Quntum Anlogs Chpter 4 Student Mnul Modeling One Dimensionl Solid Professor Rene Mtzdorf Universitet Kssel
Stud. Mn. Rev 2.0 12/09 4. Modeling one-dimensionl solid There re two different wys to explin how bnd structure in periodic potentil of solid develops. One pproch strts with free moving electron in constnt potentil tht hs prbolic dispersion reltion E(k). Introducing periodic scttering centers with smll reflection probbility results in the opening of bnd gps. The other pproch is to strt from n tom with its discrete sttes. The next steps in this pproch re the splitting of the eigensttes sttes in two-tom molecule nd further splitting in chin with n toms. With the coustic nlog, you cn study both pproches experimentlly. We will do this in the next two sections. In lter sections, we will model the electronic structure in more complex solids with superstructures (Section 4.3) nd defects (Section 4.4). 4.1 From free electron to n electron in periodic potentil To model free electron in one dimension, we re using propgting sound in tube. Since we cnnot work with infinitely long tubes, we restrict ourselves to finite tube with hrd wlls on both ends. This is ctully the sme setup we used in Chpter 1 to model the prticle in box. Due to the finite length L of the tube we get resonnces with the frequencies f : c f n = n (4.1) 2L (c is the speed of sound nd n is n integer number n=1,2,... ). The longer the tube, the denser the resonnces become. In n infinitely long tube, the resonnces would be infinitely dense. In solid-stte physics, the so-clled density of sttes is used in this context. Now let s do n experiment. Equipment Required: TechSpin Quntum Anlog System: Controller, V-Chnnel & Aluminum Cylinders, Irises Two-Chnnel Oscilloscope Two dpter cbles (BNC - 3.5 mm plug) Computer with sound crd instlled nd Quntum Anlogs SpectrumSLC.exe running Setup: First, set the ATTENUATOR knob on the Controller t 10 (out of 10) turns Using the tube-pieces, mke tube with the end-piece contining the speker on one end nd the end-piece with the microphone on the other. Attch BNC splitter or tee to SINE WAVE INPUT on the Controller. Using the dpter cble, connect the output of the sound crd to one rm of the splitter. With BNC cble, convey the soundcrd signl from the splitter to Chnnel 1 of your oscilloscope. Plug the led from the speker end of your experimentl tube to SPEAKER OUTPUT on the Controller. The sound crd signl is now going to both the speker nd Chnnel 1. Connect the microphone on your experimentl tube to MICROPHONE INPUT on the Controller. Put BNC splitter on the Controller connector lbeled AC-MONITOR. From the splitter, use n dpter cble to send the microphone signl to the microphone input on the computer soundcrd nd BNC cble to send the sme signl to Chnnel 2 of the oscilloscope. Chnnel 2 will show the ctul signl coming from the microphone. 4-1
Stud. Mn. Rev 2.0 12/09 The computer plots the instntneous frequency generted by the sound crd on the x-xis nd the mplitude of the microphone input signl on the y-xis. Configure the computer so tht Microphone or Line-In is chosen s the input You will need to djust the mgnitude of both the speker nd microphone signls to keep the microphone input to the computer from sturting. (It is the user's responsibility to ensure tht the dpter cbles re NOT used with signls greter thn 5 Volts pek-to-pek.) Refer to the Appendix 2, titled Recognizing nd Correcting Sturtion, for instructions. Mesure the resonnces in tubes of different length nd nlyze the distnce between the resonnces f = f n +1 fn s function of tube length. Convince yourself tht the resonnces become more nd more dense with incresing tube length. (As you use longer tubes, you will to increse the ATTENUATOR setting in order to get good dt.) The quntum numbers used in solid-stte physics re different from those used in tomic nd moleculr physics. In the mesurements you hve mde, you will hve noticed tht there re equidistnt resonnces, which cn be chrcterized by numbering them in the order of their frequency. From theory, we know tht they belong to stnding wves in the tube with wvelength 2L λ = (4.2) n The wvelength cn lso be expressed by nother quntity clled wve number k (in three dimensions it is the wve-vector, k r ). k 2π π = = n λ L In the cse of infinitely dense eigensttes, it is not useful to number the sttes by n integer number. It is better to use the wve-number k (or wve-vector k r in higher dimensions) to lbel the eigensttes. In tomic physics we hve chrcterized the quntum mechnicl system by energies E(n,l,m) s function of integer quntum numbers, in solid-stte physics the quntum mechnicl system is chrcterized by the energy E(k) s function of wve number. This reltion is clled dispersion reltion. We will do this nlogously in the coustic experiments. In the tube with finite length, we hve discrete eigensttes, so tht it is esy to determine the wve number by the index n of the resonnce using eqn. 4.3. This now llows us to mesure the dispersion reltion for sound wve in n empty tube. Mesure the frequencies of the resonnces in tube of length L = 600 mm nd plot the frequency s function of wve number k. Wht do you notice? (4.3) 4-2
Stud. Mn. Rev 2.0 12/09 Wht is nlogous, wht is different? Sound wves show liner dispersion with slope proportionl to sound velocity. c f ( k) = k (4.4) 2π Electrons, however, hve prbolic dispersion h E( k) k 2m 2 2 =. (4.5) Modifictions of this so clled free-electron like dispersion re observed, when electrons hve wvelength tht is comprble to twice the lttice constnt,, of the solid. In this cse, the electrons re scttered effectively by the periodic lttice. In the coustic nlog, we introduce periodic scttering centers seprted by distnce,, tht is comprble to hlf the wvelength of sound. A typicl wvelength, t resonble frequency, (3.4 khz) is λ = 10cm ( 4 inch). Therefore, we cn model lttice by periodic scttering centers t seprtion distnce of bout = 5cm ( 2 inch). Tke n overview spectrum (0-12 khz) of tube mde from 12 tube-pieces ech 5 cm long. Now, insert 11 irises with n inner dimeter of 16 mm between the pieces nd mesure spectrum gin. Wht do you observe? Due to the introduction of the periodic scttering sites, bnd structure hs developed. It shows bnds nd bnd-gps. Becuse we hve tube with finite length, the bnds consist of discrete resonnces. The bnd-gps indicte frequency rnges in which no sound cn propgte through the periodic structure. Remove the end-piece with the microphone nd put your er in its plce. Choose frequency within bnd. Then choose frequency within bnd gp. Listen to the difference in loudness. Now we wnt to study how the spectrum is influenced by vriety of prmeters (Dimeter of the irises d, number of pieces j nd length of tube-piece ). Replce the end-piece nd mesure spectr with irises of 13 mm nd 10 mm dimeter. 4-3
Stud. Mn. Rev 2.0 12/09 Now we will mesure spectr for different number of unit cells. In solid-stte physics, unit cell is the prt of spce tht is repeted periodiclly to build up the solid. In our cse, it is the combintion of tube-piece nd n iris. We hve not put 12 th iris in front of the microphone, since the end-piece reflects the sound perfectly, nywy. You my convince yourself tht the use of 12 th iris t one of the end-pieces mkes no significnt difference in the spectr. Smll chnges re due to the mount of ir within the hole of the iris. For future experiments, you my decide for yourself whether to put n iris t n end-piece. Put in the 16 mm irises gin nd mesure spectr for different numbers of tube-piece / iris. Describe the wy the spectrum chnges. Are there ny mthemticl ptterns? Now let s study how the spectrum depends on the length of tube-piece, which corresponds to the lttice constnt in solid-stte physics. Tke spectrum with 8 pieces 50mm long nd irises of 16mm dimeter. Thn replce the 50 mm long pieces by 75 mm long pieces. Wht difference in the spectr do you observe? Cn you find mthemticl pttern? 4-4
Stud. Mn. Rev 2.0 12/09 Bckground informtion: Bnd gps open up when the Brgg condition is fulfilled. You most probbly know the Brgg condition from x-ry nd neutron scttering t crystls, which re both exmples of wve reflection t periodic lttice. The Brgg conditions is fulfilled, when nλ = 2 (4.6) ( is the distnce of reflecting plnes). In our one-dimensionl cse the reflecting irises represent the reflecting plnes of solid. Reflection in the solid is so effective t this wvelength since the reflected wves from ech plne dd up constructively with perfectly fitting phse. This is the reson why wves cnnot propgte esily t this wvelength. A very convenient wy to describe the scttering phenomen t periodic structures is to use the so-clled reciprocl spce. The reciprocl spce is the spce of the wve vectors k r. In our one-dimensionl cse we hve one-dimensionl reciprocl spce with the wve-numbers k. If wve is reflected t periodic structure nd the Brgg condition is fulfilled nd the wve number k r hs chnged to k r r r r, then the difference k k = G is clled reciprocl lttice vector G r. In our one-dimensionl cse the wve hs been reflected nd k hs chnged to k with k tht fulfils the Brgg condition. π k = n (4.7) In consequence, the reciprocl lttice vectors for the one-dimensionl cse re given by 2π G = n (4.8) with n integer number n tht cn be positive or negtive or zero. In generl, the reciprocl lttice vectors re forming periodic lttice in the reciprocl spce, which is clled the reciprocl lttice. In this reciprocl lttice you cn define unit cells of the reciprocl spce tht re clled Brillouin zones. For the one-dimensionl cse the reciprocl lttice points nd the Brillouin zones (BZ) re displyed in Fig. 4.1. 1. BZ 10π 8π 6π 4π 2π 0 2π 4π 6π 8π 10π Fig. 4.1: Reciprocl lttice points (blck dots), nd Brillouin zones boundries mrked by dshed lines. 4-5
Stud. Mn. Rev 2.0 12/09 Due to the finite length of the tube, we hve discrete k-points in the reciprocl t which n eigenstte (resonnce) is observed. They re given by eqn 4.3. If we compre the smllest reciprocl lttice vector 2π G = (4.9) with the distnce of the discrete k-points in the tube of finite length L π k = (4.10) L we cn see tht there re 2L/ discrete k-points in ech Brillouin zone. Since L=j, we cn conclude tht the number of discrete k-points in Brillouin zone is twice the number of unit cells. At k=0 nd zero frequency (energy), there is no resonnce (eigenstte) for finite system. 1. BZ π π 0 L π 2π 3π L L π Fig. 4.2: Discrete k-points in reciprocl spce (blck dots), nd first Brillouin zone mrked by dshed lines. The exmple represents setup with 8 unit cells. Let us now explore the dispersion reltion in reciprocl spce. Anlyze the dt: Plot the frequency s function of wve number for resonnces in setup mde from 8 pieces 50 mm long nd 7 irises of 16 mm dimeter. Determine the wve number s given in eqn. 4.3. Where, in reciprocl spce, do the bnd gps open up? When counting the resonnces, plese note tht the little pek t 370 Hz is not resonnce. It is pek in the trnsmission function of the speker/microphone combintion. 4-6
Stud. Mn. Rev 2.0 12/09 Bckground Informtion From Bloch s theorem, we know tht wve functions in periodic structure cn be written s the product of function u k (x) tht hs the periodicity of the lttice nd exp(ikx) with the periodicity given by the wve number. ikx ψ ( x ) = u ( x) e (4.11) k A function of this form cn be written in the form i( k G) x ψ ( x) = C( k G) e. (4.12) G From this form of nottion, we see tht the wve function cnnot be ssigned to single point in the reciprocl spce. The wve function is sum with contributions from single k-point in ech Brillouin zone. All of these k-points re connected by reciprocl lttice vectors. In solidstte physics, therefore, the dispersion E(k) is usully plotted only in the first Brillouin zone. This is clled the reduced zone scheme in contrst to the extended zone scheme. Anlyze the dt: Plot the dispersion reltion E(k) in the reduced zone scheme Anlyze the dt: Anlyze the spectr for setup mde from 10 unit cells with 50 mm tubes nd 16 mm irises nd for setup mde from 12 unit cells with 50 mm tubes nd 16 mm irises. Plot the dispersion reltion into the reduced zone scheme. Note tht t higher frequencies, the first nd the lst resonnce in bnd cnnot be identified esily. You should keep in mind tht ech bnd hs j resonnces when it is build up from j unit cells. Only the first bnd hs j-1 resonnces becuse the lowest stte of tht bnd hs zero frequency nd is not visible. This is importnt when you determine the wve number from the resonnce number n. Anlyze the dt: Anlyze the spectr for setup mde from 8 unit cells with 75 mm tubes nd 16 mm irises nd compre it to setup mde from 8 unit cells with 50 mm tubes nd 16 mm irises. Plot the dispersion reltion into the reduced zone scheme. Anlyze the dt: Anlyze the spectr for setup mde from 8 unit cells with 50 mm tubes nd 16 mm, 13mm nd 10 mm irises, respectively. Plot the dispersion reltions into the reduced zone scheme. How does the dispersion depend on the iris dimeter? 4-7
Stud. Mn. Rev 2.0 12/09 In condensed mtter physics, the density of sttes (DOS) is often discussed. If the dispersion reltion is known in the complete Brillouin zone, the DOS cn be clculted from these dt. To illustrte how the DOS of one-dimensionl system looks, we will now nlyse the dt with respect to this quntity. Anlyze the dt: Let s tke the spectrum for setup mde from 8 unit cells with 50 mm tubes nd 16 mm irises nd use it to determine the DOS. Since this is system with smll number of unit cells, we cnnot simply count the number of sttes within n energy intervl. We will therefore clculte the density by one over the frequency distnce between two sttes. 1 ρ ( f ) (4.12) f i +1 f i In one-dimensionl bnd structure, there re singulrities in the density of sttes expected t the bnd edges (vn Hove singulrity), since the slope of the bnds pproches zero t zone boundries nd symmetry plnes. Due to the finite number of unit cells, the density of sttes is finite in our experiment, but significnt upturn of DOS t the bnd edges is clerly visible. 4-8
Stud. Mn. Rev 2.0 12/09 4.2 Atom Molecule Chin In the previous section, we hve seen how bnd-gps develop in free moving wve when periodic scttering sites re introduced. The other pproch to solid-stte physics strts with the eigensttes of single tom. When two toms re combined into molecule, splitting of the eigensttes into bonding nd nti-bonding sttes is observed. Finlly, bnds develop from these levels, when mny toms re rrnged into chin. In theory, this pproch is clled the tight binding model. Now we wnt to study this pproch experimentlly strting with n tom, which we will model with 50 mm long cylinder with the speker on one end nd the microphone on the other. Tke n overview spectrum (0-22 khz) in single 50 mm long tube-piece. The peks t 370 Hz, 2000 Hz nd 4900 Hz re not resonnces in the tube. They re due to the frequency response of the speker nd microphone combintion, which is not frequency independent. Below 16 khz there re 4 resonnces in the 50 mm long cylinder, which cn be described s stnding wves with 1, 2, 3 nd 4 node-plnes perpendiculr to the cylinder xis, respectively. At frequencies bove 16 khz, resonnces re observed tht hve rdil nodes (cylindricl node surfces). The inner dimeter of the tube, which is 25.4 mm (1 inch), determines the frequency of the first rdil mode. In the following, we wnt to concentrte on the resonnces below 16 khz (longitudinl modes). For these sttes, the mgnetic quntum number m is zero (σ-sttes). Mesure spectrum in longer tube-piece (75 mm). You will see tht the resonnces of the longitudinl modes shift down in energy, but the first rdil mode stys bove 16 khz. The next step is to model molecule by combining two pieces of 50 mm long tube with n iris of 10 mm dimeter (Ø10mm) between them. We re choosing to use the smllest iris becuse we wnt to model wek coupling of the toms. Tke spectrum (0-12 khz for exmple) in combintion of two 50 mm long tube-pieces with n iris Ø10 mm between them. Wht do you observe? Note tht the lowest bonding stte hs the frequency zero. The first ntibonding stte is observed t bout 1100 Hz. For the other peks the splitting in bonding / ntibonding sttes is visible clerly. Remember tht the smll peks t 370Hz nd 2000Hz re due to the frequency response of speker nd microphone. Repet the experiment with Ø13 mm nd Ø16 mm irises. Wht is different? Tke spectr with n incresing number of unit cells nd observe how bnds develop. Anlyze dt Compre the frequency difference between bonding nd ntibonding sttes with the width of the corresponding bnd in setup with lrge number of unit cells. 4-9
Stud. Mn. Rev 2.0 12/09 4.3 Superstructures nd unit cells with more thn one tom In this section, we will study the bnd structure of periodic lttice tht hs more complicted periodicity. A superstructure is periodic perturbtion of periodic lttice. The periodic perturbtion hs trnsltion vector tht is n integer multiple of the originl lttice vector. This cn be, for exmple, modifiction of every second unit cell. A superstructure results in new periodicity with lrger lttice vector, smller Brillouin zone nd smller reciprocl lttice vector. There re mny fields in condensed mtter physics where superstructures ply n importnt role. For exmple, in surfce science mny surfce structures show superstructure with respect to the bulk lttice. Another well-known exmple for superstructure in bulk lttice is Peierls distortion. We will study the effect on bnd structure by introducing periodic perturbtion into our one-dimensionl lttice. Mke setup of 12 tube-pieces 50 mm long nd 13 mm irises nd mesure spectrum. Then, replce every other iris by 16 mm iris nd mesure the spectrum gin. Wht do you observe? Plot the bnd structure for both cses. Mke setup of 5 unit cells with ech unit cell mde of 50 mm tube, 16 mm iris, 75 mm tube, nd 16 mm iris. Mesure spectrum nd plot the bnd structure. We wnt to understnd this bnd structure better by using the tight binding model nd compre therefore the energy levels with the resonnces found in the single toms. Tke spectr in 50 mm tube nd in 75 mm tube. Compre the tomic levels with the bnd structure. Wht cn you conclude? You my lso compre to spectrum mesured in single unit cell. You my now build different superstructures by yourself nd try to understnd the chnge in bnd structure due to the new periodicity. 4-10
Stud. Mn. Rev 2.0 12/09 4.4 Defect sttes In this section we will see how defects chnge the bnd structure. Defects destroy the periodicity of the lttice. They re loclized perturbtions. If the defect density is smll, the bnd structure is more or less conserved nd dditionl sttes re introduced due to the defects. The most importnt exmple for such defects sttes in condensed mtter physics is certinly the doping of semiconductors. The introduction of defect-sttes cretes the cceptor nd dontor levels tht re responsible for the unique properties of these mterils. Mke setup of 12 tube-pieces 50 mm long nd 16 mm irises nd mesure spectrum. Then, replce one tube-piece by 75 mm long piece nd mesure the spectrum gin. Wht do you observe? Plot the bnd structure for both cses. Note tht the defect-stte tht is observed in the first bnd-gp hs loclized wvefunction. Since it is loclized, it cnnot be ssigned to shrp wve-number. The stte is therefore plotted s horizontl line into the bnd structure in order to indicte tht it hs no welldefined wve-vector. You my hve noticed tht the peks within the upper bnds hve shifted little bit nd no longer show the high regulrity they did without defect. This is due to the fct tht the lttice hs lost its periodicity nd, strictly speking, it is no longer llowed to use the wve-number s good quntum number. However, from the plot of the bnd structure you see tht the defect does not chnge the bnd structure significntly. We cn tret it s smll perturbtion nd use the reciprocl spce with the Brillouin zone s we did in the periodic lttice. Put the defect t other positions within the one-dimensionl lttice nd mesure the spectr produced. Does the frequency of the defect-resonnce depend on the position? Use other tube lengths s defect. You cn try 25 mm, 37.5 mm nd 62.5 mm for exmple. In some cses you find the defect stte close to bnd edge. Such sitution is used in doped semiconductors. Donor-levels re defect sttes tht re occupied by electrons nd hve position just below the conduction bnd. The electrons cn be excited esily into the conduction bnd nd move there freely. This is very similr our cse with 62.5 mm tube s defect. Acceptor-levels re unoccupied defect sttes just bove the vlence bnd. Electrons cn be excited esily from the vlence bnd into the defect sttes nd the remining holes in the vlence bnd re responsible for the conductivity. Further experiments: You my build other setups with different types of defects. Be wre tht, within bnd gp, the propgtion of wve is suppressed strongly by reflection t the lttice. If the defects re too fr from ech other, or from speker nd microphone, they cnnot be observed. You my try using shorter setups tht hve smll number of unit cells. In this cse, it is esier to observe ll defect-sttes with sufficient mplitude. 4-11