We construct a new class of d-dimensional black hole solutions with Toroidal horizons in f(R) gravity framework which is surrounded by quintessence and string cloud configuration, whose asymptotic structures are determined by the quintessential state parameter ωq and dimension’s parameter d. We point out that the asymptotic behavior of the obtained solutions is neither asymptotically flat nor (A)dS when we have ωq+2 ωq+1 > d and d ≤ 3. Regarding the cosmological constant as a thermodynamic pressure and its conjugate quantity as a thermodynamic volume, we investigate the critical behavior of our black hole solutions. In d = 3 and Q≠0, the P–V diagrams are more complex than that of the standard Van der Waals. These diagrams behave like the Born–Infeld-AdS P–V diagrams. When d = 3 and Q = 0, the P–V diagrams behave like the standard Van der Waals system (In this work [S. Gunasekaran, D. Kubiznaak and R. B. Mann, Extended phase space thermodynamics for charged and rotating black holes and Born–Infeld vacuum polarization, J. High Energy Phys. (2012).] [S. Gunasekaran, D. Kubiznaak and R.B. Mann, Extended phase space thermodynamics for charged and rotating black holes and Born–Infeld vacuum polarization, J. High Energy Phys. (2012)], the authors believe that the BTZ black holes, contrary to the higher-dimensional case, do not show any critical behavior. In this work, we have showed that the three-dimensional (3D) black holes can also have critical behavior and coincide with those of the Van der Waals system). Besides, we have analyzed the concept of effective potential barrier by transforming the radial equation of motion into standard Schrodinger form. We figure out the effect of the coupling constant λ, the string parameter b, and the quintessential state parameter ωq on the height of the potential barrier and the other thermodynamical quantities like temperature and heat capacity. Then, we study the quasinormal modes (QNMs) of 3D black holes in the f(R) context. For this purpose, we use the WKB approximation method upto third-order corrections. We have shown the perturbation’s decay in corresponding diagrams when the string parameter b changes.

中文翻译：

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在 f(R) 重力下具有环形视界的带电黑洞解，被精髓和弦云包围：有效势垒，准正常模式

我们构建了一个新的类d具有环形视界的维黑洞解F(R)被精华和弦云构型包围的重力框架，其渐近结构由精华状态参数决定ωq和尺寸的参数d. 我们指出，当我们有ωq+2 ωq+1 > d和d ≤ 3. 将宇宙学常数视为热力学压力，将其共轭量视为热力学体积，我们研究了黑洞解的临界行为。在d = 3和问≠0， 这磷–五图表比标准范德华更复杂。这些图表的行为类似于 Born-Infeld-AdS磷–五图表。什么时候d = 3和问 = 0， 这磷–五图的行为类似于标准范德华系统（在这项工作中 [S. Gunasekaran、D. Kubiznaak 和 RB Mann，带电和旋转黑洞的扩展相空间热力学和 Born-Infeld 真空极化，J. 高能物理。(2012)。] [S。Gunasekaran、D. Kubiznaak 和 RB Mann，带电和旋转黑洞的扩展相空间热力学和 Born-Infeld 真空极化，J. 高能物理. (2012)]，作者认为，与高维情况相反，BTZ 黑洞没有表现出任何临界行为。在这项工作中，我们证明了三维（3D）黑洞也可以具有临界行为，并且与范德华系统的行为一致）。此外，我们通过将径向运动方程转化为标准薛定谔形式，分析了有效势垒的概念。我们计算出耦合常数的影响λ, 字符串参数b, 和典型的状态参数ωq关于势垒的高度和其他热力学量，如温度和热容量。然后，我们研究了 3D 黑洞的准常态模式 (QNMs)F(R)语境。为此，我们使用 WKB 近似方法进行三阶校正。我们已经在相应的图表中显示了当字符串参数时扰动的衰减b变化。