(a)
(b) Coupled problem: What are the energy levels?
Figure 2.14: (a) A schematic of two uncoupled wells separated by a large potential barrier.
(b) A schematic of two coupled potential wells.
2.5 FROM ATOMS TO MOLECULES: COUPLED WELLS
We have seen that in an attractive potential the electron (or a general particle) has discrete bound levels separated by energy gaps. We also know that, when some atoms are brought together, they attract each other to form a molecule. In this section we will examine how electronic levels in a quantum well change when another quantum well is brought close to it. This is the so-called coupled quantum well problem.
Let us consider two quantum wells separated by a large distance so that there is no coupling between them. Thus electrons occupying the bound states of each quantum well have no overlap with electrons in the other well, as shown in Fig. 2.14a. Now let us assume that the wells are brought closer to each other so that electrons in each well can sense the presence of the neighboring well, as shown in Fig. 2.14b. We are interested in finding out what happens to the allowed electronic levels in the coupled quantum wells problem.
We assume that we know the solutions for the uncoupled wells. When the quantum wells are spaced far apart and are uncoupled, let us say they each have states with energies E\ and E2, respectively, and wavefunctions <j)\ and 02, as shown in Fig.
2.14a. There may be other states in the quantum wells, but we will assume that their energies lie far from E\ and E2. We will see later that states far removed from the
ones under consideration have a smaller effect on the solutions. We are interested in the solutions of the problem where the wells are brought closer so that there is some coupling between the wells. We write the Hamiltonian of the coupled problem as
H = Ho - AU (2.86)
where AU is the correction due to the coupling of the well. The potential energy is reduced in the region between the wells, compared to the uncoupled system, so that AU is positive.
In the absence of any coupling, we have
(2.87) When the wells are coupled, the functions <j>\ and <j)2 are no longer the solutions. How- ever, the new solutions can be expressed in terms of the uncoupled solutions to a good approximation. Let us write the solution for the coupled problem as
$ = M i - f a2</>2 (2.88) where a\ and a2 are unknown parameters which we will solve. If the system is uncoupled we have the two solutions
ipi = <j>i\ a i = 1, a2 - 0 ip2 = 4>2\ a>i — 0, a2 — 1
Coming to the coupled problem, the equation to be solved has the form
H{al<i)l + a2<f)2) = E{a1<t>1 + a2<j)2) (2.89) where H represents the full Hamiltonian of the coupled problem. We now multiply this equation from the left by <f)\ and integrate over space to get
a\ / (j)\(H§ — AU)<f>id r + o>2 I ^ i ( ^ o ~~ AU)<f)2c
J J
= Ea,\ j <f)*y(f)id r -\- Ea2 I <f>^<f>2d r Using the following equations
f <f>l<f>2d3r = 0
we get
a1(E1- E) + a2H12 = 0 (2.90)
2.5. From atoms to molecules: coupled wells 71 where we have defined
J r 1
The quantity H12 is called the matrix element of the Hamiltonian between the two states 0i and (j)2. In c a s e the quantum wells are separated by a large distance the matrix element is zero. It increases as the coupling increases.
If we repeat the process described above but multiply Eq. 2.89 by ^ instead of
</>*, we get
H21ax-{-a2(E2 - E) = 0 (2.91) where
# 2 i = - f <f>*2AU<f>id3r
Assuming that the energy operator is real (energy is conserved) we have
The two coupled equations (Eqs. 2.90 and 2.91) can be written as a matrix vector product
E — Ei Hi2 a,i „ H2i E — E2 a2
To get non-trivial solutions of this equation, the determinant of the matrix must vanish.
This gives us a quadratic equation with the following solutions
The coefficients ai and a2 can now be solved for and are a\_ _ H12
a2 ~ E - Ei or
E- ^2
( M 2 )
(2.93)
The simple equation we have derived has very useful implications for understanding many interesting and important physical systems.
Coupling of identical quantum wells
Let us examine in some detail the case where the two coupled quantum wells are identical and have the same initial energies and states, as shown in Fig. 2.15. It is important to keep in mind that when we talk of "quantum wells" we are simply referring to any problem with bound states. Thus the quantum wells may be atoms and molecules as well as potential wells created by use of semiconductors. Let us write
E\ = E2 = Eo
H12 = H2l = -A (2.95)
(a)
Asymmetric state
Symmetric state
(c)
Figure 2.15: (a) A schematic of two identical uncoupled wells, (b) Coupled identical wells with energy levels and eigenfunctions. (c) Dependence of the symmetric and asymmetric eigenener- gies on well separation or coupling.
2.5. From atoms to molecules: coupled wells 73 The quantity A represents the coupling coefficient between the wells.
From the derivation given above, the energy eigenvalues of the coupled system is now
Es = Eo-A
EA = Eo + A (2.96)
For the state with energy Es the coefficients of the state are, from Eq. 2.93,
ai = a2 (2.97)
while for the state with energy EA we have
ai = -a2 (2.98)
If we normalize the state using
2 i 2 1
a\ + a2 — 1 we get the following solutions:
Symmetric state:
^s =—j=[<i>i + fa]] ES = EO-A (2.99) Asymmetric state:
^A — —7= [0i — ^2]; EA = Eo + A (2.100) The states are shown schematically in Fig. 2.15b. We see that, as a result of the coupling between the wells, the degenerate states Eo are split into two states -one with energy below the uncoupled state and one with a higher energy. Note that as the wells are brought closer to each other the coupling strength will increase and the symmetric state energy will continuously decrease, as shown in Fig. 2.15c. If the electon in the coupled system occupy the symmetric state, the system behaves as if the coupling creates an attactive interaction in the symmetric state.
Hydrogen molecule ion
As an example of the coupled quantum well problem, let us consider two hydrogen nuclei with a single electron. As shown in Fig. 2.16, when the two nuclei are far apart, the electron can be in either one of the nuclei. Let these states, which are the ground states of the hydrogen atom, be denoted as before by fa and fa, respectively. The energy of these states is just
Eo = EH = -13.6 eV
As the nuclei are brought closer together, the electron on one atom feels the po- tential due to the attractive potential of the other nucleus. This gives a matrix element, which is a function of the inter-nucleus separation, R
H12 = -A(R)
proton 2
LARGE SEPARATION
DECREASING SEPARATION
Figure 2.16: A schematic of the symmetric and antisymmetric wavefunction constructed from the ground state wioo- In the symmetric state, the electron is closer to the nuclei.
As noted above, the original state will now split into a lower energy symmetric state and a higher energy asymmetric state. These states are shown schematically in Fig. 2.16.
In Fig. 2.17a we show a plot of the change in the electronic energy as a function of the inter-nucleus separation. In the symmetric state, where the electronic function has a high probability of occupying the space between the two nuclei, the system feels an attractive interaction, since the energy is reduced as the nuclei come closer. This is the reason the ion H* is stable. For the asymmetric state, where the electron is pushed away from the center of the two nuclei, there is an effective repulsive interaction, since the energy is larger than the energy when the nuclei are separated by a large distance.
To obtain the total energy of the H^ ion as a function of inter-nucleus separation we need to add the repulsive interaction between the positively charged nuclei. This amounts to an energy
11 — 6
\j rep —
4?T6o R
This energy is plotted in Fig. 2.17a. The total energy in the ion due to the presence of the two nuclei is now (in the ground state)
(2.101) This total energy is plotted in Fig. 2.17b. We see that the energy minimizes at an
2.5. From atoms to molecules: coupled wells 75 inter-nucleus separation of Rmm- Numerical calculations show that
flmin = 1.0A (2.102) The binding energy of the ion is found to be (EH is the magnitude of the ground state energy)
Eb = -0.2EH = -2.7 eV (2.103)
The example in this subsection shows how chemical bonds are formed by sharing of electrons between two nuclei. We have seen how the attractive interaction due to the coupling of the two bound states and the repulsive potential due to the nuclear charge play a role in setting the equilibrium bond distance.
Coupling between dissimilar quantum wells
Let us examine how the attractive and repulsive interactions due to coupling of wells are influenced when the two starting potential wells are dissimilar. This would occur if, for example, we had a hydrogen and a sodium nucleus coming together instead of two H nuclei, as considered in the previous subsection. Let the starting energies of the two potentials be E\ and E2, and let E<i be larger than E\. From the derivation given above, if the separation between E2 and E\ is much larger than the coupling coefficient A, we get (see Eq. 2.92) for the symmetric and asymmetric state
EA = E2--^— (2.104)
hi 2 — hi\
If the value of E2 — E\ is much larger than the coupling coefficient A, we see that the effect of the coupling is very small. This is the reason bonding between dissimilar atoms is weak.
H2 molecule
Another important and related problem is that of attraction between atoms. Let us consider the problem of how atoms attract each other to form a chemical bond. This is obviously an important question in chemistry, material science, and solid state physics.
The problem of H^ ion discussed previously sheds some light on the problem. However, in that problem we only had to consider one electron and two nuclei. What happens when there are two electrons?
Let us consider two H atoms initially far apart, as shown in Fig. 2.18a. In this uncoupled state each atom has an electron cloud around its nucleus, just as we expect in an isolated atom. The two states <j>\ and $2 are created through an exchange of the two electrons, as shown in Fig. 2.18a. For clarity we will call the electrons 1 and 2. As the atoms come closer to each other, there is an interaction between the atoms as each electron senses the attractive potential of the neighboring nucleus and the repulsive potential of the neighboring electron. The overall coupling is again represented by A(R). Once again we have a symmetric state and an asymmetric state made from
|
C/3
I
S
(a)
repulsive energy of protons
distance between atoms Es = £0 -A
+^
-0.2 -
-0.4
-0.6 -0.8
-1.0
-1.2 (b)
F i g u r e 2.17: (a) A schematic of how the energies of the symmetric and asymmetric states vary with separation of the H-atom nuclei. Also shown is the repulsive energy arising from proton- proton repulsion; and (b) change in the energy of the H^ ion as a function of inter-proton separation.