Quantum Well: Valence Band Solution L Ev1

Ev E v 2 Ev 2 x Graphical solution: 0  2mh 2  Ev  kx  k L   2 Different red curves for Increasing Ev values  tan x       2  k k  x x 2m  h E  k2 k L 2 v x   x     cot      2  kx kx

In the limit Ev  ∞ the values of kx are: 0   3 2 5 kxL 2 kx  p L ( p = 1,2,3…….. 2 2 2

• Values of kx are quantized • Only a finite number of solutions are possible – depending on the value of Ev

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

31 Semiconductor Quantum Wells: A 2D Fermi Gas L Ev1 E1 Ev E2 E Ev 2 v 2 x 0 Since values of kx are quantized, the energy dispersion can be written as: 2 2 2k 2  kx  || E  Ev1   2mh 2mh 2 2  k|| Light-hole/heavy-hole  Ev1  Ep  p = 1,2,3…….. degeneracy breaks! 2mh 2 2   p  In the limit Ev  ∞ the values of Ep are: Ep    p = 1,2,3…….. 2mh  L  • We say that the motion in the x-direction is quantized (the energy associated with that motion can only take a discrete set of values) • The freedom of motion is now available only in the y and z directions (i.e. in directions that are in the plane of the quantum well) • Electrons (or holes) in the quantum well are essentially a two dimensional Fermi gas!

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Density of States in Quantum Wells: Valence Band 2 2   k|| E p,k  E  E   v || v1 p k 2mh || Start from: 2  Ev1 d k||  p  2  1 f E p,k  E   2 v || f Ev1  E1 p 2 Ev1  E2 And convert the k-space integral to energy space: Ef Ev1  E3 Ev1Ep  m  p  dE  h  1 f E  E    2  f p      Ev1  m   dE   h   E  E  E 1 f E  E   2  v1 p f gQW E  p     mh 3 2 This implies:   m 2 h  m   2 g E    h   E  E  E  QW  2  v1 p mh p     2  

Ev1  E3 Ev1  E2 Ev1  E1 Ev1 E

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

32 Growth of Semiconductor Heterostructures: MBE

Low pressure (10-11 Torr), near-equilibrium, chemical reaction free, layer-by-layer growth

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Growth of Semiconductor Heterostructures: MOCVD or MOVPE

PH3 CH TM-In 4

Adsorption

Growth of InP by MOCVD

Atm pressure (760 Torr) growth, involves gas flow and chemical reactions

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

33 Epitaxial Growth and Lattice Mismatch

A lattice mismatch between the epitaxial layer and the substrate means that the layer grown will be strained (biaxial strain):

a   0 Tensile strain asub  a     a sub   0 Compressive  strain asub

if the thickness h of the coherently strained layer exceeds a certain critical thickness

hc the coherent strain relaxes and this process generates crystal dislocations (crystal defects). Critical thickness is given by:  2  b 1 cos   hc  Matthews-Blakeslee hc    ln  4  1 cos    b  Formula b  a 2 for diamond and zinc-blende lattices

 Poisson ratio

 and  are both equal to 60-degrees for diamond and zinc-blende lattices

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Strain Compensation

How does one calculate the critical thickness for a multiple layer stack?

h2 h1 1h1  2h2  3h3  avg  h1  h2  h3  Substrate

2 b 1 cos    hc  hc    ln  4  1 cos    b  Strain compensation can be used to grow much thicker dislocation-free layers!

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

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