PHYSICAL REVIEW B 66, 035304 共2002兲

Green’s function for a Schro¨dinger electron in a one-dimensional superlattice miniband with axial electric and magnetic fields Norman J. M. Horing, Vadim I. Puller, and Lev G. Mourokh Department of Physics and Engineering Physics, Stevens Institute of Technology, Hoboken, New Jersey 07030

Yuriy A. Romanov Institute for Physics of Microstructures RAS, 603600 Nizhny Novgorod, Russia 共Received 23 October 2001; published 3 July 2002兲 The Green’s function for Schro¨dinger/Bloch electrons in a superlattice miniband with parallel, axial electric and magnetic fields is derived here. The electric field is taken to have arbitrary time dependence 共with specific evaluation for a constant electric field兲 and the magnetic field is arbitrarily strong with Landau quantization of orbits in the lateral plane. The resulting Green’s function is presented in both the scalar and vector potential representations for the electric field, and the magnetic phase and gauge dependencies are explicitly shown. Our results are expressed in terms of elementary functions in direct time representation; they generate the trancendental Landau eigenfunctions of the lateral plane along with the miniband Wannier-Stark-ladder Bloch states in scalar potential representation and Houston functions with accelerated Bloch states in vector potential representation for the electric field. DOI: 10.1103/PhysRevB.66.035304

PACS number共s兲: 73.21.Cd, 72.10.Bg, 75.75.⫹a

⬘ ,xB⬘ ,t ⬘ 兲 G ret 1 共 xA ,xB,t;xA

I. INTRODUCTION

The quest for smaller, faster semiconductor devices has led to intensified research on the nonlinear quantum transport properties of electrons in a miniband driven by an axial electric field—both with and without a magnetic field.1 To elucidate the dynamics of one-dimensional 共1D兲 miniband electrons, we present their retarded single-particle Schro¨dinger Green’s function in an axial electric field of arbitrary time dependence and an axial magnetic field of arbitrary strength 共constant in time兲 including Landau quantization of orbits in the lateral plane. In carrying out this derivation here we follow a technique of Schwinger2 employing operator Hamilton equations of motion to determine the miniband Green’s function with a time-dependent axial electric field. The additional part of the Hamiltonian referring to the magnetic-field electron dynamics in the lateral plane commutes with the 1D miniband part of the Hamiltonian, and the corresponding 2D magnetic field Green’s function was determined previously.3 In this situation, the two partial Green’s functions yield the full Green’s function by simply multiplying them 共with a factor iប). This is readily verified for time-independent single-particle Hamiltonians by considering the full Hamiltonian comprised of two commuting parts H (1) ⫽h A(1) ⫹h B(1) with 关 h A(1) ,h B(1) 兴 ⫺ ⫽0, and noting that the full single-particle ˆ ret Green’s operator, G 1 , is, in turn, separable because of the commutation 关the superscript 共1兲 denotes a single-particle Hamiltonian in the present consideration兴: G ret 1 共 t⫺t ⬘ 兲 ⫽⫺ ⫽⫺

i (1) i ␩ ⫹ 共 t⫺t ⬘ 兲 e ⫺ ប H (t⫺t ⬘ ) ប i (1) i (1) i ␩ ⫹ 共 t⫺t ⬘ 兲 e ⫺ ប h A (t⫺t ⬘ ) e ⫺ ប h B (t⫺t ⬘ ) . ប

i ␩ 共 t⫺t ⬘ 兲 ប ⫹ i

(1)

⫻ 具 xA兩 e ⫺ ប h A

(t⫺t ⬘ )

i

(1)

⬘ 典具 xB兩 e ⫺ ប h B 兩 xA

(t⫺t ⬘ )

兩 xB⬘ 典

⬘ ,t ⬘ 兲 G Bret 共 xB,t;xB⬘ ,t ⬘ 兲 . ⫽iបG Aret 共 xA,t;xA

共2兲

In the Appendix we verify that this result also obtains if h A(1) →h A(1) (t) and h B(1) →h B(1) (t) depend on time, provided they always commute. In regard to Schwinger’s technique for determining a retarded single-particle Green’s function, G ret 1 , we denote the single-particle time development operator by U(t,t ⬘ ) and ¨ write G ret 1 in the Schrodinger picture 共subscript s) as iបG ret 1 共 x,t;x⬘ ,t ⬘ 兲 ⫽ ␩ ⫹ 共 t⫺t ⬘ 兲 具 x兩 U 共 t,t ⬘ 兲 兩 x⬘ 典 s , and note that 关bear in mind ⫽(1/iប)H (1) S U(t,t ⬘ ) and U(t,t)⫽1兴, iប

that

共3兲

( ⳵ / ⳵ t)U(t,t ⬘ )

⳵ ret G 共 x,t;x⬘ ,t ⬘ 兲 ⳵t 1 ⫽ ␦ 共 t⫺t ⬘ 兲 ␦ (3) 共 x⫺x⬘ 兲 ⫹

1 ␩ 共 t⫺t ⬘ 兲 iប ⫹

⫻ 具 x兩 H s(1) 共 t 兲 U 共 t,t ⬘ 兲 兩 x⬘ 典 s .

共4兲 ¨ In the Schrodinger picture at hand, both 具 x兩 and 兩 x⬘ 典 are evaluated at the ‘‘initial’’ time t ⬘ . Transferring the description of 具 x兩 to time t in the Heisenberg Picture,

具 x兩 s U 共 t,t ⬘ 兲 ⫽ 具 x兩 H ⬅ 具 x共 t 兲 兩 ,

共1兲 or,

Employing direct product states associated with the mutually independent sets of variables xA , xB , associated with h A(1) , h B(1) , respectively, we have 0163-1829/2002/66共3兲/035304共5兲/$20.00

⫽⫺

具 x共 t ⬘ 兲 兩 ⫽ 具 x共 t 兲 兩 U ⫹ 共 t,t ⬘ 兲 .

共5兲

Thus, we have

66 035304-1

©2002 The American Physical Society

PHYSICAL REVIEW B 66, 035304 共2002兲

HORING, PULLER, MOUROKH, AND ROMANOV

iប

⳵ ret G 共 x,t;x⬘ ,t ⬘ 兲 ⳵t 1 ⫽ ␦ 共 t⫺t ⬘ 兲 ␦

(3)

II. 1D MINIBAND GREEN’S FUNCTION IN A TIME-DEPENDENT ELECTRIC FIELD

The 1D miniband Hamiltonian is taken in a tight-binding, nearest-neighbor approximation as 共the superlattice axis is in the z-growth direction兲,

1 共 x⫺x⬘ 兲 ⫹ ␩ ⫹ 共 t⫺t ⬘ 兲 iប

⫻ 具 x共 t 兲 兩 U ⫹ 共 t,t ⬘ 兲 H s(1) 共 t 兲 U 共 t,t ⬘ 兲 兩 x⬘ 共 t ⬘ 兲 典 .

⌬ h z(1) ⫽ 共 1⫺cos关 pˆ z d/ប 兴 兲 , 2

共6兲

Identifying the Hamiltonian in the Heisenberg picture, (1) (t) and considering t⬎t ⬘ , we obtain HH iប

⳵ ret i (1) G 共 x,t;x⬘ ,t ⬘ 兲 ⫽⫺ 具 x共 t 兲 兩 H H 共 t 兲 兩 x⬘ 共 t ⬘ 兲 典 . ⳵t 1 ប

where pˆ z is the crystal momentum operator in the z-axial direction and ⌬ and d are the miniband width and superlattice period, respectively. The presence of an axial timedependent electric field, E z (t), can be represented by the addition to h z(1) of a scalar potential ⫺eE z (t)zˆ , where zˆ is the position operator for the z-direction. While one can deal with this directly, such a choice violates lattice-translation invariance and causes inconvenience in appending boundary conditions. It is often more useful to introduce the electric field by way of a vector potential using E z (t) ⫽⫺(1/c) ⳵ A z (t)/ ⳵ t, where

共7兲

Furthermore, using Eq. 共3兲 jointly with Eq. 共5兲 and dividing Eq. 共6兲 by G ret 1 , we have 1 G ret 1 共 x,t;x⬘ ,t ⬘ 兲 ⫽⫺

⳵ ret G 共 x,t;x⬘ ,t ⬘ 兲 ⳵t 1

(1) 共 t 兲 兩 x⬘ 共 t ⬘ 兲 典 i 具 x共 t 兲 兩 H H , ប 具 x共 t 兲 兩 x⬘ 共 t ⬘ 兲 典

共8兲

冋

⫻exp ⫺

i ប

冕

t

t⬘

dt ⬙

具 x共 t ⬙ 兲 兩 H H(1) 共 t ⬙ 兲 兩 x⬘ 共 t ⬘ 兲 典 具 x共 t ⬙ 兲 兩 x⬘ 共 t ⬘ 兲 典

册

冕

e pˆ z →pˆ z ⫺ A z 共 t 兲 . 共11兲 c 0 With this, the Hamiltonian remains translationally invariant as, A z 共 t 兲 ⫽⫺c

which is readily integrated as G ret 1 共 x,t;x⬘ ,t ⬘ 兲 ⫽ ␩ ⫹ 共 t⫺t ⬘ 兲 K 共 x,x⬘ ;t ⬘ 兲

共10兲

t

dt ⬘ E z 共 t ⬘ 兲 ;

冋冉

再

冊 册冎

d ⌬ e 1⫺cos pˆ z ⫺ A z 共 t 兲 . 共12兲 2 ប c The associated Green’s function is readily obtained using the method of Sec. I. The operator Hamilton equations of motion are h z(1) ⫽

,

共9兲 where K(x,x⬘ ;t ⬘ ) is a constant of integration that is independent of t, to be determined by initial conditions. (1) Employing Eq. 共9兲 to obtain G ret 1 , one writes H H (t ⬙ ) in terms of the Heisenberg position and momentum operators xˆ(t ⬙ ),pˆ(t ⬙ ), deriving their Hamilton equations of motion and solving for xˆ(t ⬙ ),pˆ(t ⬙ ) explicitly in terms of t ⬙ ,xˆ(0) and pˆ(0) 关we set t ⬘ ⬅0 as the initial time兴. This pair of solutions (1) (t ⬙ ) in permits the elimination of pˆ(t ⬙ ) and pˆ(0) from H H ˆ ˆ favor of x(t ⬙ ) and x(0), while using the equal-time canonical commutation relations to commute xˆ(t ⬙ ) to the far left where it acts as, 具 x(t ⬙ ) 兩 xˆ(t ⬙ )⫽ 具 x(t ⬙ ) 兩 x, while xˆ(0) on the right acts as, xˆ(0) 兩 x⬘ (0) 典 ⫽x⬘ 兩 x⬘ (0) 典 , yielding an explicit function of t ⬙ ,x and x⬘ which may be integrated as in Eq. 共9兲 关with cancellation of the residual factors 具 x(t ⬙ ) 兩 x⬘ (t ⬘ ) 典 in the numerator and denominator on the right兴.

d pˆ z ⬅0⇒pˆ z 共 t 兲 ⫽const⫽ pˆ z , dt and

冋冉

d e dzˆ ⌬d pˆ z ⫺ A z 共 t 兲 ⫽ sin dt 2ប ប c which yield the solution

冕

冋冉

冊册

共13兲

共14兲

,

冊册

⌬d t e d dt ⬙ sin pˆ z ⫺ A z 共 t ⬙ 兲 . 共15兲 2ប t ⬘ ប c As pˆ z is a constant operator for the miniband it is obviously advantageous to proceed in a pˆ z representation: The derivation of G ret 1 in position representation, resulting in Eq. 共9兲, is readily repeated in momentum representation, with the result zˆ 共 t 兲 ⫺zˆ 共 t ⬘ 兲 ⫽

冋

G ret 1 共 p z ,t;p z⬘ ,t ⬘ 兲 ⫽ ␩ ⫹ 共 t⫺t ⬘ 兲 K 共 p z ,p z⬘ ;t ⬘ 兲 exp

i ⫺ ប

冕

t

t⬘

dt ⬙

具 p z 共 t ⬙ 兲 兩 H H(1) 共 t ⬙ 兲 兩 p z⬘ 共 t ⬘ 兲 典 具 p z 共 t ⬙ 兲 兩 p z⬘ 共 t ⬘ 兲 典

册

,

共16兲

where p z (t), p z⬘ (t ⬘ ) are the eigenvalues of pˆ z at time t, t ⬘, respectively. In the case at hand, pˆ z is a constant of the motion, (1) (1) and G ret 1 (p z ,t; p z⬘ ,t ⬘ ) is diagonal as a matrix in p z , p z⬘ because H H ⫽h z is diagonal. Thus we have

冋

G ret 1 共 p z ,t;p z⬘ ,t ⬘ 兲 ⫽ ␦ 共 p z ⫺p z⬘ 兲 K 共 p z 兲 exp

i ⫺ ប

where, because of the constancy of pˆ z , 035304-2

冕

t

t⬘

dt ⬙

具 p z 共 t ⬙ 兲 兩 H H(1) 共 t ⬙ 兲 兩 p z 共 t ⬘ 兲 典 具 p z共 t ⬙ 兲兩 p z共 t ⬘ 兲 典

册

,

共17兲

¨ DINGER ELECTRON IN A . . . GREEN’S FUNCTION FOR A SCHRO

具 p z 共 t ⬙ 兲 兩 H H(1) 共 t ⬙ 兲 兩 p z 共 t ⬘ 兲 典 具 p z共 t ⬙ 兲兩 p z共 t ⬘ 兲 典

具 p z共 t ⬙ 兲兩 ⫽

PHYSICAL REVIEW B 66, 035304 共2002兲

冋冉

再

⌬ d e 1⫺cos pˆ z ⫺ A z 共 t 兲 2 ប c

冊 册冎

兩 p z共 t ⬘ 兲 典

⫽

具 p z共 t ⬙ 兲兩 p z共 t ⬘ 兲 典

Consequently, in momentum representation,

冋冉

再

d ⌬ e 1⫺cos p z⫺ A z共 t 兲 2 ប c

再 冕 冠 冋冉 冠 冕 再 冋冉

共18兲

.

冊 册 冡冎

d i⌬ t e dt ⬙ 1⫺cos p z⫺ A z共 t ⬙ 兲 . 2ប t ⬘ ប c To determine K( p z ) we Fourier transform to position representation, obtaining ⬁ dp i d i⌬ t e z p z (z⫺z ⬘ ) ប e z,t;z ,t ⫽ ␩ t⫺t K p exp ⫺ dt ⬙ 1⫺cos p z⫺ A z共 t ⬙ 兲 G ret 兲 兲 兲 共 共 共 ⬘ ⬘ ⬘ ⫹ z 1 2 ␲ ប 2ប ប c ⫺⬁ t⬘ and note that the initial condition may be written as ⬁ dp i i z ⫹ K 共 p z 兲 e ប p z (z⫺z ⬘ ) ⫽⫺ ␦ 共 z⫺z ⬘ 兲 , G ret 1 共 z,t ⬘ ;z ⬘ ,t ⬘ 兲 ⫽ 2 ␲ ប ប ⫺⬁ from which it is apparent that i K 共 p z 兲 ⫽⫺ . ប The resulting 1D miniband Green’s function in vector potential representation is ⬁ dp z i p (z⫺z ) d i i⌬ t e ⬘ exp ⫺ dt ⬙ 1⫺cos p z⫺ A z共 t ⬙ 兲 eប z G ret 1 共 z,t;z ⬘ ,t ⬘ 兲 ⫽⫺ ␩ ⫹ 共 t⫺t ⬘ 兲 ប 2ប t ⬘ ប c ⫺⬁ 2 ␲ ប G ret 1 共 p z ,t;p z⬘ ,t ⬘ 兲 ⫽ ␩ ⫹ 共 t⫺t ⬘ 兲 ␦ 共 p z ⫺p z⬘ 兲 K 共 p z 兲 exp ⫺

冕

冊 册冎

共19兲

冊 册冎 冡

共20兲

,

冕

冠 冕 再 冋冉

冕

冊 册冎 冡

共21兲

共22兲

共23兲

.

Moreover, for the special case of a constant electric field, A z (t)⫽⫺cE z t, we have ( ␻ B ⫽eE z d/ប is the Bloch frequency兲, ⬁ dp z i p (z⫺z ) p zd ␻ B i i⌬ 2 ⬘ exp ⫺ t⫺t ⬘ ⫺ sin关 ␻ B 共 t⫺t ⬘ 兲 /2兴 cos , G ret eប z ⫹ (t⫹t ⬘ ) 1 共 z,t;z ⬘ ,t ⬘ 兲 ⫽⫺ ␩ ⫹ 共 t⫺t ⬘ 兲 ប 2ប ␻B ប 2 ⫺⬁ 2 ␲ ប 共24兲 in the vector potential representation.

冉

再 冋

冕

冊 册冎

III. CONCLUSIONS: MAGNETIC FIELD ROLE IN THE MINIBAND GREEN’S FUNCTION AND SCALAR POTENTIAL GAUGE

The retarded 2D Green’s function of the lateral plane in an axial magnetic field 共with Landau quantization of orbits in that plane兲 is readily extracted from its thermodynamic Green’s function counterpart which was determined previously by one of the authors3 as ¯ G ret 1 共 x,t;x⬘ ,t ⬘ 兲 ⫽⫺ ␩ ⫹ 共 T 兲

4␲ប

冉

i¯

m␻c

C 共 ¯x,x⬘ 兲 2

冊

e ⫺ ប␮0B␴3T im ␻ c 关 X 2 ⫹Y 2 兴 exp , sin共 ␻ c T/2兲 4ប tan共 ␻ c T/2兲

共25兲

where X⫽x⫺x ⬘ ,

Y ⫽y⫺y ⬘ ,

and

Z⫽z⫺z ⬘ ,

冋

C 共 ¯x,x⬘ 兲 ⫽exp

T⫽t⫺t ⬘ ,

¯x⫽ 共 x,y 兲 ,

x⬘ ⫽ 共 x ⬘ ,y ⬘ 兲

册

i e ¯x•B⫻x⬘ ⫺ ␾ 共 ¯x兲 ⫹ ␾ 共 x⬘ 兲 , ប 2c

¯ ) is an arbitrary magnetic gauge with B as the magnetic field 共and E z the electric field, both in the z-axial direction兲, ␾ (x ¯ function, ␮ 0 is the Bohr magneton, and ␻ c is the cyclotron frequency ( ␴ 3 is the third Pauli spin matrix兲. Combining this 2D magnetic field Green’s function with that of a superlattice miniband in a time-dependent electric field as determined in Sec. II, using the product form of Sec. I, we obtain the result i¯

冉

e ⫺ ប␮0B␴3T im ␻ c 关 X 2 ⫹Y 2 兴 ¯x,x⬘ 兲 C exp G ret 共 1 共 x,t;x⬘ ,t ⬘ 兲 ⫽⫺ ␩ ⫹ 共 T 兲 sin共 ␻ c T/2兲 4ប tan共 ␻ c T/2兲 4␲ប2 m␻c

冠 冕 再 冋冉 冠 再 冋 冉

冊 册冎 冡 冊 冉

冊冕

⬁

dpz i p Z eប z ⫺⬁ 2 ␲ ប

d i⌬ t e dt ⬙ 1⫺cos p ⫺ A 共t⬙兲 , 2ប t ⬘ ប z c z or, for a constant electric field E z , the p z -integral is replaced by ⬁ dp ⬁ dp i ␻B p zd ␻ B i⌬ 2 z z T⫺ sin ⇒ e ប p z Z exp ⫺ ⫹ 共 t⫺t ⬘ 兲 cos 共 t⫹t ⬘ 兲 2ប ␻B 2 ប 2 ⫺⬁ 2 ␲ ប ⫺⬁ 2 ␲ ប ⫻exp ⫺

冕

冕

035304-3

共26兲

冊 册冎 冡

.

共27兲

PHYSICAL REVIEW B 66, 035304 共2002兲

HORING, PULLER, MOUROKH, AND ROMANOV

Moreover, if the electric field vanishes, Eq. 共27兲 yields ⬁ dp ⬁ dp i p zd i⌬ z z 1⫺cos ⇒ e ប p z Z exp ⫺ 2ប ប ⫺⬁ 2 ␲ ប ⫺⬁ 2 ␲ ប

冕

冉 冊册 冎

再 冋

冕

共28兲

T .

While the vector potential gauge for the electric field has the advantage of maintaining lattice-translation invariance, there are circumstances in which the scalar potential gauge for the electric field is advantageous because it maintains time-translation invariance when the electric field is independent of time 共but not lattice-translation invariance兲. To determine the Green’s function in the scalar potential gauge, G ⬘1 , we note that under the gauge transformation A(x,t)→A⬘ (x,t)⫽A(x,t) ⫹“␭(x,t) and ␸ (x,t)→ ␸ ⬘ (x,t)⫽ ␸ (x,t)⫺1/c ⳵ ␭(x,t)/ ⳵ t, the Green’s function transforms as ie 共29兲 G ret „␭ 共 x,t 兲 ⫺␭ 共 x⬘ ,t ⬘ 兲 … G ret 1 (x,t;x⬘ ,t ⬘ )→G 1⬘ (x,t;x⬘ ,t ⬘ )⫽exp 1 共 x,t;x⬘ ,t ⬘ 兲 . បc In the vector potential gauge, A z (t)⫽⫺c 兰 t0 dt ⬘ E z (t ⬘ ), whereas in the scalar potential gauge we have a scalar potential ␸ ⬘ (x,t)⫽⫺E z (t)z. The gauge transformation between the two, which assures the vanishing of ␸ (x,t) and A z⬘ (t) is given by

再

␭ 共 x,t 兲 ⫽cz

冎

冕

t

0

dt ⬘ E z 共 t ⬘ 兲 .

共30兲

Thus, in the scalar potential gauge, the appropriate Green’s function is t ie t⬘ z dt ⬙ E z 共 t ⬙ 兲 ⫺z ⬘ dt ⬙ E z 共 t ⬙ 兲 G 1⬘ 共 x,t;x⬘ ,t ⬘ 兲 ⫽exp ប 0 0

冋 冉冕

冕

冊册

G ret 1 共 x,t;x⬘ ,t ⬘ 兲 ,

共31兲

with G ret 1 (x,t;x⬘ ,t ⬘ ) given by Eqs. 共26兲,共27兲. In the special case of a constant electric field 关Eq. 共27兲兴 there is no time dependence in the scalar potential gauge and, consequently, there is time-translation invariance which mandates that G 1⬘ (x,t;x⬘ ,t ⬘ ) be a function of T⫽t⫺t ⬘ , to the exclusion of ␶ ⫽t⫹t ⬘ . Therefore, writing t⫽(T⫹ ␶ )/2 and t ⬘ ⫽( ␶ ⫺T)/2 everywhere in Eq. 共31兲 with G ret 1 (x,t;x⬘ ,t ⬘ ) given by Eqs. 共26兲,共27兲, the fact that G 1⬘ (x,t;x⬘ ,t ⬘ ) has no dependence on ␶ means that we can evaluate it at ␶ ⫽0 and thereby obtain the result for any ␶ . This corresponds to setting t⫽T/2 and t ⬘ ⫽⫺T/2 everywhere in Eq. 共31兲 and Eqs. 共26兲,共27兲 jointly, and we obtain the result for constant electric field as G 1⬘ 共 x,x⬘ ;T 兲 ⫽⫺ ␩ ⫹ 共 T 兲

m␻c

C 共 ¯x,x⬘ 兲 exp 2

冉

冊

i¯

4␲ប ⬁ dp i p zd i⌬ 2 z e ប p z Z exp ⫺ ⫻ T⫺ cos sin共 ␻ B T/2兲 2 ␲ ប 2ប ␻ ប ⫺⬁ B

再 冋

冕

冉 册冎

ie e ⫺ ប␮0B␴3T im ␻ c 关 X 2 ⫹Y 2 兴 exp E z T(z⫹z ⬘ ) 2ប sin共 ␻ c T/2兲 4ប tan共 ␻ c T/2兲

冉 冊

冊

共32兲

.

An alternative expression may be obtained using the Jacobi-Anger formula for e iz cos ␣ leading to

冉

冊

i¯

ie e ⫺ ប␮0B␴3T ¯x,x⬘ 兲 exp C T(z⫹z ) G 1⬘ 共 x,x⬘ ;T 兲 ⫽⫺ ␩ ⫹ 共 T 兲 E 共 ⬘ z 2ប sin共 ␻ c T/2兲 4␲ប2 m␻c

⫻exp

冉

im ␻ c 关 X 2 ⫹Y 2 兴 4ប tan共 ␻ c T/2兲

冊冕

冉

⬁

冊

i d p z i p Z ⫺ i⌬T ⌬ i k e ប kdp z J k e ប z e 2ប sin共 ␻ B T/2兲 , ប␻B k⫽⫺⬁ ⫺⬁ 2 ␲ ប ⬁

兺

共33兲

where J k (x) is the Bessel function of kth order. In this form the p z -integral yields ␦ (Z⫹kd)-functions, and we obtain an alternative to Eq. 共32兲 as

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i¯

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ie e ⫺ ប␮0B␴3T im ␻ c 关 X 2 ⫹Y 2 兴 ⫺ i⌬T ¯x,x⬘ 兲 exp C T(z⫹z ) exp e 2ប G ⬘1 共 x,x⬘ ;T 兲 ⫽⫺ ␩ ⫹ 共 T 兲 E 共 ⬘ 2ប z sin共 ␻ c T/2兲 4ប tan共 ␻ c T/2兲 4␲ប2 m␻c

⬁

⫻

兺

k⫽⫺⬁

i kJ k

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⌬ sin共 ␻ B T/2兲 ␦ 共 Z⫹kd 兲 , ប␻B

共34兲

in scalar potential representation. Equation 共26兲 in vector potential representation, and Eq. 共32兲 in scalar potential representation, have the advantage of being expressed in terms of elementary functions in direct time representation 共albeit under a momentum Fourier transform兲. These elementary function representations generate the transcendental Landau eigenfunctions of the lateral plane along with the miniband Wannier-Stark-ladder Bloch states in scalar potential representation and Houston functions with accelerated Bloch states in vector potential representation. The relative simplicity of the elementary functions involved lends analytic tractability to these representations which will facilitate the analysis of more complicated problems: for example, the determination of the full nonequilibrium Green’s function 共with arbitrary time indices兲 for electrons subject to axial electric and magnetic fields in a miniband, in which the generalized-Kadanoff-Baym 共GKB兲 Ansatz involves knowledge of the retarded 共and advanced兲 function. 035304-4

¨ DINGER ELECTRON IN A . . . GREEN’S FUNCTION FOR A SCHRO

or,

APPENDIX

Addressing the case in which h A(1) and h B(1) do depend on time, h A(1) →h A(1) (t) and h B(1) →h B(1) (t), we treat a Schro¨dinger operator of the form ⳵ ⳵ ⌳ 1 ⫽iប ⫺h A(1) 共 t 兲 ⫺h B(1) 共 t 兲 ⫽iប ⫺H (1) , 共A1兲 ⳵t ⳵t where we consider single-particle Hamiltonians h A(1) (t) and h B(1) (t) that commute at all times and involve variables that are mutually independent, say sets of variables x A and x B . ⬘ ,x B⬘ ,t ⬘ ) is The associated Green’s function G ret 1 (x A ,x B ,t;x A then given by iប times the product of the individual Green’s functions associated with h A(1) and h B(1) separately as, ⬘ ,x B⬘ ,t ⬘ 兲 G ret 1 共 x A ,x B ,t;x A where

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⫽iបG Aret 共 x A ,t;x A⬘ ,t ⬘ 兲 G Bret 共 x B ,t;x B⬘ ,t ⬘ 兲 , G Aret and G Bret satisfy the equations iប

共A2兲

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⳵ ⫺h A(1) 共 x A ,t 兲 G Aret 共 x A ,t;x A⬘ ,t ⬘ 兲 ⳵t

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⳵ G ret 1 ⫺ 关 h A(1) 共 x A ,t 兲 ⫹h B(1) 共 x B ,t 兲兴共 G Aret G Bret 兲 ⫽0, 共A9兲 ⳵t since the variables in h A(1) (x A ,t) and h B(1) (x B ,t) and those in G Aret and G Bret respectively, are mutually exclusive. Thus, it is ret ret clear that G ret 1 ⫽iបG A G B satisfies the expected homogeneous equation for t⬎t ⬘ with the separable Hamiltonian h A(1) (t)⫹h B(1) (t), iប

⳵ G ret 1 ⫺ 关 h A(1) 共 x A ,t 兲 ⫹h B(1) 共 x B ,t 兲兴 G ret 1 ⫽0. 共A10兲 ⳵t

Moreover, ⫹ ⬘ ,x B⬘ ,t ⬘ 兲 G ret 1 共 x A ,x B ,t ⬘ ;x A

⫽iបG Aret 共 x A ,t ⬘ ⫹ ;x A⬘ ,t ⬘ 兲 G Bret 共 x B ,t ⬘ ⫹ ,x B⬘ ,t ⬘ 兲 , 共A11兲 where we have used the assumed form of Eq. 共A2兲. Employing Eqs. 共A5兲,共A6兲, this yields i ␦ 共 x A ⫺x A⬘ 兲 ␦ 共 x B ⫺x B⬘ 兲 , ប 共A12兲 which is the appropriate initial condition to append to Eq. ret ret 共A2兲, so that G ret 1 ⫽iបG A G B satisfies all requisite conditions to be identified as the retarded Green’s function for H (1) ⫽h A(1) (t)⫹h B(1) (t). The striking similarity of Eq. 共A2兲 to the Hartree-Fock approximation has its origin in the fact that in the case of two identical particles, A and B, h A(1) (x A ,t) and h B(1) (x B ,t) commute, as in our present considerations. To be specific, the similarity lies in the structure of the exact Hartree-Fock solution for the two-particle Green’s function in the case of noninteracting identical particles, as given by (1⫽x1 ,t 1 , etc.兲 ⫹ ⬘ ,x B⬘ ,t ⬘ 兲 ⫽⫺ G ret 1 共 x A ,x B ,t ⬘ ;x A

⫽ ␦ 共 x A ⫺x A⬘ 兲 ␦ 共 t⫺t ⬘ 兲 →0 for t⬎t ⬘ , and

PHYSICAL REVIEW B 66, 035304 共2002兲

共A3兲

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⳵ ⫺h B(1) 共 x B ,t 兲 G Bret 共 x B ,t;x B⬘ ,t ⬘ 兲 ⳵t

⫽ ␦ 共 x B ⫺x B⬘ 兲 ␦ 共 t⫺t ⬘ 兲 →0 for t⬎t ⬘ , 共A4兲 subject to the retardation condition that for t⬍t ⬘ , both G Aret and G Bret vanish, as well as the initial conditions i G Aret 共 x A ,t ⬘ ⫹ ;x A⬘ ,t ⬘ 兲 ⫽⫺ ␦ 共 x A ⫺x A⬘ 兲 , 共A5兲 ប and i 共A6兲 G Bret 共 x B ,t ⬘ ⫹ ;x B⬘ ,t ⬘ 兲 ⫽⫺ ␦ 共 x B ⫺x B⬘ 兲 , ប This assertion is readily verified by forming ⳵ / ⳵ tG ret 1 for t ⬎t ⬘ using Eq. 共A2兲 ⳵ G Aret ret ⳵ ret ⳵ ret G 1 ⫽iប G G B ⫹iបG Aret . 共A7兲 ⳵t ⳵t ⳵t B Employing the homogeneous forms of Eq. 共A3兲 and Eq. 共A4兲 for t⬎t ⬘ , we obtain ⳵ ret G ⫽ 关 h A(1) 共 x A ,t 兲 G Aret 兴 G Bret ⫹G Aret 关 h B(1) 共 x B ,t 兲 G Bret 兴 , ⳵t 1 共A8兲

G 2 共 1,2;1 ⬘ ,2⬘ 兲 ⫽G 1 共 1,1⬘ 兲 G 1 共 2,2⬘ 兲 ⫾G 1 共 1,2⬘ 兲 G 1 共 2,1⬘ 兲 . 共A13兲 Discounting the Fock term as it arises from the identity of the particles, the Hartree term is clearly of the type given in Eq. 共A2兲 associated with two commuting Hamiltonian parts, as one should expect. However, Eq. 共A2兲 has a factor iប not present in the Hartree term of Eq. 共A13兲. This ‘‘normalization’’ difference may be attributed to the fact that ⬘ ,x B⬘ ,t ⬘ ) has only one pair of time indices G ret 1 (x A ,x B ,t;x A rather than the two pairs of time indices of G 2 in Eq. 共A13兲; consequently, the initial value of G ret 1 cannot be expected to be identical with that of G 2 .

J. B. Krieger and G. J. Iafrate, Phys. Rev. B 33, 5494 共1986兲; J. B. Krieger and G. J. Iafrate, ibid. 35, 9644 共1987兲; G. J. Iafrate and J. B. Krieger, ibid. 40, 6144 共1989兲; J. He and G. J. Iafrate, ibid. 50, 7553 共1994兲; J. He and G. J. Iafrate, in Quantum Transport in Ultrasmall Devices, edited by D. K. Ferry et al. 共Plenum.

New York, 1995兲; G. J. Iafrate, J. P. Reynolds, J. He, and J. B. Krieger, Int. J. High Speed Electron. Syst. 9, 223 共1998兲. 2 J. Schwinger, Phys. Rev. 82, 664 共1951兲. 3 N. J. M. Horing and M. Yildiz, Ann. Phys. 共N.Y.兲 97, 216 共1976兲; N. J. M. Horing, ibid. 31, 1 共1965兲.

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