Surface plasmons in silver films-a novel undergraduate experiment H.J.Simon cillations in the electron density at the surface of a metal D. E.Mitchell J. G. Watson may be described in terms of these waves. onradiative surface plasma waves have been known Department of Physics and Astronomy as solutions o equations since Sommerfeld. The University of Toledo Toledo, Ohio 43606 uminum films in tota d by Turbadar; how (Received 17 October 1974; revised 21 January 1975) o surface ce plas- Surface plasmon phenomena are a topic of consi current interest. We describe a simple experiment and th ory for a senior level undergraduate investigation plasmons in silver evaporared on Light (p-polarized, the prism degrees greate sharp minimu sponding to t minimum in the refleci the resonant film. The dispersion re the reflectivity due to are calculated. Only a mo necessary to produce the required thin silver films. us and discuss n silver films evaporated at p We con- clude by suggesting extensions nt to study surface plas sion of the surface plasmon SURFACE PLASMON DISPERS We now derive a dispersion relation that will relate the propagation vector of a surface wave traveling along the INTRODUCTION surface of a metal to its angula We conside Surface plasma waves are transverse magnetic(TM) the space >0 to complex electromagnetic waves, traveling along the interface of dielectric function two different media. We will assume one of the media to dielectric constant for a be a metal and the other to be air. The waves propagete. The imaginary p ponentially attenuated in the normal direction in both the scribes the optical will assume metal and air. The wave vector,, of a surface plasmon surface plasmon on a semi-infinite dielectric bounded by vacuum is given <0 is as- by sumed to be air of he essentia assumption is to postulate the existence of only a single wave on each side of the b ectric fields are assumed to be m = ∈12 (1) E(x,2,t)=E k2(2) where e is the complex dielectric function of the medium at the angular frequency. For most metals∈a)is less The complete wave equation for this wave in each region than-1 in the visible region of the spectrum, and we see may now be written. Normally, in studying wave propa- from Eq. (1) that is greater than the wave vector of an gation in an infinite n electromagnetic wave in air at the same w. Surface plas- and ka to be give real numbers and uses the wave equation to determine the mon waves may be excited only with evanescent waves frequency of the propagating wave. Instead o this proce and are therefore classified as nonradiative. Collective os-dure, we assume that w is given and is a real number 6301 American Journal of Physics Vol. 43, No. 7, July 1975 Copyright 1975 by the American A.ssociation of Physics Teachers
to be determined For surface waves we are interested in the conditions under which there are no propagating waves in either the metal or air. We assume a and k have values such that in the metal the solution of the wave equation can be written GLAss n(w) k2=-i(k2-u2/c2 (3a) and in air k2=+i(k2-u2/c212 The signs are chosen so that the electric fields in the Fig. I. Attenuated total reflection geometry for a thin metal film be- metal and dielectric describe exponential decay normal to tween a glass prism and air the boundary metals this condition is easily satisfied in the visible region of the spectrum where o< op; in air this condition is satisfied only for evanescent waves in to- EreftectedE ,expli(w/c)n(x sin,+ 2 cose,)].(6b) tal internal reflection The boundary conditions at the plane z =0 are the standard ones: continuity of the tangential component of In the thin metal film we. write the total electric field as a E and H and of the normal component of d and B. We standing wave superposition of two exponentially damped assume the magnetic permeability is equal to unity, and B aves is equal to H. The two transverse modes can be classified by their polarizations, namely, with electric field E para met al=E expli(w/ c )ux sine,exp(z) lel or perpendicular to the plane of incidence defined by k and the z axis. For the case in which Er=Ex=0 Er'expli(w/c)nx sine, exp(-kz).(6c) -polarization, no plasma mode solution is found. The case of interest is E,=0, p-polarization. Now, using con- The transmitted wave in the vacuum is assumed evans tinuity of the tangential component of E and transversality cent of the fields, we have kE|时=k2E,|0- Transmitted=E] expli(w/c)nx sine, exp[(u/lc)n2sin201-1)4/2z](6d) above with continuity of the normal com ponent of D gives Here k is the absorption coefficient at nonnormal inci dence, which by comparison with Eqs.(2)and(3a)is written for the geometry of Fig. 1 as k=-i(w/c e-n'sin0,) Substitution from Eqs.(3a)and(3b)allows the aboy where e is the complex dielectric coet as before, n is the index of refraction glas condition to be rewritten as the dispersion relation given is the angle of incidence. The x dependence of m waves follows from Snell,'s law SURFACE PLASMON EXCITATION known amplitudes E1, Es, E2, and e be related to the incident amplitude e1' through the bound In this section we relate the amplitude of the surface ary conditions. Continuity of the tangential componen plasmon mode to the exciting incident radiation. An elec- of E and h at the i=0 and z=-d boundaries gives metal film at the hypotenuse face of the prism and at an of the fields in the metal at z=010 find the amplitude tromagnetic wave in a glass prism is incident on a thin the required four angle of incidence greater than the critical angle for total internal reflection. The attenuated total reflection geometry is shown in Fig. l. We assume the electric field vectors are p-polarized and described by monochromatic E=E1+71p2x(-2hD万 plane waves. In the glass medium we have an incident and reflected propagating wave of the form d EincidentEr' expli(w/c)n(r sine,-z cos0,))(6a) E,,Et t12Y23 exp(-2kd J. Phys. Vol. 43, No. 7, July 1975 Simon, Mitchell, and Watson/631
The Fresnel reflection and transmission amplitude fac- e〓430 e·(4t tors, with the 12 and 23 subscripts for the glass-metal and metal-air boundaries, respectively, are given by /2 cos 81+n cose2 e1/2 12/2 cos e, +n cos e 2 and (9c) Fig. 2. Polar plot of the complex Fresnel reflection factor, r23,at the cos02+E1/2cos03 silver-air interface for angles of incidence around the plasmon angle The amplitude scale is logarithmic The angles B2 and ]3 in the metal and air, respectively, linear reflectivity can be observed only for the case of are defined by nonzero absorption given by E2+0. If E2=0, the reflec cos62=(1-n2sin21/e)/2 (10a) gles of incidence including the surface plasmon resonance however, in nonlinear optics the surface plasmon reso- nance may still be observed through enhanced harmonic generation. The plasmon angle p is still defined by the cosB3=(1-n2sin201)1/ real part of Eq.(13), but r23 now assumes a large nega (10b) tive imaginary value. The amplitude and phase of rain the complex plane as a function of the angle of incidence The complex nature of these angles reflects the real expo- is plotted in Fig. 2 with a logarithmic amplitude scale.At B we now solve for E and find the ratio of the reflected we have r23 =1, but at the plasmon angle ra=-i2E1/E2 nential decay of the fields in the metal and air media optical power to the incident optical power to be given by and beyond this angle r23 rapidly approaches=-1.The amplitude and phase of r23 are a sensitive function of the r12+ regexp 1+y:2723exp( (11) The critical factor in the above formula is 23, which may be rewritten as 23sGn2sin2a1-e)1/2-t(n2sin201-1)/2 (n sin20, -E)/2+E(n ideal free electron plasma, the denominator of r23 vanishes at the plasmon angle 8p given by nsin,=Le/(E+1) which is also the condition for the surface plasmon mode given by Eq. (1). The divergence of r23 at the surface plasmon resonance was recognized by Cardonaand can be described as the ratio of a finite-amplitude reflected field to a zero-amplitude incident field at the metal-air in ect interpretation, without any diver gent factors, is to recognize directly from Eq .( 8a)that yhen 81=p then E2=0 and the only field in the metal has a simple exponential spatial decay from the metal-air boundary to the metal-glass boundary. This description FILM THICKNESS d (A consistent with the surface plasmon mode discussed viously and the surface polariton modes of Agarwal ig. 3. Normalized amplitude of the surface plasmon mode electric field The effect of the surface plasmon resonance on the evaluated at the silver-air interface versus film thickness 632/Am,J.Phys,ol,43,Mo.7,J1975 Timon Mitchell and Watson
22E2 Reflectance of the surface plasmon mode 2 ness and variable absorption col angle of incidence in this region the plasmon resonance as the absorption increases. Note In Fig. 3 we display the normalized amplitude of the that the angular half-width of these resonant curves in surface plasmon mode electric field at the plasmon angle creases with absorption. For large values of absorption evaluated at the metal-air boundary as a function of metal the plasmon resonance is heavily damped and the reflec film thickness for several values of Eg. Note the amplitude tivity minimum is no longer sharp The effect of damping scale is logarithmic. For zero absorption the amplitude of on the surface plasmon resonance has been discussed by this field increases exponentially with the thickness of the several authors. 10 The reflectivity minimum at the plas- film; however, for nonzero absorption this intemal field mon resonance is due to the increased absorption of the amplitude reaches a maximum, which may be an order enhanced internal fields throughout the volume of the of magnitude larger than the incident field. By compari- metal film. Integrating the power absorption density over son the normalized amplitude of the other field in the the irradiated volume of the film and normalizing by the metal film for zero absorption is identically zero, as pre- incident power, we have the enhanced absorptance due to viously discussed; however, for finite absorption this field the surface plasmon, given by is nonzero, although it is still several orders of magnitude smaller than the surface plasmon mode field shown in r= wLexp(2kd)-1]cosB2IE2' (14) The quantity of experimental interest is the reflectance which is shown in Fig. 4 as a function of the angle of incidence for a fixed film thickness. Again for zero ab- The above quantity is plotted in Fig. 5. Clearly, the max- sorption the reflectivity is unity but decreases sharply near ima in the absorptance match the minima in the reflec- thickness and variable ANGLE OF INCIDENCE e Am J, Phys, Vol 43, No. 7, July 1975 /63
E2 52 film thickness for tance. In Fig. 6 we show the reflectance at the plasmon was also observed. When the reflected light was detected angle as a function of film thickness in order to estimate with a RCA 1P28 photomultiplier whose output was mea- the optimum film thickness in the following section sured with a voltmeter, the results shown in Fig. 8 were obtained. The minimum in reflected intensity observed in EXPERIMENT AND RESULTS this experiment was comparable to the extinction obtained In this section we describe a simple apparatus for with crossed polarizers evaporating thin silver films, discuss the experimental re- The only experimental technique requiring a little ex- sults, and make suggestions for further experiments. Our perience is coating the correct film thickness. Referring to apparatus which produced a vacuum of 10-3 Torr con- Fig. 6, we see that films either less than 300 A thick or sisted of an Edwards 3-in. diffusion pump connected to a greater than 800 A thick will produce shallow reflectivity Vector 75 rotary roughing pump and a 6-in. bell jar with a No. 8 rubber stopper at its top through which the two electrodes entered the vacuum chamber. A coiled filament basket was made from tungsten or molybdenum wire. A 1.5-cm length of 99.999% silver wire(1-mm diameter) was placed at the bottom of the coil which was attache to the electrodes. Before the prism was placed at the bot- tom of the bell jar, the hypotenuse face was cleaned with reagent grade methanol and rinsed with distilled water. A metal shutter, magnetically coupled to the outside of the ell jar, was placed between the silver and block any impurities that might boil off while the silver was initially heated. A sketch of this apparatus is shown A current of 4-8 A supplied by a variac caused the filament to gradually glow red hot and the silver to slowly melt into a small ball at the bottom of the filament. The shutter was then opened and the coating process begun. After a few minutes of evaporation, the current was turmed off and the pressure inside the bell jar yas returned to atmospheric SHUT TER The prism was then removed from the evaporator and placed on a suitable rotary platform or a student grade spectrometer. A He-Ne laser beam polarized in the plane of incidence by a Polaroid sheet was incident on the prism, and the reflected beam could initially be observed on a white piece of paper. At the critical angle for total internal reflection, the reflected intensity approximately equaled the incident intensity, but as the external angle ASEPLATE of incidence was increased by 2-3, the reflectivity was observed first to decrease sharply and then to increase again. A simultaneous increase in the scattered-light in- Fig. 7. Sketch of simple evaporator apparatus. Prism with hypotenuse tensity from the laser spot on the back silver-air surface face up rests on the base plate 634/Am,J.Phys,.43,No.7,Jwy1975 Simon, Mitchell, and Watson
Fig. 8. Experimental reflectance of the surface plasmon mode versus angle of incidence in a silver film 560 A thick on a crown glass prism z at the He-Ne laser wavelength 6328 A AN GLE。F| NCIDENCE minima. When viewed from the hypotenuse face, too may then be compared with Eq (1). The observed back thick a film will appear opaque while too thin a film will bending in the near ultraviolet of this experimentally de- appear semitransparent; however, the apex edge of the termined dispersion curve is a result of the damping of prism should be visible through the metal film. If the sur- the surface plasmon. o Another experiment is to use other face plasmon resonance is not observed, the prism should metals which can be easily evaporated, for example be recleaned and the evaporation process repeated again. aluminum, gold, and copper. The relevant parameter for Since this process requires only a few minutes, new sam- estimating the strength of the surface plasmon resonance ples of varying thickness may be easily prepared until the is the magnitude of the ratio ey/E2. For silver in the near desired effect is achieved. Film thickness may be deter- infrared, this ratio is larger than that of the other met- mined by placing a microscope slide adjacent to the prism als. 4 The sharpest reflectivity minima will now occur for transmission of the coated slide. For the 560-A film used film, Turbadar2 gives a thickness of 125 a at a in this experiment, the transmission of the slide was ap wavelength of 5500 A. The dispersion of surface plas proximately 1% at 6328 mons in gold films has been studied by Barker. 5 The The numerical calculations for the curves presented in opportunity exists to make original observations of the Figs. 2-6 were performed on a PDP-11 time-sharing surface plasmon resonance in many metals by use of the computer. The complex dielectric constant of the silver technique we have described here film2 was E=-18.3+i0. 4 at the He-Ne laser In conclusion, we have given a theoretical description wavelength(6328 A), and the index of refraction of the of the effect of the surface plasmon mode on the reflectiv- crown glass prism was taken to be n =1.52. The optical ity of thin metal films and have demonstrated how such roperties of the silver film are sensitive to the conditions films may be prepared with a modest apparatus. The of film preparation (especially at our vacuum pressure of dramatic change in reflectivity of a silver film should not 10-3 Torr) and to the growth of silver sulfide layers on only provide a stimulus in the laboratory for physics stu the film in air; therefore, the value of e2 was allowed to dents, but should also provide a dramatic effect in a lec- take on a wide range of values to test the sensitivity of ture demonstration for a more general audience the reflectance to this parameter. We see from Fig. 4 that the qualitative features of the reflectivity minimum should be observable over a large range of eg. The results dis played in Fig. 8 were repeatable with the same film main- tained in the ambient laboratory environment over a "For a review of surface plasmon physics, see R.H.Ritchie, Surf.Sci period of several weeks. Although the films should 34,1(1973) evaporated under high vacuum 9 Torr) and remain inT.Turbad S, soc vacuum during the experiment, such stringent conditions 'A. Otto, Z. Phys. 216, 398(1968); Phys. Status Solidi 42, K37 are not necessary for the qualitative observation of surfac plasmon phenomena There are several interesting extensions of this ex ment. One experiment is to replace the He-Ne laser C. Kittel, Introduction to Solid State Physics (Wiley, New York source, which was chosen solely for its experimental con- venience, with a collimated monochromator. By measuring K, w. Chiu and JJ. Quinn, Nuovo Cimento B 10,1(1972).This the angle at which the reflectivity minimum occurs as a function of wavelength, one can experimentally determine dispersion relation the dispersion relation for the surface plasmon. This curve Borm and E. Wolf, Principles of Optics (MacMillan, New York Am, J. Phys. VoL, 43, No. 7, July 1975 Simon, Mitchell, and Watson /635
12P. B. Johnson and R. w. Christy, Phys. Rev. B 6, 4370(1972) Cardona, Am J Phys. 39, 1277(1971). 13J. M. Bennet, J. L. Stanford, and E J. Ashley, J. Opt. Soc. Am. 60, E. T. Arakawa, M. W. williams, R. N. Hamm, and R, H. Ritchie 224(1970) Phys. Rev. Lett. 31, 1127(1973):R. w. Alexander, G.S. I American Institute of Physics Handbook (McGraw-Hill, New York Kovenor, and R J. Bell, Phys. Rev, Lett, 32, 154(1974) 963), 2nd ed, Sec. 6-g Metals and evaporator supplies may be obtained from Electronic Space 15A.S. Barker, Jr, Phys. Rev. B 8, 5418(1973):Am. J. Phys. 42, Products. 854 So. Robertson Blvd Angeles, CA 90035 1123(1974) J or psi (a physicist's four-footed sonnet) Where is the thing beneath the thing? When can we say we have found it all? Is there an end or just a string That dangles down like an endless fall? Molecules, yes, and atoms, too Electrons we know and nuclei Neutrons, protons, the particle zoo And enter now the or psi Can we not try to knot the string To start from a bottom and see what grows? Plant us a seed and see what we get When unity doubles and does its thing And triples, quadruples, quintuples. Who knows What patterns will show? What world is there yet? 一 roger E. Clapp 36/Am J Phys. VoL 43, No. 7, July 1975 Simon, Mitchell, and Warson