Results And Physics Of Doped Graphene

Graphene is honeycomb lattice structure with two inequivalent carbon atoms per unit cell. The two inequivalent carbon atoms were labeled by A and B sites. The momentum space of graphene is also honeycomb lattice structure. The band structure of graphene resolved by tight-binding method is called as Dirac core, because it satisfied Dirac equa-tion and linear divergence. The point at w = 0 is called Dirac point. There are two independent Dirac cores of graphene in momentum space, their Dirac points labeled by K and K’. Experimentally, the adsorption Co atoms on heavily doped graphene lead to unusual Kondo resonances [6]. The experimental results of Ref. [6] show two-channel Kondo behavior as shown in Fig.(2.5). The QPT in doped graphene can be illustrated in a phase diagram as shown in Fig.(2.6), no Kondo screening in zero chemical potential [41].

Physical explanation for the effective low energy for magnetic impurities in graphene: ” existence of two Dirac points are related to two independent Kondo screening channels”.

Note that: someone consider doped graphene is 1CK, because they believe two Dirac

2.3. CONSTANT DENSITY OF STATES AND DOPED GRAPHENE

core are scattering to each other at low temperature. The tight-binding description shows the hybridization between electron states in graphene and impurity states preserves the A-B sub-lattice symmetry [69]. The Co is located at the center of a graphene’s hexagon [70], the inter valley scattering does not coupled to two screening channels, therefore ef-fective 2CK ensues [68]”. Theoretically, electrons in graphene provide a realization of two-dimensional Dirac electrons [41, 67]. Magnetic impurities coupled to two-dimensional Dirac electrons corresponds doped graphene [64, 65, 66], obeying the pseudogap conduc-tion electron density of states (ρc(w) ∼ |w|^r) , where exponent r=1. We use the 2CK infinite U Anderson impurity model (Eq.(2.1) and (Eq.(2.7)) to describes this model. It is worth mentioning: no vanishing gate voltage Kondo screening in graphene when weak coupling regime [41, 44, 45]. We can investigate doped graphene QPT by gate tuning as shown in Fig.(1.14.a).

Figure 2.6: The quantum phase transition in graphene can be investigated by controlling the chemical potential µ0. When µ = 0, there is no 2CK state, always in LM state.

If µ 6= 0, the quantum phase transition between LM state and 2CK can be observed.

Adapted from [41].

When chemical potential is zero (µ0 = 0), there is no Kondo screening. The QPT of graphene by gate controlled have been studied via NRG approach (see Ref. [41]).

Here, we introduce QPT out of equilibrium of doped graphene with bias voltage by NCA

2.3. CONSTANT DENSITY OF STATES AND DOPED GRAPHENE

Figure 2.7: (a) The phase diagram for 2CK-LM crossover QPT, parameter j represents coupling Γ or chemical potential µ, and j^∗ is the crossover scale for a fixed temperature T0 = 5 × 10⁻⁷D. All parameters in units of half-bandwidth D=1, and Tk and T0 are energy scales associated with the 2CK universal scaling. (b) A spin one half impurity couples symmetrically to the two sub-lattices of two graphene leads out of equilibrium, where two graphene leads are in thermal equilibrium, respectively. (c) Doped graphene in momentum space. Adapted from [60].

solutions adapted from Section 2.2 [60]. The schematic setup. corresponds to Eq.(2.7) as shown in Fig.(2.7.b), a spin one half impurity (red dot) couples symmetrically to the two sub-lattices of two graphene leads, where different chemical potentials of each graphene. The two Dirac points at two independent Dirac Core corresponds to two Kondo screening channels as shown in Fig.(2.7.c). The diagram of QPT between LM and 2CK state as shown in Fig.(2.7.a) can be controlled by Γ and chemical potential, where Γ is the coupling constant of leads and dot (|Vkσ^α2/π|). The impurity spectral function which is the convolution of the greater and lesser Green functions is calculated via NCA equations numerically at different chemical potential as shown in Fig.(2.8). NCA solutions can provide correct physic results at Dirac point as two-channel Kondo. The impurity spectral function does not present p-h symmetry. The hight of Kondo peak is related to

2.3. CONSTANT DENSITY OF STATES AND DOPED GRAPHENE

Kondo temperature (Tk), and 2CK Kondo temperature is the function as [57]:

Tk ∼ w(Γ/2π(µ − ǫ^d))^1/2exp[−π(µ − ǫ^d)/Γ]. (2.51)

The parameters Γ and µ affect the the hight of Kondo peak. To analyze 2CK conductance scaling, we start form conductance behavior of two-channel Kondo. The most important property of 2CK conductance is the square root curve (see Section 1.2). Theoretically, we analyze two-channel Kondo conductance scaling function both of equilibrium and out of equilibrium [71]. First of all, we discus equilibrium system conductance, there is no bias voltage in system, and temperature dominates physics behavior, as a function G(0,T): a temperature dependent function formulated as,

G(0, T ) − G(0, 0) = B^cT¹², (2.52)

where G(0,0) is a constant here, we have to omit this constant to observe square root curve in logarithmic scale. This analyzed function shows non-Fermi liquid behavior of two-channel Kondo in equilibrium. The non-equilibrium 2CK conductance scaling is formulated in terms of variable V and T,

G(V, T ) − G(0, T ) = B^cT¹²H(A eV

kBT). (2.53)

The function H(A_k^eV

BT) can be calculated by field theory, where H(A_k^eV

BT) ∼ (_k^eV_BT)² for

eV

kBT << 1, and H(A_k^eV

BT) ∼ (_k^eV_BT)^1/2 for _k^eV

BT >> 1. In non-equilibrium case, temperature T is a constant as T0 = 5 × 10⁻7D. From NCA equations, we numerically calculate doped graphene conductance both equilibrium and out of equilibrium. The results via NCA are shown in Fig.(2.9). Compare Fig.(2.9) to Fig.(2.5), there is comparison between theory and experiment. The Fig.(2.9.a) and Fig.(2.9.b) are the 2CK scaling of equilib-rium conductance. The Fig.(2.9.c, d, e) are the 2CK scaling at different interval out of equilibrium.

Give a summaries of 2CK graphene universal scaling in conductance: In equilibrium, there is √

T behavior when T < Tk. In non-equilibrium system, there is p

(eV /kBT ) behavior when T^′ < T < Tk, and (eV /kBT )² when T^′ < T^∗, where T^′ is a small temper-ature. We have studied QPT of 2CK pseudogap Anderson model for r=0 and r=1 via

2.3. CONSTANT DENSITY OF STATES AND DOPED GRAPHENE

NCA. In next section, we will investigate quantum criticality and find universal scaling for 0 < r < 1 both in equilibrium and out of equilibirum.

-0.2 -0.15 -0.1 -0.05 0 0.05

ω / D

2 4 6 8

ρ(ω)

µ = −0.1 µ = −0.09 µ = −0.08 µ = −0.07 µ = −0.06

6 8 10 12 14 1 / |µ|

10^-12 10^-8 10^-4 10⁰

TK / | µ |

Dirac Point

Figure 2.8: The impurity DOS via NCA v.s. different chemical potential µ in units of D. Dirac point is located at w/D = 0, and Kondo peaks are pinned near each µ.

Different levels w/D with different chemical potentials µ correspond to different Fermi levels, respectively. The parameter are T₀ = 5 × 10⁻7D, Γ = 0.2D, ǫ_d= −0.2D, where D is half-bandwidth. T0 is temperature of the system, Γ is coupling constant of leads and dot, and ǫd is dot level. Adapted from [60].

2.3. CONSTANT DENSITY OF STATES AND DOPED GRAPHENE

Figure 2.9: (a) The linear conductance in equilibrium system. It show square root behav-ior at temperature which is lower than Kondo temperature. (b) There is an additional power law behavior at high temperature in equilibrium conductance scaling. It shows T^∗ ∼ |Γ − Γ^∗|^1/µ, where Γ^∗is a small number, 0.05D and µ ∼ 0.1. (c) Nonlinear conduc-tance at Γ = 0.2D. The fixed parameters are T0 = 5 × 10⁻⁷D, µ = −0.1D, ǫ^d = −0.2D, where D = 1 is half-bandwidth. (d) The (eV /kBT )² behavior arises as (eV /kBT ) << T . (e) The (eV /kBT )¹² behavior arises as (eV /kBT ) is between T and TK. Adapted from [60].

Chapter 3

Results Of The 2CK Pseudogap Anderson Model: Quantum Phase Transition and Quantum Criticality

In this chapter, we provide our NCA results for pseudogap 2CK Anderson model within quantum critical region for 0 < r < 1. The 2CK QPT with pseudogap density of states have been investigated via renormalization group (RG) approach (see Section 1.3). Here, we investigate QPT in 2CK Anderson model by tuning exponent r via NCA approach.

From impurity density of states (DOS), we analyze the quantum phase transition between LM and 2CK state and extract the quantum critical point. The differential conductance both in equilibrium and non-equilibrium cases are analyzed in Section 3.2. From our numerical results in conductance, we find the universal scaling at and near quantum criticality. The schematic phase diagrams of equilibrium and non-equilibrium are sum-marized in Fig.(3.1). The quantum critical point is at and near r → rc ≃ 0.115. The phase diagrams describe equilibrium and non-equilibrium crossover between LM and 2CK with crossover scales being T^∗ and V^∗, respectively. Both equilibrium and out of equi-librium phase diagrams in conductance show power law divergence at quantum critical region. We also observe different universal scaling behavior between equilibrium and out

3.1. QUANTUM CRITICALITY SHOWS IN IMPURITY DENSITY OF STATES

of equilibrium systems.

Figure 3.1: The phase diagram of equilibrium and non-equilibrium conductance. The phase diagrams delineated by T^∗ and V^∗, which is crossover scale in equilibrium and out of equilibrium, respectively. The inverse of crossover scale is correlation length, ξ, which is diverge in quantum critical region, and νT = 4, νV = 0.5 are universal factor called as correlation length exponent. Due to νT, νV are different, the universal scaling in equilibrium is different from out if equilibrium. Here, critical point is at and near r_c = 0.115, even if G(0,T) and G(V,T) have slightly different parameters (Γ and ǫ_d).

3.1 Quantum Criticality Shows In Impurity Density Of States

The pseudogap DOS (ρc(w)) of conduction electrons vanishes in power law fashion at Fermi energy (ρc ∼ |ω|^r). If there is sufficient conduction electron DOS to screen the

3.1. QUANTUM CRITICALITY SHOWS IN IMPURITY DENSITY OF STATES

Figure 3.2: The r=0.05 density of states of impurity non-equilibrium system with Γ = 0.3D, ǫd = −0.3D, T = T⁰ = 5 × 10⁻⁷D, where D = 1. The impurity DOS exhibits asymmetry graph, where no peak at _d+ U due to particle-hole asymmetry. The non-equilibrium density of states with bias, V = 0.038D, Kondo peak divided into two peaks, because the Fermi energy of left and right leads are different, the width between two peaks is equal to bias.

magnetic impurity in 2CK system, the ground state is going to 2CK fixed point. On the other hand, it leads to LM fixed point if conduction electron DOS is not sufficiently large.

Here, we study quantum phase transition of pseudogap 2CK Anderson model by tuning r (0 < r < 1) at µ₀ = 0 with fixed parameters Γ and ǫ_d, where µ₀ is chemical potential of the conduction bath. Not that: There is no Kondo screening in graphene (r=1) if µ0 = 0.

We investigate quantum critical region form analyzing numerical data via NCA. In this chapter, we define two-channel Kondo temperature, T2CK. In quantum dot system, three important features arise as 2CK ground state emerges (T < T2CK),

1. The entropy of impurity spin (S(Ω)) at T → 0 is that S(Ω = 2) ≃ k^Bln√

2, where T is the temperature of the system,

2. A non-Lorentzian Kondo peak occurs at Fermi level in impurity DOS.

3. As T < T2Ck, the conductance follows the scaling function Eq.(2.51) and Eq.(2.52).

3.1. QUANTUM CRITICALITY SHOWS IN IMPURITY DENSITY OF STATES

From the 2nd feature of 2CK, we study impurity DOS for extract the critical point at and near rc in this section. The conductance behavior and the universal scaling will be discussed in Section 3.2. The non-equilibrium impurity for r=0.05 is shown in Fig.(3.2), where Kondo peak is spilt up into two peaks due to bias voltage. We find Kondo peak in impurity DOS for r=0.05 is short, because QPT between 2CK and LM leads to Kondo peak shorter as r is close to rc more and more. The impurity DOS for different r is demonstrated in Fig.(3.3.a), it exhibits QPT between LM and 2CK by varying r in Kondo peak (Fig.(3.3.b)). Numerically, we can not address zero temperature, so T0 = 5 × 10⁻⁷D is used to approach to zero temperature, where T0 is the lowest numerically accessible temperature. The Kondo peaks are shorter as r increases ,and there is a dip of Kondo peak for r > 0.13, we predict that LM state occurs while r > 0.13. The height of Kono peak is related to Kondo temperature which is as a function in Eq.(2.51). We can predict the critical point form the change of Kondo peaks. In order to study the quantum phase transition of two-channel pseudogap Anderson model both in equilibrium and non-equilibrium cases, we choose two sets of parameters to do so.

1. Equilibrium: Γ = 0.28D, ǫd = −0.2D, zero bias voltage, varying temperature T, and the lowest temperature T = T₀,

2. Non-equilibrium: Γ = 0.3D, ǫd= −0.3D, T⁰ = 5 × 10⁻⁷D, varying bias voltage V, when we study QPT in non-equilibrium conductance with bias voltage, temperature of the system is fixed at T0. The parameter Γ (∼ V²/π) is the coupling strength of leads and dot, ǫdis the dot level, and D=1 is half bandwidth. The parameters Γ and ǫdall affect 2CK Kondo temperature. But we find critical point is at and near rc = 0.115 both in parameter Γ = 0.28D, ǫd= −0.2D and Γ = 0.3D, ǫ^d = −0.3D. thus forecast quantum critical point is fixed as Γ and ǫ_d does not change too much. From the change of Kondo peaks for different r in impurity DOS, the QPT is observed and we can extract quantum critical region and critical point. Furthermore, we find the universal scaling both of equilibrium and non-equilibrium conductance in the quantum critical region, these details will be discussed in next section.