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Understanding Ferromagnetic Spin Waves and Magnons: A Comprehensive Overview

This homework explores the dynamics of ferromagnetic spin waves, considering small deviations from equilibrium. It delves into the quantization of spin waves into magnons and discusses their properties using the Bose statistics framework. Key concepts include the energy levels associated with magnons, the heat capacity in low temperatures, and the implications for superconductivity relating to antiferromagnetic conditions. Through mathematical derivations and theoretical explorations, the principles governing spin interactions, quantization, and thermodynamic behavior are examined comprehensively.

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Understanding Ferromagnetic Spin Waves and Magnons: A Comprehensive Overview

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  1. Homework

  2. Ferromagnetic spin waves • Consider a ferromagnet with all the spins line up in equilibrium. Consider small deviation from it. Write Si=S0+ Si, •  Si=Ak exp(ik t-kr), Ak=A(1, i, 0) and ~k = 2J|S0| (1-cos{k }). For k small, k~Dk2 where D=JzS02

  3. Ferromagnetic spin waves  Si=Ak exp(ik t-kr), Ak=A(1, i, 0). Take the real part. At t=0,  S is along x at r=0 and along y at k r=/2. When t=/2, S is along y at r=0 and along –x at k r=/2

  4. Magnon: Quantized spin waves • a=S+/(2Sz)1/2, a+=S-/(2Sz)1/2. • [a,a+]~[S+,S-]/(2Sz)=1. • aa+=S-S+/(2Sz)=(S2-Sz2-Sz)/2Sz=[S(S+1)-Sz2+Sz]/2Sz=[(S+Sz)(S-Sz)+S-Sz] /2Sz. • S-Sz~aa+ • Hexch=-J (S-ai+ai)(S-aj+aj)+(Si+Sj-+Si-Sj+)/2 ~ constant-JS (-ai+ai-aj+aj+aiaj++ai+aj) =kk nk。 • ~k = 2J|S0| (1-cos{k })+K

  5. Quantization: Magnons are Bosons • Eigenvalue of n=a+a is quantized with eigenfunction |n>=a+|n-1>/n0.5. (the conjugate is a|n>=n0.5|n>. • First prove that the normalization is correct: <n|n>=<n-1|aa+|n-1>/n=<n-1|(a+a+1)|n-1>/n =[(n-1)<n-2|n-2>+1]/n=1. Finally a+a|n>=a+n0.5|n-1>=n|n>. Thus the energy of the system changes by integer multiples of k

  6. Magnon heat capacity • <E>=kk<nk>=kk /(e/kBT-1) • For T<J, only magnons with small k is excited. If T>K, can neglect the gap. <E>=[V/(2)3] d3k Dk2/(eDk2/kBT-1). • <E>/V=[(kBT)5/2/(D3/242)]0xm dx x3/2/(ex-1). • At low T aprroximate xm by . Then <E>/V T5/2; C T3/2

  7. Refresher for Bose Statistics • <n>=k=0 e-kx k/Z where x= /kBT. • Z=k e-kx =1/(1-e-x). • <n>=-x lnZ=e-x/(1-e-x)=1/(ex-1).

  8. Antiferromagnetic magnons: physics related to superconductivity • H=J  SjSj+ -2BHASajz +2BHASbjz. • a=Sa+/(2Sz)1/2, a+=Sa-/(2Sz)1/2 ; b+=Sb+/(2Sz)1/2, b=Sb-/(2Sz)1/2 ; Sajz=S-aj+aj, Sblz=-S+bl+bl. • H=ek[k( ak+bk++akbk)+(ak+ak+bk+bk)]+ a(ak+ak+bk+bk)]; e=2JzS, k= exp(ik)/z, a= 20Ha. • H involves products of two creation operators!

  9. AF magnons: [ak+,H]=ek[ak+,akbk] + (e+a)[ak+,ak+ak] = -ekbk -(e+a)ak+ ; [bk,H]= ekak+ +(e+a)bk; • Define k= ukak-vkbk+ ; k=ukbk-vkak+. Look for solutions of the form k+ exp(i t). itk+=[k+,H]=-k+. • [k+,H]= uk [-ekbk -(e+a)ak+ ]-vk [ekak+ +(e+a)bk]=- ( ukak +-vkbk ). Get uk (e+a) +vkek = uk ;uk ek +vk (e+a)=- vk .

  10. uk (e+a) +vkek = uk ;uk ek +vk (e+a )= - vk • (e+a )2-2 =(ek )2 ; k2 = (e+a )2-(ek )2 • Long wavelength limit k= [(e+a+e) (e (1-k )+a)]0.5 ; • k=0= [(2e+a ) a ] 0.5 >> a; (FMR) • k (a=0)=e (1-k 2) 0.5 k. (For F, k2)

  11. Normalization [k,k+]=uk2[ak,ak+]+vk2[bk+,bk]=uk2-vk2=1. • Write u=cosh , v=sinh  • Homework : Is it true that tanh 2=-(e+a)/[e(1-k)]?

  12. Superconductivity and antiferromagnet • Superconductivity • k=ukck-vkc-k+; -k=ukc-k+vkck+ • AF-Magnon: • k= ukak-vkbk+ ; k=ukbk-vkak+.

  13. Ground state magnetization • ak=ukk+vkk+; bk=ukk+vkk+ • <Saz>=NS- ak+ak=NS- (uk2k+k+vk2kk++ off-diagonal terms). • At T=0, nk=0, NS-<Saz>= vk2=k sinh2(k)  ddk/k. Fluctuation is infinite in 1 dimension.

  14. Magnons: Holstein-Primakoff transformation • Define spin wave operators a, a+ by S+/(2S)1/2=(1-a+a/2S)1/2a; S-/(2S)1/2= a+(1-a+a/2S)1/2 a; Sz=S-a+a • Assume a+a/2S<<1, Sz~S; then [S+,S-] =2Sz=2S[a,a+]=2S if [a,a^+]=1. a behaves like a boson destruction operator.

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