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# ADVANCED STAT MECH PHYS 220

UCSB

GPA 3.8

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This 10 page Class Notes was uploaded by Hailey Halvorson on Thursday October 22, 2015. The Class Notes belongs to PHYS 220 at University of California Santa Barbara taught by Staff in Fall. Since its upload, it has received 16 views. For similar materials see /class/227136/phys-220-university-of-california-santa-barbara in Physics 2 at University of California Santa Barbara.

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Date Created: 10/22/15

MATH 220B FEBRUARY 11 2008 Groebner Bases De nition The m0n0mial ideal generated by A C ZZOYL is the ideal consisting of all sums of the form 2 04614 Where ha 6 Klz1 will and for each element in the ideal7 all but nitely many of the polynomials ha are 0 Lemma If I is the monomial ideal generated by A7 the a monomial z lies in I if and only if z is divisible by z for some a E A Theorem Dicksonls Lemma Let I be the monomial ideal generated by A Then I ltz 1 zu m for some 0417 045 in A In particular7 I has a nite basis De nition Let I C Kz1zn be a nonzero ideal The set of leading terms of elements ofI is denoted by LTI The ideal generated by the leading terms of elements ofI is denoted by ltLTI Note that ltLTI is a monomial ideal Theorem Hilbert Basis Theorem Every ideal I C Klzl will has a nite generating set That is7 I ltg17 7gt for some g17 gt 6 I Theorem For a xed monomial order7 every nonzero ideal I C Klzl will has a Groebner basis To actually nd Groebner bases7 we need an additional idea De nition Let fg E Klz1 will be nonzero polynomials7 and x a monomial order The S polynomial off and g is the polynomial SUE g10mLMfLM9 i 10mLMf LM9 LTf my 3 In this di erence7 the leading terms cancel each other out and so one has an a priori new leading term Theorem Buchbergerls Lemma The set g1L7 7g is a Groebner basis for the ideal it generates if and only if the remainder upon division of Sgigj by g1gt is 0 for every i 79739 1 2 Example 1 Consider the two polynomials in Kxpq f 963 p96 q g Bfdz 3z2 10 and the ideal I ltfggt Then h i 5029 gpz q and since f h mg we have I ltghgt Next7 3 1 S h 77 72 7quot 97 2qz3p 2 3 ASh 73 72 77 9p 291 We Claim that 97177 7 A is a Groebner basis To Check this7 we must show that all remaining S pairs are linear combinations of the generators We have 5977quot ph 597 A 39 1927 Sh7 A gqr SO A 7102A ngr Example 2 We use Groebner bases to eliminate variables and solve the system of equations z2y222 1 x222 y x2 Start with the ideal I lth17h27h3gt7 where h1x2y22271 hg z222 71 hg ziz We de ne h4h17h3y222271 I15 h2 h3 2h3222 y CHANGING RELATIVE VOLTAGEs GaTe volTage NEGATIVE r39elcn ive To semiconducTor39 ACCUMULATION VGltO EF Fermi energy 3 x5 Ec conduction band EF Fermi energy V valence band rela rive To semiconducTor DEPLETION 5 x5 Ec conduction band EF Fermi energy EF Fermi energy Ev valence band VG gtgtO CHANGING RELATIVE VOLTAGES GaTe volTage MUCH MORE POSITIVE Than semiconducTor39 ExpOSed accepTor39s 39 INVERSION EF Fermi energy elecTr ons XS Ec conduction band EF Fermi energy Ev valence band Ionized impur39ifies Fr39ee elecTr39ons Vol rage modula rion of charge curren r VgltO VggtO VggtgtO ACCUMULATION DEPLETION INVERSION channel Free carriers Fixed charge Free carriers WhaT can CAPACITANCE measuremenTs Tell us abouT The maTerial s TrucTure and The inTerfaces Change of capacitance with voltage c Ci O 0 CFBFLAT BAND a WS 0 08 CAPACITANCE A Uquot 0 6 W5 2W8 A 5 4 s 5 C b 04 quot i C m39quot C Figure 4 MOS capacitors 02 i J BREAKDOWN J 0 v V 0 me VT ant V vivorrs Fig 7 M15 capacitance voltage curves a Low frequency b High frequency 0 Deep depletion After Grove el al Re 6 By sweeping voltage from positive to negative values semiconductor surface goes through inversion depletion accumulation Voltagetunable capacitance as the depletion layer width changes The CV can give insight abou l39 The values of oxide Thickness semiconduc l39or39 doping Fig 10 CIC 3amp75102 NAixiO396cm 3 i 4 i i iO 0 IO VWOLTS Ideal MIS C V curve Solid iines for low frequencies Dashed lines for high IrequenciesAf1er Goetzberger R51 15 Impurities in Oxides Mobile ionic charge 0m SIO2 Oxide trapped charge 0m Fixed oxide QM 0 quotWequot 1 O Silicnn Melvtnvk 39 39 Charge O Nelwalktnmter mam Figure 48 Sclwmuuc ul huntmm IHLI impcrl39m lmm in SioX 3390 Interface trapped Si charge Qn Figure 414 Silicon silicon dioxide structure with mobile xed charge and interface states 1980 IEEE D l 6 Wha r affect do DEFECTS in The oxides have on CAPACITANCE CV can diagnose faul rs in The MOS s rr39uc rur39e 4EAL C V CURVE v v o 0 39 a STRETCHOUT AV DUE TO INTERFACE TRAPPED CHARGES LOW FREQUENCY HIGH FREQUENCY V Fig 17 Capacitance sireichout due to inierface trapped charges b Fig 23 C V curve shift along the voltage axis due to positive or negative fixed oxide charge a For ptype semiconducmr b For ntype semiconductor After Nicollian and Brews Ref 7 DisTor39Tion of shape and frequency shift gt fixed charge Dependence gt Traps amp mobile charge

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