Effect Of Weak Magnetic Field ( ∼ \,300 Gs) On the Intensity of Terahertz Emission of Hot Electrons in n -Ge at Helium Temperatures
aa r X i v : . [ qu a n t - ph ] N ov V.M. BONDAR, P.M. TOMCHUK, G.A. SHEPEL’SKII
EFFECT OF WEAK MAGNETIC FIELD ( ∼
300 Gs)ON THE INTENSITY OF TERAHERTZ EMISSIONOF HOT ELECTRONS IN n -Ge AT HELIUMTEMPERATURES V.M. BONDAR, P.M. TOMCHUK, G.A. SHEPEL’SKII Institute of Physics, Nat. Acad. of Sci. of Ukraine (46, Nauky Ave., Kyiv 03680, Ukraine; e-mail: [email protected]) Lashkaryov Institute of Semiconductor Physics, Nat. Acad. of Sci. of Ukraine (41, Nauky Ave., Kyiv 03680, Ukraine)
PACS 31.10.+zc (cid:13)
Experimental results of studying the effect of a weak magnetic field( ∼
300 Gs) on the intensity of the terahertz emission ( λ ≈ µ m)of hot electrons in n -Ge (crystallographic orientation h , , i ) athelium temperatures ( T ∼ ÷
1. Introduction
Recently, the peculiarities of mechanisms of generationand absorption of terahertz light attract still more at-tention of investigators [1]. In [2–4], we studied the an-gle dependences of the terahertz emission of hot elec-trons in n -Ge. This semiconductor has a cubic symme-try, but, in the case where the electric field is directednon-symmetrically with respect to valleys (minima inthe conduction band), electrons in different valleys canhave different temperatures. This results in the sym-metry violation and, consequently, the appearance ofthe polarization dependence of the hot electron emis-sion. We studied the relation between the polarizationdependences and anisotropic scattering mechanisms [4,5] characteristic of many-valley semiconductors. It wasa surprise to discover that, under certain conditions (lowtemperatures, strong electric fields), the polarization de-pendences of the hot electron emission appear in the casewhere the electric field is oriented along the h , , i di-rection, i.e. symmetrically with respect to valleys. Itwas established that, in this case, the appearance of thepolarization dependences is related to the symmetry vi- olation of the even part of the electron velocity distri-bution function under the action of an electric field. Inother words, this effect can be explained going beyondthe bounds of the traditional so-called diffusion approx-imation. As is known (see, e.g., [6]), this approximationis based upon the smallness of the ratio of the drift ve-locity of an electron to its mean thermal velocity. Atlow temperatures and strong electric fields, the diffusionapproximation appeared invalid. Another surprise wasaroused by an extremely high sensitivity of the hot elec-tron emission intensity to weak magnetic fields ( H ∼ T ∼
2. Experimental Part
All measurements were performed on the set-up de-scribed in [4] supplemented with an attachment allowingone to subject an emitting sample to the action of themagnetic field of the required direction and magnitude– from zero to the maximum value. This field was cre-ated with the use of a permanent magnet with the cor-responding devices used to regulate the field intensity.The arrangement of the units is schematically shown inFig. 1.The n -Ge samples were cut off in the crystallographicdirections h , , i or h , , i , had a standard size of7 × × , and were treated using the standard tech-nique [4]. The electric field was created by pulses witha duration of 0.8 µ s and a repetition rate of 6 Hz. Afterthat, the signal of a semiconductor detector was am-plified, integrated, and converted to the direct-currentvoltage proportional to the intensity of the hot electronemission of the sample in the region λ ≈ µ m. The ISSN 2071-0186. Ukr. J. Phys. 2010. Vol. 55, No. 9
FFECT OF WEAK MAGNETIC FIELD ( ∼
300 Gs)
Fig. 1. Diagram of the experiment. – n -Ge sample; – fil-ter limiting high frequencies; – rotating polarizer; – Ge(Ga)receiver ohmicity of the contacts to n -Ge was provided using Stalloy with a 5-% fraction of Sb.
3. Experimental Results and Their Discussion
Figure 2, a–c presents the experimental results of study-ing the effect of a weak magnetic field on the intensityof the hot electron emission in n -Ge. One can see that,at small electric fields, the magnetic field reduces theemission amplitude by almost an order of magnitude.With increase in the electric field, the effect of the mag-netic field on the emission intensity becomes consider-ably weaker. Such changes in the emission intensityunder the action of the weak magnetic field cannot beexplained by its influence on the dispersion law or scat-tering mechanisms. Simple estimates demonstrate that,during the mean free time between collisions of a carrierwith scattering centers, its trajectory changes insignifi-cantly. In this connection, it was necessary to search forother reasons of such a strong effect of the weak magneticfield on the terahertz emission intensity.For this purpose, we studied the electrophysical char-acteristics of the n -Ge samples investigating their emis-sion at helium temperatures. We took volt-ampere char-acteristics and carried out Hall measurements in orderto determine the carrier concentration starting from lowvoltages, at which not all donors are ionized [7]. Themeasuring results are given in Fig. 3 and Table. As fol-lows from the behavior of the volt-ampere characteristicof the sample in the absence and in the presence of themagnetic field (we recall that the magnetic field is weak,close to 300 Gs), the resistance of the sample in such afield grows almost by an order of magnitude at an elec-tric field of ∼ abc Fig. 2. Receiver signal: – without magnetic field – with weakmagnetic field; heating electric fields: 15 V/cm ( a ), 25 V/cm ( b ),200 V/cm ( c ) ISSN 2071-0186. Ukr. J. Phys. 2010. Vol. 55, No. 9 .M. BONDAR, P.M. TOMCHUK, G.A. SHEPEL’SKII V n × × × × × × × × Fig. 3. Volt-ampere characteristic of the sample: – withoutmagnetic field – with weak magnetic field ( ∼
300 Gs) tensity under the action of the weak magnetic field issolved, though far from exhaustively. Now, we need toexplain the large growth of the sample resistance underthe action of such a weak magnetic field at helium tem-peratures. It turned out that similar phenomena havebeen already studied and were explained by the hoppingconduction in the impurity band. According to the ex-isting conceptions (see, e.g., [8]), the main contributioninto the conduction at low temperatures (at which themajority of electrons are localized at impurities) is madeby the hopping mechanism. The effect of the weak mag-netic field on this mechanism is explained by its influenceon the “tails” of the wave function of electrons localizedat donors. The overlapping of these “tails” determinesthe probability of hops of electrons to vacancies.It is worth noting that the thermal introduction ofcarriers into the conduction band is ineffective at lowtemperatures, that is why the hopping conduction overvacancies is provided by the compensation effect. Thiscompensation is present in practically any material. As-suming that the impurity breakdown starts according tothe Zener mechanism, it is easy to understand the effectof the magnetic field on the free carrier concentration inthe conduction band on the stage where all donors arenon-ionized. In turn, this explains the influence of theweak magnetic field on the hot electron emission at lowtemperatures.In addition, the “attachment” of the carriers alreadyintroduced into the conduction band to neutral donors, the inverse process, and the dependence of both pro-cesses on the magnetic field are possible.All the above-said concerns a decrease of the emissionunder the action of a magnetic field at low electric fields10 ÷
15 V/cm (Fig. 2, a ) .At strong electric fields, this decrease is much smaller(Fig. 2, c ) and amounts to ∼
10% of the initial value.Such a behavior of the observed phenomenon can beexplained by a deformation of the velocity distributionfunction in the case where the electric field is orientedalong the h , , i direction and develops into the heat-ing one. In this case, the diffusion approximation canexplain fine characteristics of the discussed phenomenanot always, and one should use a more accurate distri-bution function.At strong electric fields (at which the concentrationof carriers in the conduction band does not change any-more), the effect of a magnetic field on the hot electronemission is related to the appearance of the longitudi-nal magnetoresistance. The latter is connected with adecrease in the electron heating and therefore a fall ofthe emission. The mechanism of the formation of thelongitudinal magnetoresistance in many-valley semicon-ductors is considered in the following section.
4. Longitudinal Magnetoresistance
For today, the general theory of galvanomagnetic phe-nomena in many-valley semiconductors with regard forthe anisotropy of the dispersion law of carriers and mech-anisms of their scattering is well developed and solidlysubstantiated (see, e.g., [9]). However, the general for-mulas of this theory are rather cumbersome. The sit-uation becomes still more complicated when trying toallow for the possibility of the electron heating by theelectric field.The purpose of this work is narrower – to explainthe reasons for the appearance of the longitudinal mag-netoresistance and estimate its magnitude in the casewhere arbitrary electric and weak magnetic fields areoriented along the direction symmetric with respect tovalleys ( h , , i in n -Ge). That is why we can employ arougher but simpler model. The essence of this approxi-mation is to characterize the hot electrons by a velocity-shifted Maxwellian distribution function [10] or (in thecase of degeneracy) by the Fermi [11] function with theeffective electron temperature. In many-valley semicon- ISSN 2071-0186. Ukr. J. Phys. 2010. Vol. 55, No. 9
FFECT OF WEAK MAGNETIC FIELD ( ∼
300 Gs) ductors, such a function is to be introduced for electronsof each valley. In the general case where there exists thepossibility of a degeneracy of the electron gas in the α -thellipsoid, we can write [11] f α = (cid:26) (cid:18) + ε ( υ ) − pu ( α ) − µ ( α ) kT ( α ) (cid:19)(cid:27) − , (1)where υ is the electron velocity, ε ( υ ) and p are its energyand momentum, respectively, T ( α ) denotes the effectiveelectron temperature, µ ( α ) is the chemical potential, and u ( α ) is the drift velocity. The quantities µ ( α ) , T ( α ) , and u ( α ) must be determined from the equations for the con-centration, energy, and momentum balance, respectively.In what follows, we will consider the case where the elec-tric field E and the magnetic field H are oriented alongthe direction symmetric with respect to valleys ( h , , i for n -Ge), so the parameters µ ( α ) and T ( α ) will be thesame for all valleys. As concerns the drift velocity u ( α ) inthe given symmetric case, it will have the same absolutevalue but different directions for different valleys. Thatis why we do not present explicitly the balance equationsfor the electron concentrations in valleys and their en-ergies and restrict ourselves to the momentum balanceequation. In the principal axes of the α -th mass ellip-soid, in which the energy dispersion law has the standardform ε ( υ ) = P ⊥ m ⊥ + P k m k , (2)the momentum balance equation is e (cid:26) E i + 1 c h u ( α ) × H i i = m ⊥ τ ⊥ u ( α ) i , ( i = x, y ) (cid:27) , (3) e (cid:26) E z + 1 c h u ( α ) × H i z = m k τ k u ( α ) z (cid:27) , (4)where e is the electron charge, c is the light velocity,while τ k and τ ⊥ are the longitudinal and transverse re-laxation times, respectively.Due to the weakness of the magnetic field, we can solveEqs. (3) and (4) using perturbation theory with respectto the parameter H . In the zero-order approximation(i.e. at H = 0) , Eqs. (3) and (4) yield n u ( α ) i o = eτ ⊥ m ⊥ E i , i = x, y, (5) n u ( α ) z o = eτ k m k E z , (6) u ( α )0 = eτ ⊥ m ⊥ E + (cid:18) eτ k m k − eτ ⊥ m ⊥ (cid:19) ( i α E ) i α , (7)where i α is the unit vector that specifies the orientationof the α -th ellipsoid (valley). From Eq.(7) (or Eqs.(5)–(6)), one can see that the direction of the drift velocity u ( α ) does not coincide with that of the electric field, un-less this field is directed along the principal axis of themass ellipsoid. As a result, the term [ u ( α ) × H ] willbe non-zero in spite of the fact that H k E . This is thereason for the appearance of the longitudinal magnetore-sistance in n -Ge.Then, one can develop the perturbation theory withrespect to H , i.e. substitute the approximate value u ( α )0 in the term [ u ( α ) × H ] neglected when obtainingEq.(7), which yields u ( α )1 and so on. As a result, weobtain the series with respect to H for the drift velocityof electrons of the α -th valley (see Appendix): u ( α ) = u ( α )0 + u ( α )1 + u ( α )2 + . . . . (8)The term linear in H in the drift velocity (8) ( u ( α )1 ) de-termines the Hall current. Due to the symmetry of theproblem, the total Hall current in all valleys is equalto zero. The component u ( α )2 determines the magne-toresistance. Its h , , i -projection and the sum overall valleys determines the addition ∆ J to the current J = J ( H = 0) . As is shown in Appendix, ∆ J J = − ( e τ ⊥ /m ⊥ c )( τ k /m k − τ ⊥ /m ⊥ ) · H τ k /m k + 2 τ ⊥ /m ⊥ ) , (9)where J = e n (cid:18) τ k m k + 2 τ ⊥ m ⊥ (cid:19) , (10)and n is the total electron concentration (in all valleys).For n -Ge, m ⊥ ≪ m k , τ ⊥ ∼ τ k . (11)In this case, formula (9) acquires the simplified form ∆ J J = − e τ ⊥ m ⊥ c H . (12)Assuming for the estimate that m ⊥ ≈ . × − g, τ ⊥ ≈ − s, and H ≈ Oe, we obtain from Eq.(12)that ∆ J / J ≈ − / , which is in good agreement withthe experiment. ISSN 2071-0186. Ukr. J. Phys. 2010. Vol. 55, No. 9 .M. BONDAR, P.M. TOMCHUK, G.A. SHEPEL’SKII
5. Conclusions
The performed studies allow us to make the followingconclusions. At low temperatures and weak electricfields, at which the majority of electrons is localized atdonor levels, the effect of a weak magnetic field on thevolt-ampere characteristics and the emission is explainedas follows. Both the hopping conduction and the Zenerbreakdown mechanism are sensitive to the influence ofa magnetic field on the “tails” of the wave function ofan electron localized at a donor. This explains the fastdecrease of the current and the emission due to the ap-plication of a weak magnetic field. In strong electricfields, all donors are ionized and the electron concentra-tion in the conduction band is constant. The effect ofthe magnetic field on the volt-ampere characteristics andthe emission in strong electric fields is much weaker. Inthis case, such a weak influence is explained by the lon-gitudinal magnetoresistance. The mechanism of the lon-gitudinal magnetoresistance is related to the anisotropyof the dispersion law of electrons in n -Ge.In conclusion, the authors express their gratitude toO.G. Sarbey and S.M. Ryabchenko for the discussion ofa number of questions. APPENDIX
With the use of Eq.(7), we obtain: [ u ( α )0 × H ] = e ( τ k /m k − τ ⊥ /m ⊥ )( i α E ) [ i α × H ] . (A1)Assuming that the vector [ i α × H ] is directed along the x axis andsubstituting Eq. (А1) into (3), one can put down u ( α )1 = e τ ⊥ m ⊥ c e ( τ k /m k − τ ⊥ /m ⊥ )( i α E ) [ i α × H ] . (A2)After that, Eq. (А2) yields [ u ( α )1 × H ] = e τ ⊥ m ⊥ c ( τ k /m k − τ ⊥ /m ⊥ )( i α E ) [( i α × H ) × H ] == e τ ⊥ m ⊥ c ( τ k /m k )( i α E ) { ( i α H ) H − i α H } . (A3)As one can see, the vector [ u ( α )1 × H ] lies in the plane specifiedby the vectors H and i α . If we take the direction [ i α × H ] in thisplane as the x axis and the direction i α as the z axis (the normal tothese vectors will be the y axis), then the components [ u ( α )1 × H ] y and [ u ( α )1 × H ] z will be non-zero. Their substitution into Eqs. (3)and (4), respectively, yields u ( α )2 y = e τ ⊥ m ⊥ c ( τ k /m k − τ ⊥ /m ⊥ )( i α E )( i α H ) { H − i α ( i α H ) } , (A4) u ( α )2 z = − e τ ⊥ τ k m ⊥ m k c ( τ k /m k − τ ⊥ /m ⊥ )( i α E ) { H − ( i α H ) } i α . (A5)Expressions (А4) and (А5) are put down in the vector form sothat to be easy-to-use in the laboratory system of coordinates and to sum up over all valleys. Thus, u ( α )2 y and u ( α )2 z are related to thegiven ellipsoid α only through the unit vector i α .As we are interested in the longitudinal magnetoresistance, itis necessary to find the addition to the current J quadratic inthe magnetic field (in the direction E k Hq ; q ≡ (1,0,0)). Thisaddition is evidently equal to ∆ J = − en X ( α ) q { u ( α )2 y + u ( α )2 z } . (A6)Here, n is the electron concentration in one valley.Substituting expressions (А4) and (А5) into (А6) and takinginto account that the unit vectors i α ( α = 1 , , , in n -Ge havethe form i = 1 √ , , , i = 1 √ − , , , i = 1 √ , − , , i = 1 √ − , − , , we obtain ∆ J = − e n m ⊥ c (cid:18) τ k m k − τ ⊥ m ⊥ (cid:19) H E . (A7)For isotropic scattering mechanisms and dispersion law, ∆ J = 0.
1. V.E. Lyubchenko, Radiotekhn. No. 2, 5 (2002).2. V.M. Bondar, O.G. Sarbey, and P.M. Tomchuk, Fiz.Tekhn. Polupr. , 1540 (2002).3. P.M. Tomchuk, Ukr. Fiz. Zh. , 681 (2004).4. V.M. Bondar and N.F. Chornomorets’, Ukr. Fiz. Zh. ,51 (2003).5. P.M. Tomchuk and V.M. Bondar, Ukr. Fiz. Zh. , 666(2008).6. I.M. Dykman and P.M. Tomchuk, Transport Phenomenaand Fluctuations in Semiconductors (Naukova Dumka,Kyiv, 1981) (in Russian).7. Yu.A. Astrov and A.A. Kastal’skii,
Peculiarities ofBreakdown of Shallow Donors in Pure n -Ge (Mantis, Vil-nius, 1972).8. B.I. Shklovskii and A.I. Efros, Electronic Properties ofDoped Semiconductors (Springer, New York, 1984).9. P.I. Baranskii, I.S. Buda, I.V. Dakhovskii, and V.V. Ko-lomoets,
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Received 03.06.10.Translated from Ukrainian by H.G. Kalyuzhna
ISSN 2071-0186. Ukr. J. Phys. 2010. Vol. 55, No. 9
FFECT OF WEAK MAGNETIC FIELD ( ∼
300 Gs)
ВПЛИВ СЛАБКОГО МАГНIТНОГОПОЛЯ ( ∼
300 Гс) НА IНТЕНСИВНIСТЬТЕРАГЕРЦОВОГО ВИПРОМIНЮВАННЯ ГАРЯЧИХЕЛЕКТРОНIВ В n -Ge ПРИ ГЕЛIЄВИХ ТЕМПЕРАТУРАХ В.М. Бондар, П.М. Томчук, Г.А. Шепельський
Р е з ю м еУ роботi наведено експериментальнi результати та їх обго-ворення у вивченнi впливу слабкого магнiтного поля ( ∼ λ ≈