A Brief Introduction to the Physical Basis for Electron Spin Resonance

Transcription

1 A Brief Itroductio to the Physical Basis for Electro Spi Resoace I ESR measuremets, the sample uder study is exposed to a large slowly varyig magetic field ad a microwave frequecy magetic field orieted perpedicularly to the applied field [1,]. Usually the measuremets are made at a X bad: a microwave frequecy 9.5 GHz. A upaired electro has two possible orietatios i the large applied field ad thus two possible orietatio depedet eergies. (From classical electricity ad magetism, the eergy of a magetic momet i a magetic field H is - H.) Magetic resoace occurs whe the eergy differece betwee the two electro orietatios is equal to Plack s costat, h, times the microwave frequecy. For the very simple case of a isolated electro, the resoace requiremet may be expressed as hv g0 eh, (1) where g0 = ad is the Bohr mageto, ad me is the electro mass. The Bohr mageto is x10-8 J/G. eh / 4m, where e is electroic charge Expressio (1) describes the resoace coditio for a electro which does ot otherwise iteract with its surroudigs. The structural iformatio provided by ESR is due to deviatios from this simple expressio. For relatively simple trappig ceters, icludig all which have bee studied i MOS systems, these deviatios are due to spi-orbit couplig ad electro uclear hyperfie iteractios. e A. Spi-orbit couplig The deviatios from expressio (1) due to spi orbit couplig come about because a charged particle, the electro, is travelig i a electric field due to the uclear charge. It therefore experieces a magetic field B=Exv/c, where E is the electric field, v is the velocity, ad c is the speed of light. The spi-orbit iteractio may be uderstood qualitatively (ad oly qualitatively) i terms of the Bohr picture: a electro moves about the ucleus i a circular orbit. It would appear to a observer o the electro that the positively charged ucleus is i a circular orbit about the electro. (It appears to a usophisticated observer o earth that the su is i a circular orbit about the earth.) The ucleus thus geerates a local magetic field which would scale with the

2 electro s orbital agular mometum, rxp, ad with the uclear charge. Oe would thus correctly surmise that spi-orbit couplig iteractios icrease with icreasig atomic umber ad orbital agular mometum quatum umber. This spi-orbit iteractio is schematically illustrated i figure. (Note: Figure is iteded to provide oly a qualitative approximatio of reality.) (a) (b) Figure. (a) A electro orbitig a ucleus i the (ot quite right but illustrative) Bohr picture (b) A observer o the electro observes the motio of the ucleus aroud the electro. This results i a curret aroud the electro ad a local magetic field. The local field icreases with uclear charge ad icreasig electro orbital agular mometum. Therefore, this cotributio to the local field reflects the chemical ature of the site. I solids, the spi-orbit iteractio is queched but a secod order effect appears from excited states. This effect scales with the applied magetic field ad depeds o the orietatio of the paramagetic defect i the applied magetic field. The spi-orbit couplig may thus be icluded i the ESR resoace coditio by replacig the costat g0 of expressio (1) with a secod rak tesor gi j. The symmetry of this tesor reflects the symmetry of the paramagetic ceter. Uder some circumstaces, the symmetry of the tesor may permit idetificatio of the defect uder study. Perturbatio theory allows calculatio (with modest accuracy) of the g tesor for the simple defects. The compoets of the g tesor are give by g ij Li k k L j g0 ij. () ( E E ) k k

3 Here g0 is the free electro value, the atomic spi-orbit couplig costat, Li ad Lj are agular mometum operators appropriate for the x, y, or z directios, ad the summatio is over all excited states k. State > ad eergy E correspod to the paramagetic groud state of the system. The g tesor ca ofte, by itself, provide structural iformatio if the system uder study cotais uclei of sigificatly differet charge. The atomic spi orbit couplig costat, as figure suggests, is a very strog fuctio of uclear charge. Thus a germaium daglig bod spectrum likely yields a much larger rage of g tesor compoets tha a silico daglig bod. B. Electro-uclear hyperfie iteractios The other importat source of deviatio from expressio (1) is the hyperfie iteractio of the upaired electro with earby uclei. Certai uclei have magetic momets; some importat examples i semicoductor device materials are 7.8% of germaium uclei, 73 Ge (spi 9/), 4.7% of silico uclei, 9 Si (spi ½), almost 100% of hydroge uclei, 1 H (spi ½), as well as essetially 100% of phosphorous uclei, 31 P (spi ½). The ear 100% abudat 14 N ucleus has a spi of 1. A spi 1/ ucleus has two possible orietatios i the large applied field; a spi 1 ucleus three possible orietatios. Each uclear momet orietatio correspods to oe local uclear momet field distributio. We evisio the uclear momet iteractig with a upaired electro residig i a wave fuctio which is a liear combiatio of atomic orbitals (LCAOs). For itrisic defects i SiGe ad related systems, the most importat cases almost certaily ivolve s- ad p-type wave fuctios. (For defects ivolvig trasitio metal impurities, aalysis will be somewhat more complicated but the basic priciples are the same.) The LCAO for a upaired electro ca be writte as a c s c p, (3) s p where s ad p represet the appropriate atomic orbitals correspodig to the th site, a represets the localizatio o the th site, ad c s ad ad p character of the wave fuctio o the th atomic site. c p represet, respectively, the amout of s

4 For a site i which the upaired electro is reasoably well localized at a sigle atom, the aalysis is particularly simple. To first order we ca iterpret ESR spectra from such a site i terms of s/p hybridized atomic orbitals localized at that cetral atom. (The geeralizatio to a defect wave fuctio ivolvig sigificat sharig by several atoms is very straightforward.[ 1]) The electro uclear hyperfie iteractio of a electro i a p orbital is aisotropic: it correspods to a classical magetic dipole iteractio as schematically illustrated i Fig. 3. The iteractio is strogest whe the field is parallel to the symmetry axis. The sig of the iteractio chages ad the magitude is decreased by oe half whe the field is perpedicular to the symmetry axis. Whe the electro ad uclear momets are aliged by a strog magetic field i the z directio, a reasoable assumptio for work discussed i this appedix, oly the z compoet of the dipolar field is importat, because the iteractio eergy ivolves a dot product. - H. This z compoet, g (1-3 cos )r -3, is averaged over the electroic wave fuctio to produce the dipolar cotributio, 1 3cos Dipolar cotributio = g 3 r. (4) (a) (b) Figure 3. Iteractio of the uclear momets dipole field (lies) with the p orbital electro (shaded areas). I (a) the applied field, ad thus the uclear momet, is aliged parallel to the orbital symmetry axis. I (b) the applied field, ad thus the uclear momet, is aliged perpedicular to the orbital symmetry axis.

5 Figure 4. Iteractio of the uclear momets dipole field (lies) with the p orbital electro (shaded areas). The iteractio of the electro ad the applied magetic field is axially idepedet because of the spherical symmetry of the s orbital. The electro uclear hyperfie iteractio of a electro i a s orbital is isotropic. This iteractio is illustrated i Fig. 4. The figure shows why the iteractio is isotropic. The spherical symmetry of the orbital results i zero iteractio with this field except for a spherical regio about the ucleus with a radius of a imagiary curret loop geeratig the uclear momet s field. Sice the s orbital has a ozero probability desity at the ucleus, a large isotropic iteractio results. The s orbital hyperfie iteractio ca also be computed from a elemetary electricity ad magetism calculatio: the magetic field at the ceter of a curret loop of radius a is give by /a 3, where m is the magetic momet of the curret loop. The probability desity of the electro varies little over the volume of the ucleus; take it to be costat, (0). Cosiderig the oly the fractio of the electro wave fuctio at the ucleus, to be be 4 a 3 3 (0), the iteractio would A (isotropic) = 4 3 a (0) 3. (5) 3 a

6 mageto, The magetic momet of the ucleus is the uclear g factor, g, times the uclear Bohr. Thus, the isotropic or Fermi cotact iteractio is give by Aiso = 8 g (0), (6) 3 where (0) represets the upaired electro probability desity at the ucleus. Both isotropic ad aisotropic hyperfie iteractios are likely preset for quite a few of the defects to be observed i SiGe systems. This is so because the upaired electro wave fuctios geerally ivolve both p-orbital ad s-orbital character, ad i some cases d-orbital character as well. The hyperfie iteractios, like the spi-orbit iteractios, are expressed i terms of a secod rak tesor. I may cases, to a pretty good approximatio, the ceters have axially symmetric wave fuctios ad thus a axially symmetric tesor is appropriate. With the magetic field parallel to the p orbital symmetry axis, the aisotropic couplig of Eq. (4) yields (4/5) g 3 r ; the field perpedicular to the symmetry axis results i a iteractio of half the magitude ad opposite sig -(/5) g 3 r. This result is ituitively satisfyig ad cosistet with the sketches of Fig. 3. The compoets of the hyperfie tesor correspod to sums of the isotropic ad aisotropic iteractios for the applied field parallel ad perpedicular to the upaired electro s orbital symmetry axis: A Aiso Aaiso ; (7) where A A iso A aiso, (8) A aiso 5 3 g N r N. (9) For a paramagetic ceter with a specific orietatio (desigated by the agle betwee the symmetry axis ad the applied field vector) the resoace coditio is

7 H 0 1 H M A, (10) where H 0 hv / g, ad M 1 is the uclear spi quatum umber, e 1 cos g si ) g ( g (11) ad 1 cos A si ) A ( A (1) Equatios (10) (1) ad oly moderately more complex geeral case expressios provide a very straightforward basis for aalyzig ESR results for defects with a specific orietatio with respect to the applied magetic field. The descriptio of ESR spectra of defects withi a poly-crystallie or amorphous film, such as a SiO passivatig layer, is more complex but still comprehesible. All defect orietatios are equally likely ad, due to the lack of log rage order, slight differeces i local defect geometry may be aticipated. The presece of defects at all orietatios leads to the cotiuous distributio of both g ad A values from g ad A to g ad A. Differeces i local geometry may also lead to slight defect-to-defect variatios i g, g, A, ad A. (a) (b) Figure 5. (a) Radomly orieted daglig bod defects lead to a relatioship with ay particular directio which is (b) ot radom at all. May more daglig bods poit i a directio perpedicular to a axis tha parallel to it. Thus, the ESR patter of a radom array of defects will yield a weak sigal correspodig to the ESR parameters for the field parallel case ad a strog sigal for the field perpedicular case.

8 Both of these complicatios are relatively easy to deal with. The radom distributio of defect orietatio ca be dealt with easily i terms of aalytical expressios foud i most ESR textbooks. Agai, let s thik about the somewhat simplified case of a axially symmetric defect wave fuctio. (The most geeral lowest possible symmetry case is a logical extesio of the higher symmetry case ad is well uderstood.) For axially symmetric ceters, far fewer ceters will have the symmetry axis parallel to the applied field tha perpedicular to it; thus the ESR spectrum itesity will be far stroger at the A ad g values tha at A ad g.) The slight defect-to-defect variatios i g ad A values lead to broadeig of the lie shapes aticipated for ubroadeed tesor compoets. (The process is illustrated i Fig. 6.) The evaluatio of ESR hyperfie tesor compoets allows for a reliable ad moderately precise idetificatio of the upaired electro s wave fuctio. A little commo sese usually allows oe to idetify the magetic uclei ivolved i observed hyperfie iteractios. (Commo sese ivolves ispectio of the fractio of the spectrum is split by the hyperfie lies, coutig the umber of hyperfie lies, if possible evaluatig the symmetry of the defect spectrum ) Havig idetified the uclear species ivolved, a first order aalysis of the upaired electro wave fuctio is extremely straightforward i terms of the LCAO picture. For defects i a crystallie eviromet, Eq. (1) ca be fit to the ESR spectrum for multiple orietatios. (For more complex defects, the extesio of Eq. (1) to the most geeral case is reasoably straightforward.[1]) For simple defects i a polycrystallie eviromet oe may fit the appropriately broadeed aalytical expressios to the ESR spectra to yield A ad A. (This process is illustrated i Fig. A5.) By applyig Eqs. (7) ad (8) to this result, oe may the obtais the isotropic ad aisotropic couplig costats Aiso ad Aaiso.

9 Figure 6. A schematic plot of ESR amplitude versus magetic field correspodig to a site with axially symmetric g ad hyperfie tesors ad a spi-1/ ucleus i a polycrystallie material, such as a polycrystallie thi film of SiGe. Note that the breadth of the lies depeds o the rage of variatio i the g ad hyperfie tesors. Tabulated values of Aiso ad Aaiso calculated for 100% occupatio probability ca the be utilized to determie the hybridizatio ad localizatio of the electroic wave fuctios. For example, the isotropic ad aisotropic couplig costats for a electro 100% localized i a silico s ad p orbital are, respectively, ao= G ad bo=40.75 G. I Fig. 7, we illustrate a ESR trace of the E ceter, the domiatig deep hole trap i high quality thermally grow oxides o silico. Figure 7. The ESR patter of E ceters from SiO. The strog cetral peak ( with about 95% of the total itegrated itesity) correspods to the 95% abudat 8 Si o-magetic uclear sites. The two weaker side peaks (with a combied itesity of about5%) are due to the 9 Si magetic uclei sites which are about 5% abudat.

10 A applicatio of the aalysis schematically idicated i Fig. 6 idicates that Aiso=439 G ad Aaiso= G. If the electro were 100% localized i a silico s orbital, we would expect a isotropic couplig costat of ao = G. We measured 439 G; thus the orbital has 439/1639 7%s. If the electro were 100% localized i a silico p orbital we would expect Aaiso=bo=40.75 G. We measured G; thus, the orbital has / %p. The aalysis idicates a localizatio o the ceter silico of about (54+7)=81%. Although the crude aalysis just discussed is ot extremely precise it is, to first order, quite reliable. Oe should realize that the isolated atomic values obtaied for ao ad bo are themselves oly moderately accurate ad that placig a isolated atom i a matrix of other atoms will ievitably alter the costats somewhat. Nevertheless, a straightforward aalysis of hyperfie parameters provides moderately accurate measuremet of hybridizatio ad localizatio. Refereces i text ca be foud o page labeled Refereces o website