Tunneling time and the post-tunneling position of an electron through a potential barrier in an anisotropic semiconductor

Superlattices and Microstructures, Vol. 24, No. 6, 1998 Article No. sm980598

Tunneling time and the post-tunneling position of an electron through a potential barrier in an anisotropic semiconductor K YOUNG -Y OUM K IM , B YOUNGHO L EE† School of Electrical Engineering, Seoul National University, Kwanak-Gu Shinlim-Dong, Seoul 151–742, Korea

(Received 14 July 1998) Tunneling time and post-tunneling position of an electron incident on a heterostructure grown on anisotropic materials are derived by solving an effective mass equation including off-diagonal effective mass tensor elements. The effects of different effective masses for a heterostructure junction are also included. c 1998 Academic Press

Key words: tunneling, tunneling time, effective mass tensor, heterostructure.

1. Introduction The tunneling time through a potential barrier has been discussed for the last half-century, and also is of present-day interest in the study of charge transport in heterostructures. The stationary-phase time model (which should not be confused with phase velocity) was introduced by Bohm [1] and Wigner [2] and though there have been other suggestions such as dwell time [3], and Lamor’s clock time [4], recently, Steinberg et al. [5, 6] proved that the stationary-phase time is a good model of the tunneling time. One of authors calculated both the tunneling time of an electron under normal incidence through a onedimensional potential barrier considering the different effective masses at a heterojunction interface [7], and the tunneling time of an electron under nonnormal incidence through a one-dimensional potential barrier assuming constant effective mass [8]. Recently, Paranjape [9] studied the two-dimensional tunneling time of an electron through an isotropic heterostructure potential barrier with different effective masses. He showed that the conservation of transverse (parallel to the heterostructure interface) momentum at a heterojunction interface with different effective masses implies the loss or gain of kinetic energy for that direction, and the energy difference is conferred on the longitudinal (perpendicular to the heterostructure interface) kinetic energy, helping or hindering the tunneling. This energy coupling was also pointed out earlier by Doezema and Drew [10]. However, besides the energy coupling characteristic due to the heterostructure, such coupling of the longitudinal motion to the transverse motion also exists in anisotropic materials via the off-diagonal effective mass tensor elements. Moreover, the off-diagonal effective mass tensor elements are valley-dependent, which suggests the coupling effects are not identical in all valleys. Therefore, it is necessary to investigate its effect on the tunneling phenomenon quite rigorously. Recently, the authors derived the appropriate boundary conditions † E-mail: [email protected]

0749–6036/98/060389 + 09

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c 1998 Academic Press

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for a heterojunction interface composed of anisotropic materials and successfully calculated the transmission coefficient of an electron through a potential barrier grown on an anisotropic material for the first time [11]. In this paper, as an extension to this, we calculate the tunneling time and the post-tunneling position of an electron through a potential barrier grown on an anisotropic material.

2. Calculations Though the procedure of the transmission coefficient calculation will be briefly reported elsewhere [11], we will discuss it here in detail because it is an essential part of this paper. The Hamiltonian for the general anisotropic materials is as follows [12] H=

1 T p · α(r) · p + V (r), 2m 0

(1)

where m 0 is the mass of free electron, p is a momentum vector, and (1/m 0 )α is the inverse effective mass tensor. It is noteworthy that in considering the spatially varying effective mass tensor, we simply place it without rigorous justification between the two derivative (momentum) operators [11] as an extension of the scalar effective mass case. Fig. 1 shows the potential profile we assumed in the normal direction (z-direction) to the layer. The electron is incident from region I to the potential barrier. The effective mass of electron and potential are dependent only on the z-direction. Therefore, the solution of the effective mass equation involving eqn (1) in region I and III is [12]  [A exp(ik 0 z) + B exp(−ik 0 z)] exp(−iγ1 z) exp(ik x x) exp(ik y y), z < 0 (2) ψ1 (x, y, z) = z>d F exp(ik 0 z) exp(−iγ1 z) exp(ik x x) exp(ik y y), where αzz,1

h¯ 2 (k 0 )2 h¯ 2 =E− (βx x,1 k x2 + 2βx y,1 k x k y + β yy,1 k 2y ), 2m 0 2m 0 k x αx z,1 + k y α yz,1 , γ1 = αzz,1 αi z,1 α j z,1 , (i, j ∈ {x, y}). βi j,1 = αi j,1 − αzz,1

(k 0 > 0),

(3a) (3b) (3c)

Here subscript 1 denotes the region I and III. Later, subscript 2 will be used for region II. In eqn (3a) E is the energy given by X h¯ 2 αi j,1 ki k j , (3d) E= 2m 0 i, j∈{x,y,z} which is the eigenvalue for the Hamiltonian of eqn (1) with V = 0. In eqn (3d), k z is related to k 0 in eqn (3a) by k z = −γ1 ± k 0 .

(3e)

Equation (3e) is instructive in that k z is separated into two parts: one (−γ1 ) linked to k x and k y via eqn (3b) and the other (±k 0 ) which is independent of the momentum in the x- or y-direction. In eqns (3b) and (3e) we see that if the electron has crystal-momentum h¯ k x , h¯ k y , respectively in the x, y-direction, then the momentum is coupled to the z-direction by the off-diagonal effective mass tensor element ratios αx z /αzz and α yz /αzz . The values of αx z and α yz are dependent on which valley the electron belongs to. In most cases, the magnitudes are the same for all valleys but the signs are not the same. Therefore, even though the electrons have the same momentum in the x, y-direction, the linked z-direction momentum is dependent on the valley where the electrons are, i.e. the magnitude of the linked z-direction momentum is independent of the valley but its

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Region I

391

Region II

Region III

Potential energy

V0

V0=216 meV 0 Si(110)

Si(110)

Si0.5Ge0.5 0

d = 50 Å

Position (z) Fig. 1. The model used in the numerical calculation.

direction is valley-dependent. In short, if the electron has momentum in the x, y-direction, it also has linked momentum in the z-direction. Therefore the motion in x, y-direction is not independent of that in z-direction, but they are mutually coupled by the off-diagonal effective mass tensor elements. In region II, the wavefunction is ψ2 (x, y, z) = [C exp(−γ 0 z) + D exp(γ 0 z)] exp(−iγ2 z) exp(ik x x) exp(ik y y),

0
(4)

where γ2 is defined similarly to eqn (3b) with subscript 2 instead of 1, and (γ 0 )2 =

2m 0 1 αzz,1 0 2 1 V0 − (k ) − 2 αzz,2 αzz,2 h¯ αzz,2

X

(βi j,1 − βi j,2 )ki k j ,

(γ 0 > 0)

(5a)

i, j∈{x,y}

which comes from the conservation of energy, i.e. h¯ 2 (k 0 )2 h¯ 2 (βx x,1 k x2 + 2βx y,1 k x k y + β yy,1 k 2y ) + αzz,1 2m 0 2m 0 2 2 h¯ (γ 0 )2 h¯ = (βx x,2 k x2 + 2βx y,2 k x k y + β yy,2 k 2y ) − αzz,2 + V0 , 2m 0 2m 0

E=

(5b)

where βi j,2 is defined similarly to eqn (3c) with subscript 2 instead of 1. In eqns (2) and (4), note that k x and k y are conserved through all regions as in Ref. [9], (see Appendix). We see that eqns (2) and (4) are dependent on the valley by way of valley-dependent off-diagonal effective mass tensor elements. As will be discussed later, this dependency makes the tunneling time, post-tunneling position, and tunneling speed of an electron valley-dependent. PThe boundary conditions used here are derived from the continuity condition of ψ and that of [11] j∈{x,y,z} αz j ∂ψ/∂ j:



∂ψ1 ∂ψ1 + αzy,1 ∂x ∂y  ∂ψ2 ∂ψ2 + αzy,2 αzx,2 ∂x ∂y αzx,1

ψ1 (x, y, 0− ) = ψ2 (x, y, 0+ ), ψ2 (x, y, d − ) = ψ1 (x, y, d + ),    ∂ψ1 ∂ψ2 ∂ψ2 ∂ψ2 = αzx,2 , + αzz,1 + αzy,2 + αzz,2 ∂z z=0− ∂x ∂y ∂z z=0+    ∂ψ2 ∂ψ1 ∂ψ1 ∂ψ1 = αzx,1 , + αzz,2 + αzy,1 + αzz,1 ∂z z=d − ∂x ∂y ∂z z=d +

(6a) (6b) (6c) (6d)

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which can be written as follows. A + B = C + D, [C exp(−γ 0 d) + D exp(γ 0 d)] exp(−iγ2 d) = F exp(ik 0 d) exp(−iγ1 d), αzz,1 [ik 0 A − ik 0 B] = αzz,2 [−γ 0 C + γ 0 D], αzz,2 [−γ 0 C exp{−(γ 0 + iγ2 )d} + γ 0 D exp{(γ 0 − iγ2 )d}] = αzz,1 ik 0 F exp{i(k 0 − γ1 )d}.

(7a) (7b) (7c) (7d)

In the derivation of eqns (7c) and (7d), eqn (3b) and a similar one with subscript 2 are used. In eqns (7b) and (7d), if we define F 0 ≡ F exp(i(γ2 − γ1 )d), then the equations become, respectively, [C exp(−γ 0 d) + D exp(γ 0 d)] = F 0 exp(ik 0 d), αzz,2 [−γ 0 C exp(−γ 0 d) + γ 0 D exp(γ 0 d)] = αzz,1 ik 0 F 0 exp(ik 0 d).

(7e) (7f)

Equations (7a), (7c), (7e), and (7f) have the same form as the equations obtained in isotropic cases [7, 9]. Hence by using an analogy to the results for the isotropic cases [9], we obtain the transmission coefficient T as follows [11]: T ≡ Ta∗ Ta , F F0 = exp[i(γ1 − γ2 )d] = G exp(i8), Ta ≡ A A

(8a) (8b)

where 2k 0 γ 0

, [P 2 sinh2 (γ 0 d) + 4(k 0 )2 (γ 0 )2 cosh2 (γ 0 d)]1/2   P 0 tanh(γ d) − k 0 d + (γ1 − γ2 )d, 8 = tan−1 2k 0 γ 0 αzz,1 0 2 αzz,2 0 2 (k ) − (γ ) . P= αzz,2 αzz,1 G=

(9) (10) (11)

The stationary-phase tunneling time of the electron wave packet under nonnormal incidence can be derived from the phase component of the electron wavefunction for the most probable crystal momentum after tunneling [8, 13]. The phase component ξ of the wavefunction after tunneling is ξ = 8(k x , k y , k 0 ) + k x x + k y y + (k 0 − γ1 )z −

E(k x , k y , k 0 ) t. h¯

(12)

By taking the gradient of eqn (12) in k-space and making it equal to zero, we can get the post-tunneling position in the x, y-direction (1x , 1 y , respectively: See Fig. 2) and tunneling time (τ ). (In this derivation x, y, t in eqn (12) should be replaced with 1x , 1 y , τ , respectively [8].) In this paper, we differentiate eqn (12) with respect to k x , k y , k z and set each result equal to zero. Then we obtain: αx z,2 τ ∂E αx z,1 +d + , h¯ ∂k x αzz,1 αzz,2 α yz,2 α yz,1 τ ∂E +d + , 1 y = −T2 B y − T1 h¯ ∂k y αzz,1 αzz,2 1x = −T2 Bx − T1

τ=

T1 , 1 ∂E h¯ ∂k z

(13a) (13b)

(14)

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y [110] y

k ϕ

x

v ϕ'

Post-tunneling electron

x [001]

θ θ'

1y

z 0

1x

z [110]

Incident electron (direction of v)

Fig. 2. The coordinates used in this paper and the physical definition of the post-tunneling position.We take where the electron hits the barrier, as the origin of the coordinate system.

where

    αzz,1 αzz,1 αzz,1 2 αzz,2 0 4 + 1 + (k 0 )4 + (γ ) sinh(2γ 0 d) − 2(k 0 )2 γ 0 d P αzz,2 αzz,2 αzz,1 αzz,2 , (15a) T1 = γ 0 [4(k 0 )2 (γ 0 )2 cosh2 (γ 0 d) + P 2 sinh2 (γ 0 d)]   1 1 1 2k 0 (γ 0 )2 + Pk 0 P sinh(2γ 0 d) − 2k 0 γ 0 d αzz,1 αzz,2 αzz,2 , (15b) T2 = γ 0 [4(k 0 )2 (γ 0 )2 cosh2 (γ 0 d) + P 2 sinh2 (γ 0 d)] 

(k 0 )2 (γ 0 )2



Bx = δβx x k x + δβx y k y , B y = δβx y k x + δβ yy k y , δβi j = βi j,1 − βi j,2 . The effective tunneling speed is defined as follows [8]. q d 2 + 12x + 12y . s≡ τ

(16a) (16b) (16c)

(17)

From eqns (13a), (13b), (14) and (17), it can be easily seen that the tunneling time, the post-tunneling position and the effective tunneling speed are valley-dependent by way of the valley-dependent effective mass tensor. These results are quite interesting in that if we can measure the post-tunneling position quite accurately, it can be a method of determining the valley to which the electrons belong and also the valley occupancy (the relative population of the electrons in that specific valley) of the electrons.

3. Results and discussion The model used as an example in the numerical calculation is shown in Fig. 1. There is a Si0.5 Ge0.5 potential barrier grown on Si(110). The width of the barrier is 50 Å and the conduction band discontinuity is taken as 216 meV [14]. The electron has 50 meV of energy for all calculations. There are four equivalent valleys (X4 valley) in the conduction band of Si(110) [14]. The effective mass

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Superlattices and Microstructures, Vol. 24, No. 6, 1998 Table 1: The inverse effective mass tensors used in the numerical calculation.

Valley 1

2

Region I, III   5.26 0 0  0 3.14 2.12  0 2.12 3.14   5.26 0 0  0 3.14 −2.12  0 −2.12 3.14

Region II   6.45 0 0  0 4.56 2.74  0 2.74 4.56   6.45 0 0  0 4.56 −2.74  0 −2.74 4.56

tensor elements of these four valleys are not the same. There are two groups of valleys in Si(110), each of them has two valleys. In both group of valleys, αx z is zero (see Fig. 2 for the definition of directions). This means that the x-directional group velocity component is conserved in our problem (see eqn (A3)). However, one group has positive α yz , while the other has negative α yz [15]. Therefore, the calculated results are dependent on the group to which the electron belongs. For simplicity, we denote the group which has positive α yz as valley 1 and the other as valley 2. The effective masses used in our example are shown in Table 1 [11]. Figure 2 shows the chosen coordinate system and the physical definition of post-tunneling position. We take where the electron hits the barrier, as the origin of the coordinate system. Note that k (wavevector of the incident electron) and ν (velocity vector of the incident electron) are not necessarily parallel. The θ, ϕ and θ 0 , ϕ 0 are the angle coordinate components of k and ν, respectively. As shown in Fig. 2, they can be defined as follows: X h¯ 2 αi j,1 ki k j E= 2m 0 i, j{x,y,z} =

h¯ 2 {αx x,1 K 2 sin2 θ cos2 ϕ + α yy,1 K 2 sin2 θ sin2 ϕ + αzz,1 K 2 cos2 θ 2m 0 +2(αx y,1 K 2 sin2 θ cos ϕ sin ϕ + α yz,1 K 2 sin θ cos θ sin ϕ + αzx,1 K 2 sin θ cos θ cos ϕ)},

(18)

θ 0 = ± tan−1 ({νx2 + ν y2 }1/2 /|νz |),

(19a)

ϕ 0 = ± tan−1 (|ν y |/|νx |).

(19b)

In eqns (19a) and (19b), νx , ν y , νz are defined as in the Appendix (eqns (A2a)–(A2c), respectively) and the sign can be determined physically with reference to Fig. 2. The tunneling time versus incident angle is given in Fig. 3. The incidence angles are θ 0 , ϕ 0 , but we fix ϕ 0 to π/2 for simplicity and change only θ 0 . The solid line is for an electron in valley 1, and the dash-dotted line is for an electron in valley 2, calculated assuming a constant effective mass tensor (that of Si(110)). The dotted line is for an electron in valley 1, and the dashed line is for an electron in valley 2, calculated with a spatially varying effective mass tensor. It is noteworthy that, in all valleys, the tunneling time is not symmetric with the change of sign of the incidence angle (θ 0 → −θ 0 ), which confirms the anisotropy of the material. We also see that the position-dependent effective mass tensor gives about 20 ∼ 25% reduction of tunneling time. This is reasonable due to the fact that the lighter mass is introduced in the barrier region after considering spatially varying effective mass tensor. Figure 4 shows the post-tunneling positions in the y-coordinate versus incident angle. The x-position is not shown because the incident electron is assumed not to have x-directional velocity and it is conserved because αx z is zero in all valleys. The post-tunneling positions of electrons for valley 1 and valley 2 are apparently

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valley 1, constant m* valley 1, varying m* valley 2, constant m* valley 2, varying m*

100 90 Tunneling time (fs)

80 70 60 50 40 30 20 10

–80

–60

20 40 –40 –20 0 Incident angle (θ') (degree)

60

80

Fig. 3. The tunneling time versus incident angle (electron energy is 50 meV).

100 Post-tunneling position (y) (Å)

80 60

valley 1, constant m* valley 1, varying m* valley 2, constant m* valley 2, varying m*

40 20 0 –20 –40 –60 –80

–100

20 40 –80 –60 –40 –20 0 Incident angle (θ') (degree)

60

80

Fig. 4. The post-tunneling positions in the y-coordinate versus incident angle (electron energy is 50 meV).

different. Here we also point out that the position-dependent effective mass tensor gives about 20 ∼ 25% change of the post-tunneling position. Figure 5 shows the effective tunneling speed of an electron versus incident angle. Due to the valleydependent post-tunneling position it also becomes dependent on the valley. As mentioned above, this suggests a method of determining the valley to which the electrons belong and also the valley occupancy (the relative population of the electrons in that specific valley) of the electrons, which indicates a possibility for future practical purposes.

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10

Tunneling speed (105 m s–1)

9 8 7 6 5 4 3 valley 1, constant m* valley 1, varying m* valley 2, constant m* valley 2, varying m*

2 1 0 –80

–60

20 –40 –40 –20 0 ' Incident angle (θ ) (degree)

60

–80

Fig. 5. The effective tunneling speed of the electron versus incident angle (electron energy is 50 meV).

4. Conclusion In this paper, for the first time to authors’ knowledge, we discussed the tunneling time and post-tunneling position of an electron through a potential barrier grown on an anisotropic material under nonnormal incidence. We included the effect of different effective masses at heterojunction interfaces. Numerical calculations were done with a Si0.5 Ge0.5 potential barrier grown on Si(110). The calculation shows interesting results that the tunneling time and post-tunneling position (including the transmission coefficient) are dependent on the valley, which indicates a possibility for future practical purposes.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

}D. Bohm, Quantum Theory. (Englewood Cliffs, NJ, Prentice-Hall, 1951) p. 257. }E. P. Wigner, Phys. Rev. 98, 145 (1955). }M. Buttiker and R. Landauer, Phys. Rev. Lett. 49, 1739 (1982). }M. Buttiker, Phys. Rev. B27, 6178 (1983). }A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, Phys. Rev. Lett. 71, 708 (1993). }A. M. Steinberg and R. Y. Chiao, Phys. Rev. A51, 3525 (1995). }B. Lee, Superlattices Microstruct. 14, 295 (1993). }B. Lee and W. Lee, Superlattices Microstruct. 18, 177 (1995). }V. V. Paranjape, Phys. Rev. B52, 10740 (1995). }R. E. Doezema and H. D. Drew, Phys. Rev. Lett. 57, 762 (1986). }K.-Y. Kim and B. Lee, Phys. Rev. B58, 6728 (1998). }K. S. Yi and J. J. Quinn, Phys. Rev. B27, 1184 (1983). }B. Lee and W. Lee, Inst. Phys. Conf. Ser. 145, 861 (1996). }C. Lee, Intersubband absorption in conduction bands of silicon and germanium quantum wells, UCLA, (1994). [15] }K. S. Yi and J. J. Quinn, Phys. Rev. B27, 2396 (1983).

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Appendix In this Appendix, we will prove the conservation of k x , k y in anisotropic materials when the potential and effective mass tensor are dependent on only the z-direction. The ‘classical’ velocity in each direction of an electron can be defined as 1 ∂E , i ∈ {x, y, z}, (A1) νi ≡ h¯ ∂ki which can be written as follows: h¯ (βx x k x + βx y k y + αx z k 0 ), (A2a) νx = m0 h¯ (β yx k x + β yy k y + α yz k 0 ), (A2b) νy = m0 h¯ αzz k 0 . (A2c) νz = m0 Here we use Newton’s law,



  νx α 1  xx d  νy  = α yx dt m0 νz αzx

αx y α yy αzy

  αx z Fx α yz   Fy  , αzz Fz

(A3)

where Fi = −∂ V /∂i, therefore Fx and Fy are zero and d(m 0 /αzz · νz ) dk 0 dpz = = h¯ . (A4) dt dt dt The time derivatives of νx , ν y are not necessarily zero though the force is exerted only in the z-direction, and from eqns (A2a), (A2b), (A3), and (A4) we see that they must satisfy the following equations for all kx , k y ,   dk y dk x dk 0 h¯ 1 dk 0 dνx αx z h¯ = + βx y + αx z = , (A5a) βx x dt m0 dt dt dt m0 dt   dk y dν y h¯ 1 dk 0 dk x dk 0 = + β yy + α yz = . (A5b) α yz h¯ β yx dt m0 dt dt dt m0 dt Fz =

Therefore, the time derivatives of k x , k y must vanish, which proves the conservation of k x , k y in anisotropic materials when the potential and effective mass tensor are dependent on only the z-direction.