Quantum Tunnelling Quantum Physics 2002 Recommended Reading: R.Harris, Chapter 5 Sections 1, 2 and 3

Potential Barrier: E < U 0 Region I Region III Region II Potential where and x U = U 0 E = K.E. III x =0 U III x =L 1 E = K.E.

Region I: 2 Wavefunctions IncidentReflected Region II: 3 Region III: Transmitted Left Moving No term because there is no particle incident from the right. 4 Must keep both terms. Do you see why?

Boundary Conditions Match wavefunction and derivative at x = Match wavefunction and derivative at x = L.

Boundary Conditions We now have 4 equations and 5 unknowns, Can solve for B, C, D and F in terms of A. This is left as an exercise, A lot of algebra but nothing complicated!! Again we define a Reflection and Transmission coefficient: Since k 1 = k 3. Substituting for B and F in terms of A gives: Reflection Coefficient R 9

R and T Coefficients: We can write k 1 and k 2 in terms of E and U 0. Transmission Coefficient T This then gives Recall that sin(i  ) =sinh(  ).

similarly we can find an expression for the Reflection coefficient 11a Dividing across by gives 12 or rearranging 12a

Graph of Transmission Probability U 0 = 0.1 eV U 0 = 1.0 eV U 0 = 5.0 eV U 0 = 10.0 eV T E/U 0 Transmission curves for a barrier of constant width 1.0 nm with different heights U 0 L = 1.0 nm L = 0.5 nm L = 0.1 nm T E/U 0 Transmission curves for a barrier of constant height 1.0 eV for a series of different widths L

Wavefunction E U0U0

Optical Analog If second prism is brought close to the first there is a small probability for part of the incident wave to couple through the air gap and emerge in the second prism. If reflection angle is greater than the critical angle then the light ray will be totally internally reflected evanescent wave

Limiting Case Tunnelling through wide barriers: Inside the barrier the wavefunction is proportional to exp(-  x) or exp(-x/  ), where  = 1/  is the penetration depth (see Potential step lecture). If L   then very little of the wavefunction will survive to x = L. The condition for a ‘wide barrier’ is thus The barrier can be considered to be wide if L is large or if E << U 0. Making this approximation we see that and then so for a thick barrier equation 11 reduces to The probability of tunnelling is then dominated by the exponentially decreasing term

Example An electron (m = 9.11  kg) encounters a potential barrier of height 0.100eV and width 15nm What is the transmission probability if its energy is (a) 0.040eV and (b) eV? We first check to see if the barrier is thick (equation 13). for E = 0.04eV = 18.8 >> 1  thick barrier and for E = 0.060: L/  = 15.5 >> 1  thick barrier  we can use equation 14 for the transmission probability Very small in both cases!! Can we observe this in a real stuation

Field Emission metal electrons bound by potential step at surface Tunnelling through potential barrier Cathode Anode +V 0

Field Emission Displays (FED)

Scanning Tunnelling Microscope (STM)

Pt Surface Si (111) Surface

Pentacene molecules on Silicon Sample negativeSample positive

The Tunnel Diode see p-typen-type EFEF EFEF donors acceptors p-typen-type EFEF eV Conduction Band Valence Band

EFEF eV 0 - eV ext p-typen-type - + V ext + - p-typen-type V ext + - forward biased reversed biased EFEF eV 0 + V ext The Tunnel Diode Conduction Band Valence Band

The Tunnel Diode

Uranium 238 Thorium 234 Alpha particle Strong Nuclear Force Electrostatic repulsion To escape the nucleus the  -particle must tunnel. Alpha Decay of Nuclei