Quantum Mechanics Course Tunneling Part1
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5.
TUNNELING
In the previous chapter we discovered that all particles with E
Resonant tunneling diodes, which are used as switching units in fast electronic circuits. Scanning tunneling microscope (STM), based on the penetration of electrons near the surface of a solid sample through the barrier at the surface. These electrons form a "cloud" of probability outside the sample. Although the probability of detecting one of these electrons decays exponentially with distance (from the surface), one can induce and measure a current of these electrons and attain a magnification factor of 100 million - large enough to permit resolution of a few hundredths the size of an atom. Gerd Binning and Heinrich Rohrer won the Noble Prize in Physics in 1986 for the invention of The STM.
In the rest of this section, we will first describe the tunneling phenomenon through the example of a single barrier (section 5.1). Then, in section 5.2, we will talk about doublebarrier case, a type of barriers that we have in resonant tunneling diodes. The description of the WKB approximation is given in section 5.3. Esaki and resonant tunneling diodes are a subject of analysis of section 5.4. We will finish this section with a description of periodic potentials. 5.1
Tunneling through a single-barrier
Consider the potential barrier shown in Figure 1, for which the potential energy term appearing in the 1D TISE is of the form:
0, x < 0 V ( x ) = V0 , 0 ≤ x ≤ L 0, x > L Following the steps outlined in the previous section, it is easy to show that for energies E
− ψ 1 ( x ) = Aeikx + Be ikx
2mE
x x , where k = ψ 2 ( x ) = Ce − γ + De γ
h
ψ 3 ( x ) = Eeikx + Fe − ikx
2
2m (V0 − E )
and γ=
h2
.
V(x) Region 2 (classically forbidden)
Region 1 (classically allowed)
Region 3 (classically allowed)
V0
E E=
h 2 k12 2m
V0 − E =
h 2 κ22 2m
x
L Figure 1. Single potential barrier.
The application of the continuity conditions of the wavefunction at the boundaries x = 0 and x = L , leads to the following relationship between the unknown constants: ψ 1 ( 0) = ψ 2 ( 0 ) → ψ
' 1 ( 0)
= ψ
ψ 2 ( L) = ψ ' ψ 2 ( L) = ψ
' 2 ( 0)
3 ( L) ' 3 ( L)
A+ B = C + D
→ ik ( A − B ) = − γ (C − D ) L − γ
→
Ce
→
− γCe
(
+ De
L − γ
L γ
= Ee
− De
L γ
ikL
+ Fe
)= ik (Ee
− ikL
ikL
− Fe
− ikL
)
Using the above four equations, we can find the relationships between various coefficients, i.e. using matrix representation these relationships can be represented as: 1 1 + A = 2 B 1 1 − 2 1 1 − C = 2 D 1 1 + 2
γ i k γ i k
1 1 − 2 1 1 + 2
k (ik + γ) L i e γ k (ik − γ) L i e γ
γ i k C C D = M 1 D γ i k 1 1 + 2 1 1 − 2
k − (ik − γ) L i e γ E = M 2 E F F k − (ik + γ) L i e γ
In other words, we have the following relationship between the coefficients A and B, and the coefficients E and F:
A = M C = M M E = M E , 1 D 1 2 F B F where the matrix M has elements mij. Therefore, for coefficients A and E (using the asymptotic condition that F=0) we have the following simple relationship: A=m11E, i.e. thetransmission coefficient is simply given by:
E T (E) = A
2
=
1 m11
2
.
After a rather straightforward calculation, we arrive at the following expression for the transmission coefficient for particle energies less than the barrier height: T ( E ) = 1 +
−1
2 2 2 γ + k sh 2 ( γ L) 2 kγ
In the case of a weak barrier (γ L<<1), the expression for the transmission coefficient simplifies to: 1 T (E) ≈ . 1 + (kL / 2)2 In the opposite limit, i.e. when the barrier is very strong (γ L is very large), we have the following approximate expression for the transmission coefficient: 2
4 kγ exp(− 2 γ T ( E ) ≈ L) . 2 k 2 + γ
For energies larger than the barrier height, i.e. E>V0, using that γ =ik2, gives: T ( E ) = 1 +
−1 2 2 2 k − k 2 sin 2 ( k L) . 2 2kk 2
The later result is similar to the one obtained in the previous section, i.e. the transmission maxima (T(E)=1) occur for k 2 L = nπ . In Fig. 2a, we show several results for a potential barrier. The barrier height equals V0=0.4 eV, whereas the barrier width is L=6 nm. We also show how the transmission coefficient varies with the width of the barrier, for fixed E and V0 (Fig. 2b). We consider two cases: particle energies smaller and larger than the barrier height.
1
T(E)
0.8
0.6
L=6 nm, V =0.4 eV 0
0.4
-32
m=6x10
kg
0.2
0 0.0
0.5
1.0
1.5
2.0
Energy [eV]
1
0.8
T(E)
0.6
E=0.2 eV E=0.6 eV
0.4
0.2
0
-0.2 0.0
5.0
10.0
15.0
20.0
25.0
30.0
Barrier thickness L [nm]
Figure 2. (a) Variation of the transmission coefficient with energy. (b) Variation of the transmission coefficient with the barrier thickness.
Discussion of the results: • Classical physics would predict that no particles with energy EV0 are transmitted; quantum mechanics shows that this condition - called total transmission - occurs only at a few discrete energies. An incident particle with E>V0 that lies between these special values, determined by the condition k 2 L = nπ , may be reflected. The probability of reflection decreases very rapidly with increasing the energy of the particle E (see Fig. 2a). • For another perspective on transmission and reflection by a barrier, now lets look at the results shown in Fig. 2b. Here, the energy of the particle E and the barrier height are fixed and T(E) is plotted as a function of the barrier width L. This figure shows another bizzare result: for a given energy E, only barriers of certain width will transmit all particles of this energy (transparent barriers). But there is no value of the width such that a barrier of this width reflects all incident particles, because for all values of L, the reflection coefficient R(E) is less than one. • Because of the hyperbolic decay of the eigenfunction in the classically forbidden region, the amplitude of the eigenfunction in the detector region is reduced from its value in the source region.
TUNNELING
In the previous chapter we discovered that all particles with E
Resonant tunneling diodes, which are used as switching units in fast electronic circuits. Scanning tunneling microscope (STM), based on the penetration of electrons near the surface of a solid sample through the barrier at the surface. These electrons form a "cloud" of probability outside the sample. Although the probability of detecting one of these electrons decays exponentially with distance (from the surface), one can induce and measure a current of these electrons and attain a magnification factor of 100 million - large enough to permit resolution of a few hundredths the size of an atom. Gerd Binning and Heinrich Rohrer won the Noble Prize in Physics in 1986 for the invention of The STM.
In the rest of this section, we will first describe the tunneling phenomenon through the example of a single barrier (section 5.1). Then, in section 5.2, we will talk about doublebarrier case, a type of barriers that we have in resonant tunneling diodes. The description of the WKB approximation is given in section 5.3. Esaki and resonant tunneling diodes are a subject of analysis of section 5.4. We will finish this section with a description of periodic potentials. 5.1
Tunneling through a single-barrier
Consider the potential barrier shown in Figure 1, for which the potential energy term appearing in the 1D TISE is of the form:
0, x < 0 V ( x ) = V0 , 0 ≤ x ≤ L 0, x > L Following the steps outlined in the previous section, it is easy to show that for energies E
− ψ 1 ( x ) = Aeikx + Be ikx
2mE
x x , where k = ψ 2 ( x ) = Ce − γ + De γ
h
ψ 3 ( x ) = Eeikx + Fe − ikx
2
2m (V0 − E )
and γ=
h2
.
V(x) Region 2 (classically forbidden)
Region 1 (classically allowed)
Region 3 (classically allowed)
V0
E E=
h 2 k12 2m
V0 − E =
h 2 κ22 2m
x
L Figure 1. Single potential barrier.
The application of the continuity conditions of the wavefunction at the boundaries x = 0 and x = L , leads to the following relationship between the unknown constants: ψ 1 ( 0) = ψ 2 ( 0 ) → ψ
' 1 ( 0)
= ψ
ψ 2 ( L) = ψ ' ψ 2 ( L) = ψ
' 2 ( 0)
3 ( L) ' 3 ( L)
A+ B = C + D
→ ik ( A − B ) = − γ (C − D ) L − γ
→
Ce
→
− γCe
(
+ De
L − γ
L γ
= Ee
− De
L γ
ikL
+ Fe
)= ik (Ee
− ikL
ikL
− Fe
− ikL
)
Using the above four equations, we can find the relationships between various coefficients, i.e. using matrix representation these relationships can be represented as: 1 1 + A = 2 B 1 1 − 2 1 1 − C = 2 D 1 1 + 2
γ i k γ i k
1 1 − 2 1 1 + 2
k (ik + γ) L i e γ k (ik − γ) L i e γ
γ i k C C D = M 1 D γ i k 1 1 + 2 1 1 − 2
k − (ik − γ) L i e γ E = M 2 E F F k − (ik + γ) L i e γ
In other words, we have the following relationship between the coefficients A and B, and the coefficients E and F:
A = M C = M M E = M E , 1 D 1 2 F B F where the matrix M has elements mij. Therefore, for coefficients A and E (using the asymptotic condition that F=0) we have the following simple relationship: A=m11E, i.e. thetransmission coefficient is simply given by:
E T (E) = A
2
=
1 m11
2
.
After a rather straightforward calculation, we arrive at the following expression for the transmission coefficient for particle energies less than the barrier height: T ( E ) = 1 +
−1
2 2 2 γ + k sh 2 ( γ L) 2 kγ
In the case of a weak barrier (γ L<<1), the expression for the transmission coefficient simplifies to: 1 T (E) ≈ . 1 + (kL / 2)2 In the opposite limit, i.e. when the barrier is very strong (γ L is very large), we have the following approximate expression for the transmission coefficient: 2
4 kγ exp(− 2 γ T ( E ) ≈ L) . 2 k 2 + γ
For energies larger than the barrier height, i.e. E>V0, using that γ =ik2, gives: T ( E ) = 1 +
−1 2 2 2 k − k 2 sin 2 ( k L) . 2 2kk 2
The later result is similar to the one obtained in the previous section, i.e. the transmission maxima (T(E)=1) occur for k 2 L = nπ . In Fig. 2a, we show several results for a potential barrier. The barrier height equals V0=0.4 eV, whereas the barrier width is L=6 nm. We also show how the transmission coefficient varies with the width of the barrier, for fixed E and V0 (Fig. 2b). We consider two cases: particle energies smaller and larger than the barrier height.
1
T(E)
0.8
0.6
L=6 nm, V =0.4 eV 0
0.4
-32
m=6x10
kg
0.2
0 0.0
0.5
1.0
1.5
2.0
Energy [eV]
1
0.8
T(E)
0.6
E=0.2 eV E=0.6 eV
0.4
0.2
0
-0.2 0.0
5.0
10.0
15.0
20.0
25.0
30.0
Barrier thickness L [nm]
Figure 2. (a) Variation of the transmission coefficient with energy. (b) Variation of the transmission coefficient with the barrier thickness.
Discussion of the results: • Classical physics would predict that no particles with energy E
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