10. Optics of metals - plasmons

Transcription

1 1. Optics of metals - plasmons Drude theory at higher frequencies The Drude scattering time corresponds to the frictional damping rate The ultraviolet transparency of metals Interface waves - surface plasmons lasmon excitations of metal nanoparticles

2 A few announcements Reminder: no class on Monday (Winter recess) roblem set 3 due today. roblem set 4 will be posted later today. It will include a note reminding you to sign up for an exam time slot. Sign-up for Exam 1 will be posted some time on Friday. AFTER it is posted: - You need to Dr. Mittleman to sign up for a time slot. - only. No other method is acceptable. - Include a 1 st choice AND a nd choice. - Time slots assigned on a first- , first-allocated basis. - Sign-up deadline is Friday Feb Exam dates: March -3. See sign-up page for more info. - IF YOU WILL BE UNAVAILABLE ON THOSE DATES: contact Dr. Mittleman IMMEDIATELY to arrange an alternative.

3 Our guess from the last lecture Recall the inhomogeneous wave equation: E 1 E z t t c As a guess, we replaced d/dt with the current density J(t) = E(t): 1 c E E J E z t t t from which we found the complex refractive index: n 1 j e j 4 n large

4 When is this guess likely to be wrong? Because is REAL, our guess, J(t) = E(t), implies that the current is always in phase with the incident wave! Recall: in the Drude model, the electrons are free to move - they are not bound to atoms by springs. So, for low or moderate frequencies, our guess is ok. But at a high enough frequency, it MUST fail. Even without springs, the electrons must take some time to respond, so a very high frequency oscillation must leave them lagging: J must eventually be out of phase with E. So, we are back to the inhomogeneous wave equation. E 1 E z t t c what goes here?

5 The polarization when there is current Let s go back to our forced oscillator model. Newton s law F=ma gave us: d xe t dxe t ee e jt xe t dt dt me spring constant m e resonant frequency of the spring frictional damping force due to the incident light field From this, we found the polarization: (see lecture 7) () t Ne / m e j E t For a Drude metal, there is no spring holding the electrons. So what if we simply take =?

6 The plasma frequency () t E t j Ne / m e Define a new constant, the plasma frequency : Thus () t Ne m e j E t We now use this as a new and improved guess: plug this in for the polarization term in the wave equation.

7 Back to the wave equation 1 z c t t j t E E E But this is the same as a problem we solved earlier: E 1 1 E z c j t This is the wave equation for a wave propagating in a uniform medium, if we define the refractive index of the medium as: n metal 1 j This is our new result for the (complex) refractive index of a metal.

8 The new and improved result Instead of making a guess that we make the better guess that t () d dt E t, j E t and then the Drude model (plus the wave equation) predict the optical properties of metals as: n metal 1 j or metal 1 j It is instructive to compare the guess from last lecture to this new result, and see where they are similar.

9 How does this compare to our earlier result? For this new result, consider the low-frequency limit, << : 1 1 j j But in the last lecture, our guess gave this result: 1 n j 1 j Ne In the Drude model, we have, so thus: m e Ne m e With our earlier definition of, we find: Ne m e Ne 1 m The two results are consistent at low frequency! e

10 High frequency dielectric of metals How does this dielectric function behave at higher frequencies, e.g., >>? For high frequencies we find: 1 The dielectric function becomes purely a real number. And, it is negative below the plasma frequency and positive above the plasma frequency. Some numbers: 1 j Recall from Drude theory, that ~ 1-14 sec, so ~ 1/ ~ 1 14 Hz. For a typical metal, is 1 or even 1 times larger. (corresponding to the frequency of infrared light) (corresponding to the frequency of ultraviolet light)

11 Dielectric function of metals 1 A plot of Re() and Im() for some example values: j ()/ Im() Frequency (cm -1 ) linear scale Re() = 4 cm -1 = cm -1 ()/ Re() Re() Im() log scale Frequency (cm -1 ) imaginary part gets very small for high frequencies real part has a zero crossing at the plasma frequency real and imaginary parts are equal in magnitude at

12 Drude theory: it works pretty well large and negative (below ) real / small and positive imag /

13 but not perfectly dramatic departure from Drude model Drude model lasma frequency ~ 9 ev. So the stuff at ~4 ev is not due to. It is due to inter-band (valence-to-conduction band) transitions of the bound electrons (our Drude model analysis ignored bound electrons).

14 High frequency optical properties In the regime where >>, we find: n 1 For frequencies below the plasma frequency, n is complex, so >. The wave is attenuated and does not propagate very far into the metal. For high frequencies above the plasma frequency, n is real. The metal becomes transparent! It behaves like a non-absorbing dielectric medium. reflectivity drops abruptly at the plasma frequency This is why x-rays can pass through metal objects.

15 FM radio Another example: the ionosphere AM radio the uppermost part of the atmosphere, where many of the atoms are ionized. There are a lot of free electrons floating around here For N ~ 1 1 m -3, the plasma frequency is: Ne 9 MHz m Radiation at frequencies above 9 MHz is transmitted, while radiation at lower frequencies is reflected back to earth. That s why AM radio broadcasts can be heard very far away. e

16 An observation For real metals, there is a very broad range of frequencies for which Im() ~ and Re() is negative. ()/ Im() Re() Frequency (cm -1 ) linear scale = 15 cm -1 = cm -1 This has interesting implications.

17 Waves trapped at an interface x z Consider a wave at the interface between two semi-infinite media. z= Is there a solution to Maxwell s equations describing a wave that propagates along the surface? We can guess a solution of the form: 1 E1 E1x,, E1z e e subscript 1 refers to medium #1 1 B, B, e e z jkx t y 1 1 z j kxt This propagates along the interface, and decays exponentially into both media. (and similar for medium #) (Note: this is not a transverse wave but that s ok )

18 Interface waves In order to exist, the wave must satisfy Maxwell s equations: j B E j B E 1 1y 1 1x y x c c and also the continuity boundary conditions at z=: z 1 z z 1 z y y x x B B E E It is easy to show that these conditions can only be satisfied if: 1 1 Since 1 and are always positive, this shows that these interface waves only exist if 1 and have opposite signs. We just showed: in a metal, < for frequencies less than.

19 Surface plasmon polaritons surface plasmon polariton (S) - a surface wave moving along the interface between a metal and a dielectric (e.g., air) The electrons in the metal oscillate in conjunction with the surface wave, at the same frequency. In fact, an S is both an electromagnetic wave and a collective oscillation of the electrons.

20 Surface plasmon sensors Surface plasmons are very sensitive to molecules on the metal surface. A commercial S-based biosensor

21 Surface plasmons on small objects Instead of considering a semi-infinite piece of metal, what if the metal object is small? e.g., a metal nanosphere We can still excite a plasmon, but in this case it does not propagate! The electrons just collectively slosh back and forth. excess negative charge excess positive charge There is a restoring force on the electron cloud! Once again, we encounter something like a mass on a spring, with a resonance

22 Surface plasmon resonance The sloshing electrons interact with light most strongly at the resonant frequency of their oscillation..8 nm copper nanoparticles edersen et al., J hys Chem C (7) gold nanoparticles give rise to the red colors in stained glass windows

23 Controlling the surface plasmon resonance The frequency of the plasmon resonance can be tuned by changing the geometry of the metal nano-object. Halas group, Rice U. gold nano-shells Changing the color by changing only the shell thickness!