基于格林函数的纳米器件模型分析

基于格林函数的纳米器件模型分析一．The Green’s function1.定义A Green's function, G(x, s), of a linear differential operatorL = L(x) acting on distributions over a subset of the Euclidean space R n, at a point s, is any solution of(1)where is the Dirac delta function. This property of a Green's function can be exploited to solve differential equations of the form(2)As a side note, the Green's function as used in physics is usually defined with the opposite sign; that is,（3）If the operator is translation invariant, that is when L has constant coefficients with respect to x, then the Green's function can be taken to be a convolution operator, that is,（4）In this case, the Green's function is the same as the impulse response of linear time-invariant system theory.2.推算Loosely speaking, if such a function G can be found for theoperator L, then if we multiply the equation (1) for the Green's function by f(s), and then perform an integration in the svariable, we obtain;（5）The right hand side is now given by the equation (2) to be equal to L u(x), thus:（6）Because the operator L = L(x) is linear and acts on the variable x alone (not on the variable of integration s), we can take the operator L outside of the integration on the right hand side, obtaining;（7）And this suggests;3.格林函数在纳米器件中作用（8）（9）对比方程（1），可知：带入方程（9），可知：其中，这就是格林函数表示电荷密度和密度矩阵，其中[A(E)]是谱函数（spectral function）,谱函数A的物理意义：Indeed the diagonal elements of [A（E）]/2*pi in the realspace representation give us the local density of states atdifferent points in space (a quantity that can be measuredwith scanning probe microscopy)4.自能矩阵（self-energy）（1）物理含义The concept of self-energy is used in many-body physics to describe electron–electron and electron–phononinteractions. In the present context, however, we are usingthis concept to describe something much simpler, namely, theeffect of a semi-infinite contact.（2）自能矩阵的推导在考虑了电极时，沟道的总哈密顿量为τ是电极对沟道的作用矩阵H是电极的哈密顿矩阵R总的格林函数为从上面的公式可得：其中电极作用的等价表示：This shows that the effect of the coupling to the reservoir can be accounted for by adding a self-energy matrix to the Hamiltonian H This is a very general concept that allows us to eliminate the huge reservoir and work solely within the device subspace whose dimensions are much smaller自能矩阵的求解：The indices m, n refer to points within the device while μυ，refer to points inside the reservoir.表面格林函数：t he coupling matrix τ couples the points withinthe device to a small number of points on the surface ofthe reservoir, so that we only need (,)R g μυ for points (,)μυ that are on the surface.)(][1ββαi i i i gs I E gs ++---= （）it should be noted that the periodic boundary conditions merely get rid of end effects through the artifact of wrapping the device into a ring while the self-energy method treats the open boundary condition exactly. An open system has a continuous energy spectrum, while a ring has a discrete energy spectrum.It might appear that the self-energy method is just anothermethod for handling boundary effects.（3）自能矩阵的性质与推演Firstly, they are energy dependent.Secondly, they are not Hermitian.自能矩阵性质的影响The point we want to make is that the self-energy terms have two effects. One is to change the Hamiltonian from H L to L H ∧which changes the eigenstates and their energies. But more importantly, itintroduces an imaginary part to the energy determined by the‘broadening’ functions （扩展函数）1Γand 2Γ . The former represents a minor quantitative change （量变）; the latter represents a qualitative change （质变）with conceptual implications.[H +∑] has complex eigenvalues and the imaginary part of the eigenvalues both broadens the density of states and givestheeigenstates a finite lifetime.扩展矩阵：We have often made use of the fact that we can simplify ourdescription of a problem by using the eigenstates of the Hamiltonian[H] as our basis. For open systems we would want to use arepresentation that diagonalizes [H +H ∑]in our energy range ofinterest.If the same representation also diagonalizes [Γ], then the problem could be viewed simply in terms of many one-level devices in parallel.本征能量为where 1,2,εγγare the corresponding diagonal elements 1,L H ∧ΓΓ2，, respectively122212+())((+)/2)E E γγδεεγγ-−−→-+（ 5. 态密度（density of states ）和局域态密度(Local density of states)ε has a density of statesa system with a set of eigenvaluesαgiven by从这个公式发现能级的态密度权重为1，但实际发现不同的能级态密度的权值不同，这主要是原来的态密度没有考虑空间态密度分布（spatial distribution of the states）,所以为了知道沟道的局域态密度，我们要乘入属于沟道波函数平方，即If we look at the local density of states in the channel we see a series of energy levels with varying heights, reflecting the fraction of the squared wavefunction residing in the channel局域态密度定义：局域态密度更普通的概念：the diagonal element(divided by 2π)of the spectral function [A(E)] 同样的电荷密度是密度矩阵的对角元素因为所以可证明二．相干传输（Coherent transport）1.传输系数（Transmission）传输函数：One could view the device as a “semi-permeable membrane” that separates two reservoirs of electrons (source and drain) and the transmission function T (E)asa measure of the permeability of this membrane to electrons with energy E.传输模型：In the transmission formalism (sometimes referred to as the Landauer approach) the channel is assumed to be connected to the contacts by two uniform leads that can be viewed as quantum wires with multiple modes or subbands having well-defined E–k relationships传输定理：This allows us to define an S-matrix for the device analogous to a microwave waveguide where the element tnm of the t-matrix tells us the amplitude for an electron incident in mode m in lead 1 to transmit to a mode n in lead 2两端器件：The Green’s function formalism provides a simple way to separate the total spectral function TAU into a left spectral function [A1] and aright spectral function [A2]:朗道定理：I-V 特性分析：自洽计算二维泊松方程在沟道碳原子构成的正六边形网格上的差分格式为U bdy与两端源漏电势有关平衡态电压分布U(r)。