臺灣博碩士論文加值系統

所謂的量子相變(quantum phase transition) 是指在絕對零度經由調整某些參數所以引起的基態的連續相變。然而這些相變是因為量子擾動而非熱擾動。另一方面, 近藤效應(Kondo effect) 也是凝態物理中的重要課題, 它是一個磁性雜質被電子自旋屏蔽而引起的物理現象。因為奈米科技的進步, 近藤效應在量子點(quantum dot) 系統的想法得以實現, 也因此, 與近藤效應破滅相關的量子臨界現象(quantum criticality) 也變成了重要的課題。如果我們可以讓磁性
雜質同時和兩個獨立的電子庫耦合, 雙渠道的近藤效應(two-channel Kondo) 就能被實現。雙渠道的近藤效應會造成非費米液體行為(non-Fermi liquid), 這是有別於一般費米金屬的行為。
本論文中, 我們使用的是雙渠道贗能隙安德森雜質模型(two-channel pseudogap single impurity Anderson model), 所謂的贗能隙能態密度(pseudogap density of states) 就是能態密度(ρ(ω) ∼ |ω|r, 0 ≤ (r ≤ 1) 在接近w = 0時會呈現冪次方的消逝。然而贗能隙能態密度的指數r
也分別對應不同的磁性物質參雜材料, 例如加入磁性雜質的石墨稀(doped graphene) 就是對應r=1, 而常數能態密度則對應r=0。如果指數r 過於大, 能態密度會消失過快而沒有足夠的電子去屏蔽雜質, 因此雙渠道的近藤效應會破滅, 取而代之的是矩限態(local moment, LM). 只要r夠小的話, 系統就會停留在雙渠道的近藤效應基態(two-channel Kondo ground state)。我們
研究雙渠道贗能隙安德森雜質模型(0 ≤ (r ≤ 1) 在化學能為零(μ0 = 0) 的情況下, 靠著調整指數r 來觀察雙渠道近藤效應和矩限態的量子相變。我們只用slave-boson 大N 近似法(large N approach) 去自洽的解雜質電子的格林函數, 其中我們忽略所有有不交叉的費曼圖,所以又稱不交叉近似法(NCA)。從磁性雜質的能態密度, 我們可以觀察到量子臨界點(r = rc), 更進一步的, 同時在平衡態和非平衡的導電度(conductance) 上找出統一標度律(universal scaling)。
本論文可以提供未來研究非費米液體及量子臨界行為的理論基礎。

Quantum phase transition (QPT) are the continuous phase transition of ground states by tuning couplings in the quantum system. They are due to zero-temperature quantum
fluctuations, not thermal fluctuations. Meanwhile, Kondo effect is an important phenomenon in condensed matter systems, which is an effect describing the screening of
magnetic impurity by the spin of conduction electrons in magnetical doped metals. Due to the advances in nano-technology, Kondo effect in quantum dots (QD) have been realized in single electron tunneling transistor (SET), therefore QPT associated with the broken down of the Kondo effect becomes an interesting subject. If two independent electron reservoirs exist, two-channel Kondo effect (2CK) becomes possible. It leads to non-Fermi
liquid (NFL) behavior, which shows different electric transport from Fermi liquid metals.
In our study, we use the 2CK pseudogap Anderson impurity model to describe the system
where the single impurity is coupled to 2CK pseudogap electron bath, where its density
of states (ρ(ω)) vanishes in a power law fashion (ρ(ω) ∼ |ω|r, 0 ≤ r ≤ 1) for ω → 0. The exponent r of pseudogap density of states varies with different materials. The magnetical doped graphene (r=1) is an example of 2CK pseudogap Anderson single impurity model system, and two-channel quantum dot system with constant density of states corresponds to r = 0 case. If r is too large, there is no sufficient electron density of states to screen
the impurity spin, 2CK state is broken down, resulting in unscreened local moment (LM) ground state. Let r be small enough, two-channel Kondo effect becomes possible. We
study QPT in 2CK pseudogap single impurity Anderson model at zero-chemical potential by tuning r (0 < r < 1) both of equilibrium and out of equilibrium. We use slaveboson
large-N approach to self-consistently solve Green’s functions of electron on the dot by including all non-crossing diagrams, so-called non-crossing approximation (NCA). We extract the quantum critical point (rc) from impurity density of states, and find the universal
scaling both in equilibrium and non-equilibrium conductances near rc. This thesis provides theoretical basis for further study in Kondo break down, quantum criticality and
non-Fermi liquid behavior in condensed matter systems.

1 Introduction 1
1.1 Kondo Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Kondo Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Kondo Effect In Quantum Dot System . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Coulomb Blockade Oscillations With Kondo Effect . . . . . . . . . 5
1.3 Quantum Phase Transitions (QPT) . . . . . . . . . . . . . . . . . . . . . . 9
1.3.1 Two Channel Kondo (2CK) Physics . . . . . . . . . . . . . . . . . . 12
1.3.2 The Pseudogap Kondo Problems . . . . . . . . . . . . . . . . . . . 16
2 Large N Approaches To 2CK Anderson Model 24
2.1 Methods To N-fold Degenerate Anderson Model . . . . . . . . . . . . . . . 25
2.1.1 The Foundation Of Slave Boson Representation . . . . . . . . . . . 25
2.1.2 Non crossing approximation (NCA) approach . . . . . . . . . . . . 26
2.2 The 2CK Pseudogap Anderson Model out of equilibrium . . . . . . . . . . 27
2.3 Constant Density of States And Doped Graphene . . . . . . . . . . . . . . 36
2.3.1 Results Of Constant Density Of States Anderson Model . . . . . . . 36
2.3.2 Results And Physics Of Doped Graphene . . . . . . . . . . . . . . . 38
3 Results Of The 2CK Pseudogap Anderson Model: Quantum Phase
Transition and Quantum Criticality 44
3.1 Quantum Criticality Shows In Impurity Density Of States . . . . . . . . . 45
3.2 Conductance near LM-2CK Quantum Critical Point . . . . . . . . . . . . . 48
3.2.1 Equilibrium Conductance G(0.T) . . . . . . . . . . . . . . . . . . . 49
3.2.2 Non-equilibrium Conductance G(V,T0) . . . . . . . . . . . . . . . . 52
4 Summary And Conclusion 58
Appendix 61
ix

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