Abstract
The authors deal with the tunneling of electrons across an inhomogeneous delta-barrier defined by the potential energy \(V\left( r \right) = \left[ {\eta + \mu \left( {x^2 + y^2 } \right)} \right]\delta \left( z \right)\) (where \(\eta >0\) and \(\mu >0\) are two constants). In particular, the perpendicular incidence of an electron with a given value \(k_0 \) of the wave vector \(k_0 = \left( {0,0,k_0 } \right)\)is considered. The electron is forward-scattered into the region behind the barrier (region 2: \(z >0\)), i. e. the wave function \(\psi _2 \left( r \right)\) is composed of plane waves with all wave vectors \(k_2 \) such that \(\left| {k_2 } \right| = k_0 \) and \(k_{2z} = \sqrt {k_{_0 }^2 - q^2 >\left. 0 \right)} \)) (where \(q = \left( {k_{2x} ,k_{2y} ,0} \right),q = \left| q \right|\)). Therefore, if \(z >0\), the wave function of the electron is represented as \(\psi _2 \left( r \right) = \int {d^2 qU_2 \left( q \right)\exp \left[ {{\text{i}}\left( {q.u + \sqrt {k_{_0 }^2 - q^2 } } \right)z} \right]} \), where \(u = \left( {x,y,0} \right)\). An approximate formula is derived for the amplitude \(U_2 \left( q \right)\). The authors pay a special attention to the flow density \(J_2 \left( r \right) = \left( {\hbar /m} \right)\operatorname{Im} \psi _{_2 }^* \left( r \right)\nabla \psi _2 \left( r \right)\) and calculate this function in two cases: 1. for the plane \(z = 0\) and 2. for high values of \(R = \left| r \right|\left( {z = R{cos}\vartheta ,{i}{.e}{.}\vartheta \in \left( {0,{\pi}/2} \right.} \right)\) is the diffraction angle). The authors discuss the relevance of their diffraction problem in a prospective quantum-mechanical theory of the tunneling of electrons across a randomly inhomogeneous Schottky barrier.
Similar content being viewed by others
References
E. Burstein and S. Lundqvist, Eds.: Tunneling phenomena in solids, Plenum Press, New York, 1969 (Russian translation: Mir, Moscow, 1973).
E. L. Wolf: Principles of electron tunneling spectroscopy, Oxford University Press, New York, 1985 (Russian translation: Naukova dumka, Kiev, 1990).
M. Ožvold: Phys. Status Solidi A 132 (1992) 517.
J. H. Werner and H. H. Güttler: J. Appl. Phys. 69 (1991) 1522.
R. T. Tung: Appl. Phys. Lett. 58 (1991) 2821.
J. P. Sullivan, R. T. Tung, M. R. Pinto, and W. R. Graham: J. Appl. Phys. 70 (1991) 7403.
R. T. Tung: Phys. Rev. B 45 (1992) 13509.
E. Dobročka and J. Osvald: Appl. Phys. Lett. 65 (1994) 575.
J. Osvald and E. Dobročka: Semicond. Sci. Technol 11 (1996) 1198.
J. Osvald: J. Appl. Phys. 85 (1999) 1935.
Zs. J. Horváth, A. Bosacchi, S. Franchi, E. Gombia, R. Mosca, and A. Motta: Materials Sci. Eng. B 28 (1994) 429.
Zs. Horváth, A. Bosacchi, S. Franchi. E. Gombia, R. Mosca and D. Biondelli: Vacuum 46 (1995) 959.
Zs. Horváth: Vacuum 46 (1995) 963.
F. A. Padovani: in Semicoductors and semimetals (Eds. R. K. Willardson and A. C. Beer), Vol. 7 (Applications and devices), Part A, Academic Press, New York, 1971, p. 75.
E. H. Rhoderick and R. H. Williams: Metal-semiconductor contacts, Clarendon Press, Oxford, 1988.
S. Flügge: Practical quantum mechanics I (Die Grundlagen der mathematischen Wissenschaften, Vol. 177), Springer, Berlin, 1971.
A. Galindo and B. Pascual: Quantum mechanics I, Springer, Berlin, 1990.
M. J. Goovaerts, A. Babcenko, and J. T. Devreese: J. Math. Phys. 14 (1973) 554.
A. Lacina: Czech. J. Phys. B 30 (1980) 668.
B. Gaveau and L. S. Shulman: J. Phys, Math. Gen. 19 (1986) 1833.
S. W. Lawande and K. V. Bhagvat: Phys. Lett. 131 (1988) 8.
S. M. Blinder: Phys. Rev. A 37 (1988) 973.
C. Grosche: J. Phys., Math. Gen. 23 (1990) 5205.
C. Grosche: Phys. Rev. Lett. 71 (1993) 1.
C. Grosche: Ann. Phys. 2 (1993) 557.
V. Bezák: Acta Physica Univ. Comenianae 36 (1995) 179.
V. Bezák: J. Math. Phys. 37 (1996) 5939.
V. Bezák: Czech. J. Phys. 47 (1997) 223.
V. Bezák: Czech. J. Phys. 47 (1997) 237.
I. Yanetka: Physica B 270 (1999).
S. H. Patil: Am. J. Phys. 68 (2000) 712.
I. S. Gradshtein and I. M. Ryzhik: Tables of integrals,sum s, series and products, Nauka, Moscow, 1971, formula 6.532.4 (in Russian).
M. Abramowitz and I. A. Stegun: Handbook of Mathematical Functions, Nauka, Moscow, 1979, Chapter 9 (in Russian).
E. Jahnke, F. Emde, and F. Lösch: Tafeln höheren Funktionen, Teubner, Stuttgart, 1960, Chapter 13; (transl. Nauka, Moscow, 1964).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bezák, V., Selim, M.M. Tunneling Across an Inhomogeneous Delta-Barrier. Czechoslovak Journal of Physics 51, 829–852 (2001). https://doi.org/10.1023/A:1011678501341
Issue Date:
DOI: https://doi.org/10.1023/A:1011678501341